Founding Mathematics on Semantic Conventions (Synthese Library, 446) 303088533X, 9783030885335

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Synthese Library 446 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

Casper Storm Hansen

Founding Mathematics on Semantic Conventions

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 446 Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, USA Editors Berit Brogaard, University of Miami, USA Anjan Chakravartty, University of Notre Dame, USA Steven French, University of Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, The Netherlands Darrell P. Rowbottom, Lingnan University, Hong Kong Emma Ruttkamp, University of South Africa, South Africa Kristie Miller, University of Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. Besides monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

More information about this series at https://link.springer.com/bookseries/6607

Casper Storm Hansen

Founding Mathematics on Semantic Conventions

Casper Storm Hansen Institute of Philosophy Chinese Academy of Sciences Beijing, China

ISSN 0166-6991 Synthese Library

ISSN 2542-8292 (electronic)

ISBN 978-3-030-88533-5 https://doi.org/10.1007/978-3-030-88534-2

ISBN 978-3-030-88534-2 (eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Luisa, a stranger when this project began; by the grace of fortune, no more

Preface

This is my second attempt at formulating a nominalistic philosophy of mathematics. The first was my dissertation, which I wrote at the Northern Institute of Philosophy at the University of Aberdeen between 2011 and ’14. I had barely defended it before I became convinced that it could not be defended. That conviction was triggered by an epiphany about the liar paradox. About that, in the ’14–’15 academic year, I wrote the first draft of what has now, countless revisions later, become Chap. 5 of this book. Initially, I thought that I could plug that solution to the liar paradox into the philosophy of mathematics presented in my dissertation, make some minor adjustments, and then be done. However, I soon realized that a rewrite almost from scratch was required, and that project has been underway, on and off, since the summer of 2015. The result has few similarities to the theory of my dissertation. During this long process, I have come to appreciate how extremely complex a field the philosophy of mathematics is, and how very many data points from pure mathematics and the application of mathematics in the empirical sciences one has to take into account when attempting to provide a comprehensive philosophy of mathematics. For that reason, and because my theory has many moving parts, I feel that some preface-paradoxical humility is in order. Even in this second attempt, I probably did not get everything right. I feel differently about the solution to the liar paradox. That one I would happily bet my life on for the chance of winning a spoonful of sand! I have had interesting discussions with and received useful feedback from many people. They include Claire Benn, Sharon Berry, Thomas Brouwer, Andrés Eduardo Caicedo, Elvira Di Bona, Eric Epstein, Filippo Ferrari, Hartry Field, Andreas Fjellstad, Alain Genestier, Peter Gerdes, Eirik Gjerstad, Anil Gupta, Joel David vii

viii

Preface

Hamkins, Gerard Hough, Silvia Jonas, Asaf Karagila, David Kashtan, Leora Katz, Øystein Linnebo, Shay Logan, Toby Meadows, Marco Panza, Graham Priest, Dave Renfro, Martin and Naja Fogt Pollas Rønberg, Daniel Schepler, Noah Schweber, James Shaw, Olla Solomyak, Georgie Statham, Daniel Telech, Robert Trueman, Mark van Atten, Crispin Wright, the anonymous reviewer, and many conference participants whose names I was not diligent enough to record. My sincere thanks to all of you. I am also grateful to the series editor, Otávio Bueno, for accepting this work into the same book series that is home to the best piece of philosophy to have been created in recent times,1 and to the good people of Springer and VTeX: Lucy Fleet, Christopher Wilby, Palani Murugesan, Svetlana Kleiner, Werner Hermes, and Vygintas Vilimas—in particular for going out of their way to accommodate my unusual typesetting needs. In addition, I owe thanks to Daniel MacCannell for proofreading the entire manuscript. My work in the ’14–’15 academic year, which (like my Ph.D. research) took place in Aberdeen, was funded by a grant from the Analysis Trust. In the following year, I worked at the Institute for the History and Philosophy of Science and Technology in Paris, and was funded through Gerhard Heinzmann, Hannes Leitgeb, and Marco Panza’s Mathematics: Objectivity by Representation project. My work since then has taken place at the Polonsky Academy for Advanced Study in the Humanities and Social Sciences at the Van Leer Jerusalem Institute, and has thus been funded through the generous philanthropy of Leonard Polonsky. I’m grateful for all the adventures—philosophical, mathematical, and otherwise—over these years. Chapter 3 of this book has previously been published (with slight differences) under the title “Choice Sequences and the Continuum” (Hansen 2020a). And while Chap. 5 has not already been published, I have already published a defence of the solution to the liar paradox therein in a paper titled “The Signalman against the Glut and Gap Theorists” (Hansen 2020b). While the arguments in that paper are structured around an analogy with the titular signalman, I have taken a more direct approach in this book. Beijing September 2021

Casper Storm Hansen

1 I am of course referring to Atkinson and Peijnenburg’s (2017) solution to Agrippa’s Trilemma.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview and Guide to Partial Reading . . . . . . . . . . . . . . .

2

Classical Mathematics and Plenitudinous Combinatorialism 2.1 Large Cardinal Axioms and Theorems of Arithmetic . . . 2.2 Transfinite Ordinals . . . . . . . . . . . . . . . . . . . . 2.3 Transfinite Cardinals . . . . . . . . . . . . . . . . . . . . 2.4 The Continuum Hypothesis . . . . . . . . . . . . . . . .

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Intuitionism and Choice Sequences . . . . . . . . . . 3.1 General Introduction . . . . . . . . . . . . . . . . 3.2 Brouwer on Freely Proceeding Choice Sequences 3.3 Constitution of Free Choice Sequences . . . . . . 3.4 Evaluation of Brouwer’s Claim . . . . . . . . . . 3.5 Verificationism and Intuitionistic Logic . . . . . .

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From Logicism to Predicativism . . . . . . . . . 4.1 Frege . . . . . . . . . . . . . . . . . . . . . 4.2 Russell . . . . . . . . . . . . . . . . . . . . 4.3 Weyl . . . . . . . . . . . . . . . . . . . . . 4.4 Weyl’s Failure to Include All Real Numbers

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Conventional Truth . . . . . . . . . . . . . . . 5.1 The Obvious Solution to the Liar Paradox . 5.2 Conventional Truth Conditions . . . . . . 5.3 The Dogma . . . . . . . . . . . . . . . . .

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Semantic Conventionalism for Mathematics . . . . 6.1 Needs Assessment . . . . . . . . . . . . . . . . 6.2 Simple Arithmetic as a Conventional Language . 6.3 Quine’s Anti-Conventionalism . . . . . . . . . 6.4 Rule-Following . . . . . . . . . . . . . . . . . 6.5 Choice of Logic . . . . . . . . . . . . . . . . .

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A Convention for a Type-free Language . . . . . . . . . . . . . . . 7.1 The Kripke Convention and Its Shortcomings . . . . . . . . . . . 7.2 Reformulating the Kripke Convention . . . . . . . . . . . . . . . 7.2.1 Collapsing Truth and Satisfaction of View-From-Nowhere Truth Conditions . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Kleenification . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Kripke Recursion . . . . . . . . . . . . . . . . . . . . . 7.3 Adding a Conditional with Supervaluational Semantics . . . . . 7.3.1 Supervaluation over All Possibilities . . . . . . . . . . . 7.3.2 View-From-Nowhere Truth Conditions for the Strong Conditional . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 If the Supervaluation Criterion is Not Satisfied . . . . . . 7.3.4 Ensuring Quantification over All Possibilities in the Presence of Supervaluation . . . . . . . . . . . . . 7.3.5 Iteration of the Strong Conditional . . . . . . . . . . . . 7.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Denoting Terms for Applied Mathematics . . . . . . . . . . . . 7.5 Meta-Theorems . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Possible Language Conventions . . . T-schemas and Expressive Strength . Dialectical Situation . . . . . . . . . The View from Nowhere . . . . . . . Comparison with Chihara’s Position Revenge . . . . . . . . . . . . . . .

Basic Mathematics . . 8.1 Logic . . . . . . . 8.2 Natural Numbers . 8.3 Integers . . . . . . 8.4 Rational Numbers 8.5 Classicality So Far 8.6 Classes . . . . . .

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Contents

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8.7 An Example of Applied Mathematics . . . . . . . . . . . . . . . . 165 9

Real Analysis . . . . . . . . . . . . . . . . . . . . . 9.1 Functions . . . . . . . . . . . . . . . . . . . . . 9.2 Real Numbers . . . . . . . . . . . . . . . . . . 9.3 Exponentiation . . . . . . . . . . . . . . . . . . 9.4 Completeness . . . . . . . . . . . . . . . . . . 9.5 Suprema, Infima, and Roots . . . . . . . . . . . 9.6 Continuity . . . . . . . . . . . . . . . . . . . . 9.7 Operations on Functions . . . . . . . . . . . . . 9.8 Differentiation . . . . . . . . . . . . . . . . . . 9.8.1 Calculating Derivatives . . . . . . . . . 9.8.2 Uniform Differentiability . . . . . . . . 9.8.3 Mean Value Theorem and Monotonicity 9.9 Integration . . . . . . . . . . . . . . . . . . . . 9.10 Unbounded Intervals and Piecewise Continuity . 9.11 Completifications of Functions Generalized . . . 9.12 Another Example of Applied Mathematics . . . 9.13 Diagonalization . . . . . . . . . . . . . . . . .

10 Possibility . . . . . . . . . . . . 10.1 All Possible Real Numbers 10.2 Modal Metaphysics . . . . 10.3 Conclusion . . . . . . . . .

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225 225 231 235

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Chapter 1

Introduction

The motivating intuition behind this study is that mathematics ought to be accountable for using only entities whose existence—and states of affairs whose obtaining—we have good reasons to believe in, independently of considerations pertaining to mathematics. Mathematics is divided into pure and applied branches, neither of which would seem, at least prima facie, to give us reasons for extending our ontology with sui generis mathematical objects. Pure mathematics is an intellectual activity that we should be able to account for by means of the ontology that must be admitted to account for intellects. It seems impotent when it comes to providing evidence for numbers having a human-independent existence; for how could mathematicians moving some symbols around on a piece of paper while sitting in their armchairs show us that there is a wholly separate category of entities in existence? Applied mathematics, for its part, is a means of describing and reasoning about reality. Yet, such describing and reasoning is, again, an intellectual activity, and the described and reasoned-about reality does not seem to be mathematical. We can, for instance, use mathematics to describe the trajectories of heavenly bodies, and we then seem to describe a reality that involves a lot of particles and some forces acting on them, but not any mathematical objects. The heavenly bodies would follow the same trajectories whether or not mathematical objects existed. While the above considerations omit a great deal of detail that certainly merits consideration (and which it will receive in due course), I think they suffice to establish something about the burden of proof: we should not conclude that there are specifically mathematical objects among the basic constituents of the world until we have firmly ruled out that a reductionist philosophy of mathematics can save the phenomena. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C.S. Hansen, Founding Mathematics on Semantic Conventions, Studies in Epistemology, Logic, Methodology, and Philosophy of Science, https://doi.org/10.1007/978-3-030-88534-2_1

1

2

Introduction

Contemporary, classical mathematics can not be justified reductively, I believe, but classical mathematics is not itself the phenomenon that needs saving. So if our starting point is that we must make do with previously admitted ontology unless and until we are forced to do otherwise, large parts of mainstream, classical, set-based mathematics are suspicious. Is there a set consisting of exactly the numbers 3, 8, and 21? Well, I can think of those three numbers as a set, and if we accept this thought as constituting a set, then we have accounted for this little piece of mathematics using something we have good reasons to believe in independently of mathematics: namely, thoughts. But what about an infinite set of natural numbers that defies definition and therefore cannot be identified with a thought? Such a set does not seem to fit into a view of ontology that is unbiased by the apparent needs of mathematics. It is nevertheless a common intuition among philosophers of mathematics that whenever there are some things, there is a set of those things, independently of whether the set can be defined—and indeed, that nothing more is required for a set of some things to exist then that those things exist. Together with a few other assumptions, this intuition leads to paradox,1 but the usual reaction to that fact is to salvage as much as possible of the intuition by placing the absolute minimum of restrictions on the otherwise jointly inconsistent assumptions.2 I think that the intuition is fundamentally misguided. According to it, if there are seven apples, there is also, automatically and in the absence of any action or creation process, an eighth object, the set of those seven apples. Even if we start out with an empty universe, an infinity of sets are supposed to pop up all by themselves, since the sufficient condition for the existence of the empty set is itself completely empty; and the empty set gives rise to the set containing only the empty set, and so on. I do not know why that assumption would be true, as it postulates ontology that requires a lot of saving, even in the absence of antecedent phenomena. It is one thing to conclude, after lengthy studies, that there is no way we can make do without lots of mindindependent sets in our best theory of the world, but starting out from that intuition strikes me as deeply unreasonable: like a belief in magical set creation. I therefore will not proceed from that assumption, and attempt to build a plausible case that we do not have to end up accepting it either. What kinds of entities and facts do we have reason to believe in, independently of mathematics? There are physical entities with physical properties, and there are facts about these. There are also, as Descartes would put it, res cogitans that produce

1 See

Linnebo (2010). e.g., Gödel (1947), Parsons (1975), Fine (2005), Yablo (2006), and, again, Linnebo

2 See,

(2010).

Introduction

3

thoughts; and there are facts about these, too.3 That seems pretty basic. You would almost have to be a philosopher to deny it. I also think that there are facts about how the world could have been different from how it is. This is not entirely uncontroversial and, unlike the existence of physical and mental entities, I will therefore devote a little bit of space to discussing it later. But it seems like a fairly safe thing to assume, and, importantly, our reasons for believing that there are non-trivial modal facts are not predicated on mathematics: it is not because of mathematics that we think that it could have rained, even though it does not. I do not want to make any further assumptions about what there is. In particular, I suspect that there are no abstract entities. I will not argue for that directly. However, by attempting to demonstrate that mathematics can be accounted for without such an assumption, this monograph can provide part of an argument for that negative thesis in combination with, among other premises, Ockham’s Razor. In addition, I share Aristotle’s (1930), Brouwer’s (1908), and Hilbert’s (1926) doubt that any actual infinities exist, and so I likewise think that the assumption that some do exist must be avoided when accounting for mathematics. I am also unwilling to make extremely liberal assumptions about what is possible in order to accommodate mathematics. If one does make such assumptions, then classical mathematics can be justified without the need to assume that undefinable sets actually exist, and mathematics can just be considered the science of possible structures (Putnam 1967; Hellman 1989, 1996). However, if the actual world features no “magical” set-creating mechanisms that combine things into sets that cannot be described or thought of, then it would seem that the world would have to be radically different from how it is for there to be such sets. Flying pigs and living in a four-dimensional space would be nothing in comparison: this would be a world that contains a fundamentally different category of entities, namely, abstract mathematical objects that can somehow “contain” an actual infinity of other objects. I am not sure that is even possible. This places me in opposition to another common thesis. It is often assumed that the range of possibilities satisfies a principle of plenitude, or very roughly,4 that anything that is consistent is possible.5 I am not willing to bet on that. The foundation for our mathematics ought to have a very high degree of certainty, and it does not strike me as highly certain that a world could have been created in which 3 Unlike Descartes, I will not take a stand on whether the physical and the mental are ultimately the same kind of thing. 4 Setting aside as irrelevant, for present purposes, that water must necessarily be H O, etc. 2 Kripke (1980). 5 Principles of plenitute in this spirit have been formulated by, e.g., Lewis (1986) and Kment (2014).

4

Introduction

φ is true just because we cannot formally derive a contradiction from φ. I am only comfortable with basing mathematics on the possibility of entities and facts that are relatively familiar. This includes thinking beings having other thoughts than the ones they actually have, and thoughts that complexity-wise go far beyond what any actual human being is capable of, but excludes abstract objects and actual infinity. Several different systems of mathematics that can be justified as legitimate on this ontological basis exist in the literature. They include Bishop’s constructivism (Bishop and Bridges 1985), Russian constructivism (Markov 1954), Weyl’s (1918) predicativism, and formalism. There are also systems of mathematics that have to be rejected if my skeptical attitude towards abstract objects, actual infinity, and the possibility thereof is to stand. They include classical mathematics6 (understood nonformalistically) and Brouwer’s intuitionism. However, there is no existing system that exploits the mentioned ontological basis to the fullest but avoids overstepping its limits. Or so I will argue. The purpose of this monograph is to create such a system and defend it philosophically. I will now provide an overview of the salient details of this system, leaving arguments and most explanation for later. It is a system based on language. Human beings have the ability to commit to conventional truth conditions for a potential infinity of sentences. Via a finite number of finite thoughts, it is possible to determine the truth conditions and therefore, indirectly, the truth values of infinitely many possible sentences. By exploiting this ability, I will create a language that can not only be used to communicate about mathematics, but that will itself constitute the structure that mathematics is about. There will be no need for the terms of that language to refer to anything: the structural properties of the language itself suffice (at least for the purpose of pure mathematics). While there is an obvious similarity between these ideas and formalism, consisting in the identification of mathematics with mathematical language, formalism is based on stipulated axioms and stipulated inference rules, not on stipulated truth conditions. This limits a formalistic system to what can be proved, and so, because of incompleteness (Gödel 1931), formalistic arithmetic is weaker than classical, bivalent arithmetic. My linguistic-mathematical system, on the other hand, will encompass a full reconstruction of classical arithmetic. To contrast my position against formalism, I will call it “semantic conventionalism”. As the last half of that name suggests, there are many different linguistic6 The most famous version of nominalism, Field’s (1980) fictionalist justification of the use of classical mathematics, also falls for the demand that the possibility of abstract objects and actual infinity must not be relied on. As Colyvan (2001, 135) points out, Field’s account of how classical mathematics can be useful, even though false, presupposed that it is only contingently false.

Introduction

5

mathematical systems that I consider philosophically acceptable, but which can differ in terms of their usefulness. I will develop one such system in extensive detail. A quick comparison with constructivism is also useful as an initial indication of what semantic conventionalism is. Historically, the idea that mathematics is a human construction has been strongly correlated with the verificationist thesis that mathematical truth is limited to what is provable (or even to what is proved). I am suggesting that we drive a wedge between them. Our power to determine truth conditions extends beyond what we can survey. We can bring ourselves to a point where we have fixed the truth value of each of a potential infinity of instances of a universally quantified sentence, and therefore also the truth value of that sentence, but are unable to find out what it is. If you infer, from the premise that mathematical truths are facts about human constructions, that all truths are known or knowable, you are assuming that the only relevant human constructions are proofs. But there are others. Language is a human construct, and I claim that something can be true by virtue of the finite rules of a language, even if there is no (possible) proof by which we can know that. The big challenge for this project is how to handle real numbers, which are where the undefinable sets first show up and demand to be admitted into mathematics. And both the classical mathematician and the intuitionist will tell you that they should be admitted, on pain of us being left without a “full” set of real numbers. I will insist on turning them away anyway, as I reject the notion that restricting ourselves to definable real numbers somehow leaves us with something incomplete. However, the mere rejection of undefinable real numbers cannot be the whole story. The evidence that is interpreted by the platonist as showing that there are more real numbers than there are natural and rational numbers cannot be undermined by our rejection of platonism, but only reinterpreted. The way I reinterpret it is that the family of real numbers is indefinitely extensible. That is, whenever you have some real numbers, you can define more real numbers in terms of them. Prima facie, indefinite extensibility seems to lead to a hierarchy of levels of real numbers, which means that quantification over all the real numbers of a system becomes illegitimate: an unacceptable outcome that makes any serious real analysis impossible. The first half of the solution to that problem is to reject one of the premises of the argument that a hierarchy is necessary, namely, that classical logic is the uniquely correct logic to use for mathematics. I will argue that there is a sense in which it is not. The typical reason for using a non-classical logic in connection with mathematics is the verificationism that I have already disavowed; the need to avoid hierarchies is another reason, as is well known from the attempts to avoid hierarchies of truth when dealing with the paradoxes of truth. However, this is a motivation for

6

1.1 Overview and Guide to Partial Reading

wanting to avoid classical logic, and not itself sufficient as a justification for the claim that we are allowed to use another type of logic. For classical logic is, after all, traditionally seen as the One True Logic. I will provide a justification for doing so, arguing that the conventionality of language extends as deep as its logical structure. If one’s ambition is to have a system of mathematics that is in some reasonable sense about all real numbers (among other things), then a language that avoids a type-hierarchy by means of a non-classical logic is not enough in itself. The indefinite extensibility of the real-number system is so strong that the totality of real numbers that can be defined in any fixed linguistic system cannot exhaust the space of possible real numbers: it is always possible to diagonalize out. Therefore, instead of a fixed language, I will employ an open-ended one. That is, while it will not at any given point contain all possible real numbers, any possible real number can potentially be added to it. The use of an open-ended language leads to a different way of doing mathematics in general: in a situation where a classical mathematician would introduce a new definition that facilitates new ways of talking about a static universe, a mathematician operating under the guidelines I propose will stipulate that some new linguistic form is to be counted as a new form of term or new form of sentence in our system, and stipulate new truth conditions. To accommodate this open-endedness, theorems will have to live up to a stronger modal criterion than in classical mathematics, namely, that a sentence is only a theorem if it has been proved to be true under any possible extension of the language.

1.1

Overview and Guide to Partial Reading

The next three chapters discuss and criticize some existing systems of mathematics. Chapter 2 is about classical mathematics as inspired by Cantor, and especially its use of undefinable sets. I will refute some arguments for actual infinity, and suggest that the picture the Cantorians provide of a realm of higher infinities is so unclear that there are good reasons to doubt that they are even describing a possibility. Chapter 3 deals with intuitionism. According to Brouwer, it is possible to have a mathematics with an austere ontological basis of mental acts, and nevertheless accept undefinable real numbers. That is allegedly made possible by freely-proceeding choice sequences, i.e., sequences created through repeated random choices of elements by a creating subject, in a potentially infinite process. I will argue that we are not as fortunate as Brouwer thought. While those two chapters deliver an entirely negative message about their respective theories, the conclusions of Chap. 4 are more complex. Its subjects are Frege’s logicism, Russell’s type theory, and Weyl’s predicativism. While I will criticize all

Introduction

7

of them, there is also valuable inspiration to be drawn from them about what a mathematical system based on language ought to look like. After that, we will turn to the positive development of the new proposal. However, the comprehensive overview of that proposal will be delayed until Chap. 6, as Chap. 5 is concerned with just one issue relevant to it: paradoxes of self-reference. Since mathematics is to be based on truth conditions, the distinction between semantic and mathematical paradoxes proposed by Ramsey (1925) collapses, and we will need to obtain clarity on that entire field of problems if we want to use a hierarchyfree language. I will propose a solution. Unlike most other proposed solutions, it does not consist in a claim about how truth values are distributed over some language that allows for self-reference. Instead, it takes one step back: to what truth conditions are in general, namely, a conventional system instituted by thinking subjects for the purpose of communicating with one another; to why our collective intentions for that system cannot be fully satisfied; and to how we tend to be naïve about that limitation, resulting in discrepancies between the actual truth conditions of some sentences and the truth conditions we think they have. This solution is inspired by Lewis’s (1969) theory of conventions and Nagel’s (1986) concept of a view from nowhere. One important consequence of that solution is that we are—in a sense to be explained—free to choose our logic. We can, for instance, choose between classical logic, a gappy logic like the one that results from Kripke’s (1975) work on the paradoxes, or a glutty, dialetheist logic. Chapter 6 discusses the indispensability argument for the existence of abstract mathematical objects, Quine’s argument for the non-conventional status of logic, and the rule-following problem. It also contains a sandbox version of a semanticconventionalist system, and my explanation for why I chose a gappy logic for the full system. Finally, it briefly compares my approach to formalism. The three chapters that follow it lay out an extensive semantic-conventionalist system, and are therefore technical in nature. First, in Chap. 7, I develop the specific gappy logic that I think is most useful for a mathematical system that is to include real numbers. Though it is strongly inspired by Kripke’s gappy logic, the latter is just a little too gappy. This problem is alleviated by adding a conditional with a special kind of supervaluational semantics that validates hypothetical proofs. The framework thus provided is then filled up with mathematical content, starting from arithmetic and other relatively basic areas of mathematics in Chap. 8. I hope to be able to convince the reader that enough mathematics to satisfy the needs of the empirical scientist can be reconstructed in this setting. Of course, actually reconstructing that much is not feasible in just one book, so the strategy is to deal with what, prima facie, would seem to be the biggest stumbling block, namely real anal-

8

1.1 Overview and Guide to Partial Reading

ysis. If sufficient aspects of real analysis can be reconstructed, it will be a strong indication that all of the mathematics needed for science can as well. Hence, real analysis is the focus of Chap. 9. (It is clear that there are some parts of mathematics that are not needed in the sciences that cannot be reconstructed. Notable among them are the higher reaches of set theory, but in that regard, I am happy to be a revisionist and reject it as an artifact of the currently accepted foundation of mathematics.) Finally, in Chap. 10, I return to the important philosophical question of what is possible, in order to clarify and support the motivation for this project I have given above. There are several proper parts of this monograph that can be read in isolation from the rest. In particular, Chap. 5, about the liar paradox, is written like a standalone essay and can be of interest to someone with no interest in the philosophy of mathematics. While Chaps. 2, 3, and 4 are not written like stand-alone essays, and contain some foreshadowing of ideas that will only be unpacked later in the book, the critiques of classical mathematics, intuitionism, and logicism/predicativism in those chapters may very well be of independent interest, and can be read in isolation both from the rest of the book and from each other (except that the introduction to Chap. 2 should be read before reading Chap. 3). Finally, it is possible for the reader who wants to come to grips with my positive contribution as quickly as possible to skip ahead to Chap. 5—but then, much of the motivation for that original proposal will be missed.

Chapter 2

Classical Mathematics and Plenitudinous Combinatorialism

The mainstream position in mathematics and the philosophy of mathematics originates with Cantor, who managed the impressive feat of turning a consensus against actual infinity that had been intact since Aristotle into a consensus for it. Today, it is the position that is associated with an acceptance of ZFC as a true theory of a universe of sets that is the subject matter of mathematics. As with any other camp that hosts a large number of philosophers, there are internal disagreements about the details in this mainstream camp. Since I want to explain my doubts about the entire camp, I will therefore not formulate a precise thesis to serve as the target of my criticism. However, I can do a little better than merely associating the position with the person of Cantor and the axiom system ZFC, by characterizing it as one of plenitudinous combinatorialism. Combinatorialism is the thesis that sets can exist independently of minds, independently of a creation event, and independently of there being a property or predicate to “bind” their elements together; instead, whenever there are some objects, there is a set of those objects. The modifier “plenitudinous” is added to indicate that there are, so to speak, lots and lots of these sets. As mentioned in the Introduction, the thesis that whenever there are some objects, there is a set of those objects leads to contradiction when combined with other premises that the scholars in this camp usually want to hold on to. “Plenitudinous” is meant to indicate the belief that as few restrictions as possible should be placed on set existence. In particular, the existence of undefinable sets is not threatened by the inconsistencies, so we should believe in such sets. And, indeed, as many of them as possible. In particular, if an infinite set S exists, then any subset S  that could be created by a demon who could go through the elements of S and decide individually for each one whether or not to include it in S  actually exists. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C.S. Hansen, Founding Mathematics on Semantic Conventions, Studies in Epistemology, Logic, Methodology, and Philosophy of Science, https://doi.org/10.1007/978-3-030-88534-2_2

9

10

2.1 Large Cardinal Axioms and Theorems of Arithmetic

An important part of this camp’s attempt to avoid contradiction is the thesis that a set presupposes its elements, which leads to the idea that the sets are organized in levels in a cumulative hierarchy. Each level consists of every possible (in the demon-plenitudinous sense) subset of the set of all the sets at the previous level. Or in any case, that is what a level consists of if it is the immediate successor to another level. But by the principle of plenitude, there are levels beyond the initial omega-sequence of levels. Each level that is not a successor level contains every set that exists at any lower level. Another restriction on plenitude that is needed to avoid contradiction is that not every plurality of levels has a level above it. Instead, it might be claimed that for any plurality of levels that could possibly exist, another level could possibly exist. As I believe that mathematics can very well be based on merely possible objects, I will not take issue with this move by my opponents. Indeed, whether any sets actually exist is—in spite of the impression that I have given in the last few paragraphs— beside the point. The point is whether actually1 infinite sets, i.e., infinite sets that cannot be defined, could possibly exist, and whether an infinite set could have the “full” powerset consisting of all the “demon” subsets of that set. In each of the four sections of this chapter, I will either present an argument against my opponents or counter an argument that they might make. Section 2.1 concerns the relationship between arithmetic and large cardinal axioms; Sect. 2.2, one alleged route into the uncountable, namely ordinals; Sect. 2.3, the other, namely cardinals; and Sect. 2.4 is about how these two routes interact. I should make it clear upfront that I do not have a knockdown argument against my opponent. Actually, my strongest reason for doubting her is the one already explained in the Introduction: namely, that her possibilia are too alien, and that does not amount to a knockdown argument. Instead, the arguments of this chapter are intended to show that the case for Cantor-inspired, ZFC-believing plenitudinous combinatorialism is not very strong.

2.1

Large Cardinal Axioms and Theorems of Arithmetic

There is an alleged obstacle to maintaining the position that classical arithmetic is in order while not accepting the higher infinities. Since I will later defend and 1 Following two different traditions, I am using the word “actually” in two different senses. Here, it is used in the Aristotelian sense, whereby actual infinity is opposed to potential infinity, while in the previous sentence, it was used in the modal sense, whereby actual existence is opposed to merely possible existence.

2

Classical Mathematics and Plenitudinous Combinatorialism

11

reconstruct classical arithmetic, the plenitudinous combinatorialist may claim that my position is unstable. I will begin by countering this claim. There is a surprisingly intimate relationship between the top and the bottom of the set-theoretic hierarchy. Using ZFC, one can only prove the existence of some of the cardinals that the plenitudinous combinatorialist believes to (possibly) exist. In order to prove the existence of more of them, axioms have to be added. There is by now a fairly long list of suggestions for such large cardinal axioms.2 What is surprising is that adding large cardinal axioms to ZFC means that more arithmetical propositions become provable. This is prima facie quite baffling; you would think that these areas of mathematics could be considered separately; that the decision to “admit” into your mathematical universe the natural numbers and arithmetical relations between them can be made first, and the decision about whether to believe in higher infinities made later and independently. If nothing else, you would think this because it is suggested by the plenitudinous combinatorialist’s own hierarchical picture of the sets: the higher levels depend on the lower levels, but not the other way around. But instead, it seems like you have to buy the entire classical store in order to get a fully functioning version of just one of its products. Let me explain in some detail why that is, before responding to the challenge. It has to do with the fact that ZFC cannot prove the consistency of ZFC. While this follows from Gödel’s Second Incompleteness Theorem, it will for present purposes be more useful to look at the reason for this impossibility from a different angle. Because of the completeness and soundness of the first-order predicate calculus, proving that ZFC is consistent is the same as proving that ZFC has a model. Consider some model M of ZFC. To prove that something has a given property in a set theory, that something has to be a set. So to prove in ZFC that M is a model of ZFC, M has to be a set. But it cannot be proved that M is a set if it is possible that M is the entire universe of things. And to say that M is a model of ZFC means, exactly, that it is consistent with ZFC that M is the entire universe of things. Ergo, it cannot be proved in ZFC that M is a model of ZFC. This leads us almost us to the conclusion that ZFC cannot prove the consistency of ZFC, for we see that there is no specific model of ZFC that forms a set in all models of ZFC. To reach all the way to that conclusion we just have to rule out the possibility that all models of ZFC contain some other model as a set. But this follows from the well-foundedness of sets: among the models of ZFC that are sets in models of ZFC that are sets in models of ZFC etc., there has to be a minimal one.

2 For

an overview, see Kanamori (2003).

12

2.1 Large Cardinal Axioms and Theorems of Arithmetic

However, an effect of adding a large cardinal axiom to ZFC is that any model of ZFC-plus-that-large-cardinal-axiom must contain a model of ZFC as a set. For, letting ℵ be the smallest cardinal that is not guaranteed to exist by ZFC alone, but which must exist according to the enlarged axiom system, the set of all sets below level ℵ is a model of ZFC. Hence, the enlarged axiom system—let us call it ZFC+ —proves the consistency of ZFC. Now, consider the impact of the above point about models upon consistency as a syntactic property of formal systems. That ZFC is syntactically consistent can be expressed as an arithmetical sentence because the axiom system, the deducibility relation, and a formal inconsistency can all be arithmetized. This arithmetical sentence is not provable from ZFC, but it is from ZFC+ . Similarly, adding an axiom that implies the existence of even larger cardinals makes it possible to prove an arithmetical sentence expressing the consistency of ZFC+ , which cannot be done with ZFC+ alone. And so on. This prompted Cohen (1966, 80) to write that In general, each new axiom will allow one to prove the consistency of a weaker system. Thus each axiom will prove a statement in elementary number theory not previously provable. To dismiss these axioms as irrelevant is thus to essentially give up any hope of proving these statements. There thus appears to be a pressure, once one has accepted classical arithmetic, to also accept the higher reaches of set theory. The argument for doing so would go like this: Each statement of arithmetic is either true or false. The falsity of the consistency of ZFC (or ZFC+ , etc.) is not very plausible, since it would be detectable through a finite deduction, and we have not found one in spite of searching for it for a long time. So, it is probably true, and if it is, it ought to be a theorem of our most comprehensive system of mathematics. And because the large cardinal axioms deliver that, they should be accepted. Cohen’s point can be countered by pointing to the difference between proof in the technical, mathematical sense and in the epistemic sense. Granting, for the sake of argument, that ZFC is more likely to be syntactically consistent than to be inconsistent, our evidence for the consistency is empirical and inductive in nature, in the manner already explained: many attempts to find an inconsistency have failed. But this is very different from the axiom system, ZFC+ , we are asked to accept. Our evidence for the mere syntactical consistency of the weaker system ZFC is certainly not that we know the stronger system ZFC+ to be (possibly) true. That is, believing that no formal inconsistency can be derived from ZFC is a far cry from believing in

2

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13

the (possible) existence of all the sets postulated by ZFC+ . Thus, adding the large cardinal axiom in order to obtain the arithmetical sentence saying that ZFC is consistent as a theorem does not correspond to the epistemic order of things.3 If one, for some reason, wants the consistency sentence as a theorem in one’s mathematical system, one should just add that sentence itself as an axiom. This might seem like cheating, but such a move would not reflect a dogmatism, but rather, the fact that our evidence is extra-mathematical. That is, we cannot introduce into the formal system the premise that many years of searching for an inconsistency have failed to deliver. We can only state this premise and reason from it outside of formal mathematics, and then deliver the result, fully finished, to formal mathematics in the form of what will then be, from the perspective of formal mathematics, an axiom.4 Cohen’s aversion to “giv[ing] up any hope of proving these statements” should just lead him to broaden his view of what kinds of justification are available, and to accept that as a result we might need to, in a merely formal sense, introduce the desired theorem by fiat. It should not, however, lead him to think that there is instability in the position claiming both that classical arithmetic is legitimate and that higher set theory is not.

2.2

Transfinite Ordinals

Among many other, much stronger claims, the Cantorian plenitudinous combinatorialist believes that a structure consisting of all finite ordinals, all countable ordinals, and the first uncountable ordinal, ω1 , could exist. Let us name the proposition that this structure does exist “φ”. I have no knockdown argument to the effect that possibly-φ is false. I am agnostic. But it seems quite reasonable to demand of the classical mathematician that she can produce some positive reasons for believing in possibly-φ. The epistemic possibility of the metaphysical possibility of φ consisting in the absence of proof of the metaphysical impossibility of φ is too shaky a foundation for mathematics. My general contention here is that, to be warranted in making a positive assertion about the metaphysical possibility of some proposition, one must at least be able to describe, or point to, or imagine—or something along those lines—a scenario in 3A

similar point is made by Feferman (1987). may cite another reason for believing in the consistency of ZFC: having an intuitive picture of the cumulative hierarchy. But because of my restrictive views about metaphysical possibility (see Chaps. 1 and 10), I would also only accept that as evidence for syntactical consistency, and not for (possible) truth. 4 One

14

2.2 Transfinite Ordinals

which the proposition is true. That is a fair demand of a defense lawyer in a trial claiming the possibility that someone other than his client is the murderer, if he wants to meet the mild standard of reasonable doubt; and it is likewise fair in the present context, in which we are looking for a foundation for mathematics, and should therefore be expected to live up to a higher standard. I will explain why such a scenario has not been provided in the case of φ. Let us set out on the way to Cantor’s Paradise. The journey begins with all the finite ordinals: 1, 2, 3, . . . Could they all exist together? I am already skeptical, for that would already be actual infinity. But at least it is pretty clear what I am being asked to believe the possibility of: I understand the structure of an omega sequence. So, for the sake of argument, let us assume that they could all exist together. If that is possible, it is certainly also possible that one more number, ω0 , coexists with them. Then, we are into the transfinite; and having temporarily waived reservations about actually infinite structures we can comprehend, it seems undeniable that we must also accept the possibility of another long stretch of the road, such as:5 ω0 , . . . , ω0 · 2, . . . , ω0 · 3, . . . Again, the classical mathematician must be deemed to have given an adequate description of the possibilia that she wants us to acknowledge. Each element of the sequence so far has a name and can be thought of individually; and the system of notation makes the structure clear. The same holds further on: ω

ω0

ω02 , . . . , ω03 , . . . , ω0ω0 , . . . , ω0 0 , . . . Indeed, it holds quite a bit further on, for when we run out of possibilities with exponentiation, there are techniques for bringing us further that still allow naming of each ordinal and that bring no new philosophical troubles, resulting in the socalled Veblen ordinals (Veblen 1908). However, no matter how creative we are in devising systems of notation for initial segments of the ordinals, we have only progressed an infinitesimal part of the distance to the first uncountable ordinal. The reason, of course, is that any system of notation can only name countably many ordinals, and yet by definition there are uncountably many below ω1 . So even if, for any system of ordinals you may describe 5 For definitions of ordinal addition, multiplication, and exponentiation see Cantor (1897, §14 and §18).

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Classical Mathematics and Plenitudinous Combinatorialism

15

to me, I accept for the sake of argument that they could all exist, you have still not explained to me what it is you want me to believe when you ask me to believe φ. It’s like planning to drive from Paris to Beijing and asking for directions, but only ever receiving more and more detailed instructions on how to get out of your driveway, together with a cheerful assurance that after that accomplishment, you should just proceed in the same way and you shall find the Forbidden City. Let me contrast this case with the case of the finite ordinals. There, I have a conceptual understanding—through the definitions of “ordinal” and “finite”—and, in addition, would claim that I have an intuitive understanding. That is, I have something that is close enough to a picture of what that specific structure is like. For the countable ordinals, on the other hand, I only have a conceptual understanding, through the definitions of “ordinal” and “countable”. Any attempt at creating a mental picture that is specifically of that alleged structure fails: the best I can do is to picture a countable initial segment thereof, or a non-specific picture of a generic ordinal sequence. That we cannot explain what the structure of the ordinals up to and including ω1 looks like ought to make us suspicious of whether it is a real possibility. This is far from a knockdown argument, however, for the plenitudinous combinatorialist can respond that it follows from her position that the structure cannot be described, because it is exactly the thesis that that structure is too big to be described in the way I demand. That is true; but the result is a standoff, and not a situation in which confidence in her position has been made warranted. The standoff is similar to that between an atheist who complaints that there is no evidence of God’s existence, and a theist who claims that God has on purpose ensured that there is no such evidence. Let me make the point about my skeptical attitude towards possibly-φ in a slightly different way: namely, by taking a closer look at the method for obtaining larger and larger ordinals. Cantor “created” the ordinals using two “principles of generation”. The first states that for every already-formed and existing number, a new number can be created by adding a unit to it. The strong principle is the second one, according to which “if any definite succession of defined integers is put forward of which no greatest exists, a new number is created by means of this second principle of generation, which is thought of as the limit of those numbers; that is, it is defined as the next number greater than all of them”.6 Cantor did not mean it literally when he used dynamic vocabulary here. Instead, we must understand the 6 “[W]enn irgendeine bestimmte Sukzession definierter ganzer realen Zahlen vorligt, von denen

keine größte existiert, auf Grund dieses zweiten Erzeugungsprinzips eine neue Zahl geschaffen wird, welche als Grenze jener Zahlen gedacht, d.h. als die ihnen allen nächst größere Zahl definiert wird” (Cantor 1883, §11). (The emphasis here is taken from the quoted text. The same holds of emphasis in all other quotes in the present book, unless otherwise stated.)

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2.2 Transfinite Ordinals

demand on a succession of numbers to be that those numbers can exist together. As such, the second principle becomes “For every possibly existing, definite succession of numbers of which none is the greatest, another number could exist together with them, defined as the next, greater than them all”. Given that interpretation, I have no objection to the two principles. Between them, they just say that if some numbers can exist together, then those numbers plus one more could too. Only a strict finitist would deny that. Rather, the question is, what could exist together. When we have a system of notation for a set of ordinals, it is relatively clear what believing in their possible co-existence amounts to. Hence, the application of the second principle to such sets is not as objectionable as the more liberal applications that Cantor allows. However, as noted above, it does not remove us from the domain of the countable. The more liberal and daring way of using the second principle is to consider a succession “definite” simply because it allegedly consists of exactly those ordinals that satisfy a given predicate. For example, the way to get ω1 using Cantor’s principles is to take the predicate of being a countable ordinal, turn it into a set by comprehension, and apply the second principle to that. Cantor gives us no good reason to believe that such a set could exist, but merely presupposes it. It is of course not legitimate to use the second principle in this way on any predicate. If the predicate applies to all possible ordinals, as for instance the vacuous predicate of being self-identical does, we run into Burali-Forti’s (1897) paradox: the new ordinal that the second principle produces must be larger than all ordinals including itself. How do we know that “countable” is not such a predicate? We do not. Only if there is a possible uncountable ordinal does “countable” not apply to all the possible ordinals that “self-identical” applies to. So, the possible existence of such an ordinal is only guaranteed by Cantor’s method in a viciously circular way. Cantor’s (1883, §12) attempt to convince us of the possible existence of ω1 in effect takes the following form. Let ψ be a predicate such that we do not know whether all possible ordinals satisfy ψ. Consider the set of all ordinals that satisfy ψ, apply the second principle to it, and call the resulting ordinal α. By reductio it follows that α does not satisfy ψ. Ergo, ψ does not apply to all possible ordinals. This is conviction created ex nihilo. From the mere absence of a known contradiction in the assumption that all ordinals satisfying ψ could exist together, Cantor wants us to conclude that they can.7 7 In

contemporary set theory this route into the uncountable has been abandoned. Instead, the existence of ω1 is proved using (most notably) the powerset axiom and the axiom schema of replacement, as follows. The former implies the existence of P (ω0 ×ω0 ), which is the set of all binary relations on subsets of ω0 . Any countable well-ordering is isomorphic to an ordering of a subset of ω0 , and an ordering of a subset of ω0 is a binary relation on that subset of ω0 . So the function

2

Classical Mathematics and Plenitudinous Combinatorialism

17

I think we should remain skeptical about the possible truth of φ because we have only the flimsiest of ideas of what the truth of φ would amount to. No intuitive understanding of the structure of the ordinals up to and including ω1 can be provided. The grasp that we have of φ is purely conceptual and purely negative; it is based on nothing but the negation of the property of being countable. The fact that no contradiction has, as yet, been deduced from the linguistic description of φ, is to my mind not sufficient assurance to justify basing mathematics on it. The plenitudinous combinatorialist might reply: “You are saying that we have no way of knowing that the predicates countable ordinal and self-identical ordinal are not co-extensional. But we do. Cantor also proved that the powerset of a countable set is uncountable, so we know that there are numbers beyond the countable.” There are two points I want to make in response to that. First, even if there are uncountable sets, that does not by itself imply that there are uncountable ordinals. The two things are only related if uncountable sets can, in general, be well-ordered. However, well-orderings of uncountable sets cannot, in general, be described,8 so any appeal to them would beg the question. In set theory, the possibility of wellordering an uncountable set is, in effect, simply assumed in the form of the Axiom of Choice, which is equivalent to the proposition that any set can be well-ordered (and very much an expression of plenitudinous combinatorialism). And even when it is thus assumed that any set can be well-ordered and thus “match up” with the system of ordinals, how it matches up with the ordinals remains unclear. This will be the subject of Sect. 2.4. Second, it is not clear that Cantor’s Theorem shows that there are numbers beyond the countable. At any rate, this does not follow from Cantor’s Theorem alone, but only from Cantor’s Theorem together with a plenitudinous-combinatorialist assumption. If that extra assumption is reversed, the Theorem does not show that there are more real numbers than there are natural numbers, but merely that the set of all real numbers cannot be well-ordered. That is the subject of the next section.

that takes any well-ordering to its corresponding ordinal maps the subset of all well-orderings in P (ω0 × ω0 ) onto the set of all countable ordinals—its sethood is guaranteed by the axiom schema of replacement. Since an ordinal is the set of all its predecessors, this set is ω1 . (This is a special case of the proof of Hartogs’s Theorem (Hartogs 1915).) The powerset axiom will be discussed in the next section. 8 Loosely speaking, a well-ordering of an uncountable set can only be described if the set is well-ordered by design or defined from other sets that are. For instance, if we help ourselves to ω1 , a well-ordering thereof can trivially be described using “∈”. However, such a trivial case does not help the plenitudinous combinatorialist here.

18

2.3

2.3 Transfinite Cardinals

Transfinite Cardinals

The decisive moment in Cantor’s development of the theory of the transfinite occurred when he proved that there are more real numbers than natural numbers.9 That discovery would seem to decide the debate in favor of the combinatorialist: we can only define countably many entities individually in language, so if there are more than countably many real numbers, then some of them are combinatorial. The well-known proof is indirect: Assume that there are only as many real numbers as there are natural numbers. Then the real numbers can be ordered in an ω-sequence. But then the real numbers, so ordered, can be diagonalized into a new real number that is not in the sequence, contradicting the assumption. However, there is a circularity here. An anti-combinatorialist might claim that there are no more real numbers than there are natural numbers (because at any given time only finitely many of each are in existence, and both classes are potentially infinite) and that, nevertheless, the real numbers cannot be ordered in an ω-sequence (because a bijection between the natural numbers and the real numbers cannot be described). Hence, the inference from the reductio-assumption of the equinumerosity of the reals and the naturals to the existence of an ω-sequence containing all the real numbers is based on the following plenitudinous-combinatorialist assumption: whenever two sets are of the same size, there must be a “demon” function mapping one element from the first set to one element from the second set, another element from the first set to another element from the second, etc., until all the elements from both sets are used up. Hence, the argument begs the question. As a means of convincing us of combinatorialism, it holds no dialectical power. “Wait a minute,” the combinatorialist will say, “the supposition that the real numbers can be ordered in an ω-sequence is exactly what it means for the real numbers and the natural numbers to be equinumerous. You are, just like the many cranks who again and again claim to have disproved Cantor’s Theorem,10 confusing the concepts. There is nothing wrong with Cantor’s proof!” This I will have to concede. There is indeed nothing wrong with Cantor’s Theorem and its proof when the mathematics and the philosophy are clearly distinguished. And we do need to be more careful with the terminology. But my philosophical point will survive this objection. So let us try again to formulate the Theorem and its proof. The Theorem is that there is no bijection between the set of natural numbers and the set of real numbers. The proof begins with the negation of the Theorem: there is a bijection between the two sets. This does, indeed, imply by definition that the real numbers can be 9 This was first proved in Cantor (1874), but the version of the proof that has since become the best known is given in Cantor (1891). 10 See Hodges (1998).

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19

ordered in an ω-sequence. And we can diagonalize this ω-sequence of real numbers to produce a real number that is not in the sequence. This contradicts the assumption, and the Theorem is proved. Thus formulated, both the Theorem and its proof are common ground between me and my combinatorialist opponent. Technically, the difference between the two formulations is merely a substitution of definiendum for definiens. However, the choice of the word used for the definiendum is misleading. It is not reasonable to use the expression that a set is “larger” than the set of natural numbers, instead of the two sets being “equinumerous”, if it is possible for such a “larger” set to be a subset of a set that is “equinumerous” with the set of natural numbers. And the proof of Cantor’s Theorem does nothing to rule out that situation. Let me describe such a situation in more detail. Imagine a mathematical universe consisting of just the terms and sentences of a (formal) language. There is a counterpart for each of the platonist’s natural numbers, but the real numbers are just the available descriptions of Cauchy-sequences of rational numbers. That is, there are no so-called arbitrary sequences of rational numbers that exist despite their resistance to being captured in language. In this universe, Cantor’s proof goes through just as well as in the universe that the platonist believes to be available to her. If it is assumed that a given function is a describable bijection from the natural numbers to the real numbers, it can be demonstrated that, from that description, it is possible to construct a description of the diagonalization of the bijection; and that this description of a diagonalization is itself a description of a real number that is not in the range of the bijection. Hence, in that universe, there is also no bijection between the set of natural numbers and the set of real numbers—our common-ground theorem. However, the formal language within which all of this happens only has countably many terms in total, and the real numbers are among those terms, meaning that one has to abuse the expression “larger than” to conclude by saying that the set of real numbers is larger than the set of natural numbers. This means that, to use Cantor’s Theorem in an argument for combinatorialism, it is necessary to add the premise that if there are no more real numbers than natural ones, then the former can be ordered in an ω-sequence. But that is, in disguise, the assumption of plenitudinous combinatorialism that the Theorem was supposed to convince us of. As a rhetorical move, the idea of defining vocabulary in such a way that the absence of a bijection between two sets can be expressed by saying that there are “more” elements in one set than the other is brilliant. It disguises the fact that a negative result—there is something that does not exist—is being offered as a proof of plenitude. Really, one has to assume the plenitude of combinatorial real numbers and bijections separately if one is to justify such a plenitudinous interpretation of

20

2.4 The Continuum Hypothesis

Cantor’s discovery. With a belief in a plentitude of combinatorial sets it would seem that only a difference in size could prevent the existence of a bijection, but not otherwise. Cantor’s Theorem does not by itself inflate our ontology with higher infinities; on the contrary, it says that there is something that doesn’t exist. The universe I just described is defined and investigated in detail in the second half of this book. In particular, Sect. 9.13 contains a proof that diagonalization also goes through in that setting. However, you don’t have to go there to convince yourself that it is possible for the real numbers to live inside a countable universe. That is a well-known fact which follows from the Löwenheim-Skolem Theorem.11 However, that aspect of that Theorem is usually ignored, and its significance is instead typically taken to be that it reveals a shortcoming of first-order logic: namely that a first-order axiomatization of a theory, in particular set theory, cannot be categorical if it is to have any infinite models. For example, first-order ZFC cannot distinguish between the intended “fat” model that contains the full powerset (as they believe it to be) of the set of natural numbers,12 on the one hand and on the other, leaner models such as Gödel’s constructible hierarchy, which we will explore in Sect. 2.4. Second-order logic, in contrast, is supposed to be categorical, but only because the “full” powerset is assumed to be already available when the semantics of the logic is defined. You only get as much actual infinity out as you put in.

2.4

The Continuum Hypothesis

According to Cantor, there are two routes into the transfinite cardinals. One is via the ordinals. First, we take the predicate of having finitely many predecessors, negate it, and turn it into an ordinal using Cantor’s second principle. That gives us the first ordinal that has infinitely many predecessors, ω0 , and as a consequence the first infinite cardinal, ℵ0 .13 Then, similarly, we take the predicate of having at most ℵ0 predecessors; negate it; and turn it into the first ordinal to have more than ℵ0 predecessors, 11 The Löwenheim-Skolem Theorem states that a theory that is axiomatized in first-order logic with countably many axioms and that has an infinite model, has a model of any infinite cardinality (Löwenheim 1915; Skolem 1920). ZFC is an example of such a theory. Hence, the first-order powerset axiom, ∀x ∃y ∀z (z ⊂ x → z ∈ y), can hold in a countable model, and thus does not capture what the plenitudinous combinatorialist aims at with the term “powerset”. 12 I do not dispute the (elementary) classical theorem according to which there is a bijection between the set of real numbers and the powerset of the set of natural numbers. Hence, just like in classical mathematics, we can use them interchangeably for purposes of discussions about size. I have used the former so far and will use the latter from here to the end of this chapter. 13 In Cantor’s approach, ℵ is the result of abstracting from the order of the predecessors of ω . 0 0 In contemporary set theory, ℵ0 is simply the same object as ω0 .

2

Classical Mathematics and Plenitudinous Combinatorialism

21

ω1 , with its associated cardinal ℵ1 . And so it continues: ℵ2 , ℵ3 , . . . , ℵω , ℵω+1 , . . ., to name the first “few”. The second route is via the notion of powersets. The procedure of going through the elements of a collection, and selecting or deselecting each one to form a subset, is idealized to infinite sets; and it is further assumed that there can be a set of all the possible results of this infinitary “procedure”. Because Cantor’s Theorem as interpreted by the plenitudinous combinatorialist tells us that the powerset of any set is larger than that set, repeatedly taking powersets is an alternative engine of transfinite cardinal production: the set of natural numbers gives us one cardinal, its powerset another, the powerset of the powerset of the set of natural numbers a third cardinal, etc. If one believes that we are dealing with a clear conception of the infinite, and in particular that we are in possession of a definite idea about what the world would be like if φ (as defined in Sect. 2.2) were true, then one must believe that these two routes lead to the same place. Any cardinal “produced” by the ordinal method must be comparable to any cardinal “produced” by the powerset method; i.e., it must be the case that the first is smaller than the second, or that they are identical, or that the first is larger than the second. In particular, there must be an answer to the question of whether the cardinal number of the powerset of the set of natural numbers is identical to ℵ1 . But of course, to ask that question is to ask about the question of the truth value of the continuum hypothesis. And that question does not seem to have an answer. That is, as far as we know, Cantor’s generative approach and the powerset approach to the transfinite do not match up.14 Cantor discovered the powerset route first, and the ordinal route later, intending the latter to be the canonical measuring stick for cardinality. Given that background, the most basic “proof of concept” you should demand before accepting the theory ought to be that it can measure the first uncountable cardinality found on the powerset route. But it cannot.15 Hence, the theory of the transfinite exists in a condition in which it is not at all clear what it is we are asked to believe in the possibility of. (As Quine (1969, 23) put it: “No entity without identity”.) The continuum hypothesis is not just an interesting open research 14 In

footnote 7, it was explained that the contemporary approach to establishing the existence of uncountable ordinals employs power sets. You might think that having one of the routes to higher infinity go via the other would help in comparing the two routes. But no, the problem remains. 15 In fact, it cannot measure anything: among the uncountable sets that were first thought of by mathematicians independently of set theory, such as—in addition to the set of real numbers— the set of complex numbers, the set of functions from real numbers to real numbers, its subset of differentiable functions, etc., not a single one has had its size determined by one of Cantor’s alephs. The system of alephs is supposed to be a measuring system for infinite sizes, but except for the countable, it is a measuring system that can only measure itself. As such, it is a complete failure.

22

2.4 The Continuum Hypothesis

question; its undecidability is a philosophical problem for the plenitudinous combinatorialist. Of course, it may be the case that the question is not undecidable, but just undecided. If that changes in the future, the plenitudinous combinatorialist will have a somewhat stronger case. However, in the present state of affairs, the openness of the question is another reason for being skeptical. The previous paragraph contains the conclusion of this section. But that conclusion has not yet been substantiated, as the reasoning in that paragraph was very naïve. We therefore need to disambiguate different meanings of “the continuum hypothesis”, identify the meaning that is relevant, and then give a proper argument for our conclusion based on that meaning. One thing that can be meant by “the continuum hypothesis” is the hypothesis that, from ZFC, one can deduce the formal sentence in the language of set theory asserting that there is no cardinal larger than ℵ0 and smaller than the cardinality of the power set of ℵ0 . The truth value of that hypothesis is well known (provided one assumes that ZFC is consistent), as Cohen (1963, 1964, 1966) proved that it is false. However, as Gödel (1940) had previously shown, the negation of the formal sentence does not follow from ZFC either, so Cohen’s result does not constitute the kind of answer that would show us that the two routes into the infinite match up. On the contrary, it indicates that they do not. However, this formal independence of ZFC does not imply, by itself, that the two routes cannot be shown to match up. Here, we are interested in whether they do so in the case of the iterative hierarchy. This is a question about a certain structure, which according to the plenitudinous combinatorialist possibly exists, and not a question about what follows from formal axioms. Formal first-order axioms are notoriously impotent when it comes to characterizing structures, so we should not look to them to settle this question. The relevant meaning of “the continuum hypothesis” is the hypothesis that the cardinality of the power set of ℵ0 is the second-smallest cardinal in the iterative hierarchy, i.e., the intended model of ZFC.16 There might be an answer to this question that we simply haven’t found yet. But I take the lack of an answer after so much work has been done to be an indication that there is no answer. And at least until I have been proved wrong in that regard, I do not think anyone should consider Cantor’s theory to be more than a tentative work-in-progress. That history has, instead, elevated it to the status of standard foundation is in my view very imprudent. 16 There is not actually a unique intended model as the “the” indicates, because equally legitimate

contestants can arguably differ with regard to the height of the hierarchy; but that is not relevant to a question about the second-smallest cardinal.

2

Classical Mathematics and Plenitudinous Combinatorialism

23

Now, we do know whether the continuum hypothesis is true of certain models of ZFC. Maybe comparing those models to the intended model could give us a clue about whether it is true of the latter? Both Gödel and Cohen proved their respective theorems about lack of formal derivability by investigating such models, so let us have a look at those. Gödel considered a model consisting only of so-called constructible sets. Given a set A, the set F(A) of constructible subsets of A is the set of all subsets of A of the form {x ∈ A | ψ(x)}, where ψ(x) is a first-order formula with constants only from A, and with x as its only free variable. The class of constructible sets is defined by iterating the operation of forming all constructible subsets of what is already available. This iteration is done over the class On of all ordinals of some given model V of ZFC. Thus, L0 = ∅, Lα+1 = F(Lα ) for all α ∈ On, and Lγ = ∪α 0 ∨ x ≤ 0). Brouwer gives examples of real numbers for which we cannot assert that it is one or the other. A prerequisite for those examples is the intuitionistic notion of real numbers. With the exception of the strict finitist, all parties to the debate agree that a real number is an infinitary object. Either it is an ordered pair of actually infinite sets of rational numbers (Dedekind 1872), or an actually infinite, converging sequence of rational numbers (Cauchy 1821; Heine 1872)—or, if you ask Brouwer, a potentially infinite, converging sequence of rational numbers.6 That is, a real number is the process whereby a creating subject constructs more and more elements of a so-called choice sequence. The elements can be freely chosen 5 For

more on this, see Hansen (2016a). the second and third cases it should really be “an actually/potentially infinite equivalence class of actually/potentially infinite, converging sequences of rational numbers”. In the interest of avoiding cumbersome formulations, I will pretend that a sequence is a real number, rather than an element of one. 6 In

32

3.1 General Introduction

by the subject, or he can decide to follow a rule when choosing elements. In the latter case, it must be possible to calculate each element in a finite amount of time for which an upper bound is known in advance. (Foreshadowing: while I will reject freely proceeding choice sequences, I will not place this restriction on definable sequences.) The specific details of the definition of “real number” can be filled out in several different, intuitionistically acceptable ways. For the purpose of this chapter, let us define a real number as a choice sequence q1 , q2 , q3 , . . . of rational numbers, such that |qn − qn+1 | ≤ 2−n for all natural numbers n, and such that each qn is of the form m · 2−n−1 for some integer m. A real number that, at present, can neither be asserted to be positive nor to be nonpositive can be constructed using a so-called fleeing property, defined by Brouwer (1955, 114) as follows. A property f having a sense for natural numbers is called a fleeing property if it satisfies the following three requirements: (i) For each natural number n, it can be decided whether or not n possesses the property f ; (ii) no way is known to calculate a natural number possessing f ; (iii) the assumption that at least one natural number possesses f , is not known to be contradictory. An example of a fleeing property P is, for a given finite sequence of digits not yet found in the decimal expansion of π and not yet proved not to occur in it, that that sequence occurs beginning at the n’th decimal. Then, let the real number a be defined as the choice sequence that begins with the elements 1/4, −1/8, 1/16, . . . , (−1/2)n+1 , . . ., and continues like that as long as no n has had the property P , and stays constant at (−1/2)n+1 from the first n that has the property P onwards (if such an n is found). Then, at any given point in the construction where the choice sequence is still “oscillating”, the creating subject is not in possession of a truth maker for either the sentence “a > 0” or the sentence “a ≤ 0”. This invalidity of a classical theorem leads to the validity of a non-classical one: namely, that all functions from R to R are continuous (Brouwer 1924). For instance, this is an illegitimate definition of such a function,  f (x) =

0 if x ≤ 0 1 if x > 0 ,

3

Intuitionism and Choice Sequences

33

because it would have to map a to a choice sequence f (a). The first two elements of f (a) could both be equal to 12 , for that is consistent with subsequent elements of f (a) converging to 0, and also consistent with subsequent elements of f (a) converging to 1. However, as we cannot make it the case that a > 0 or a ≤ 0 via a finite calculation with an upper bound on time consumption that is known in advance, there is no way, at present, to choose a third element of f (a), for any possible choice would either be too far away from 0 or too far away from 1 to allow the sequence to converge to that value if a subsequently attains a specific value (i.e., if a natural number is determined to have the property P , or it is determined that it is impossible for any natural number to have it). Thus f is not a total function on the real numbers, but only a partial function defined for those real numbers that are either positive or non-positive. It would thus seem that the foundation of mathematics on mental constructions comes at a heavy price compared to platonism: we are no longer allowed to reason using tertium non datur when we are not in a position to assert either the proposition in question or its negation; we are not permitted to use non-constructive proofs; and working with discontinuous real functions is also banned. One important point that this monograph makes is that the price of mentalism is much more modest than the intuitionist would have it seem. The reason is that the cost is to a large extend incurred because of the auxiliary doctrine of verificationism, not because of mentalism itself.

3.2

Brouwer on Freely Proceeding Choice Sequences

The specific elements of intuitionism that I want to focus on are arbitrary real numbers and freely proceeding choice sequences. The aspect of Brouwer’s intuitionism that distinguishes it the most from other types of constructivism is its use of choice sequences: sequences created in time via successive choices of new elements by a creating subject (Brouwer 1952, 142). At any point in time, only a finite initial segment has been constructed. The sequence is, therefore, never finished, but always in a state of expansion. According to Brouwer, basing mathematics on such objects eliminates the need to assume that something actually infinite exists. The creating subject can choose to pick the elements according to an algorithm: for example, one that selects rational numbers that are increasingly better approximations of π. But that is not required. The subject can also create a sequence in which each choice of an element is made at random. The subject may grant himself

34

3.2 Brouwer on Freely Proceeding Choice Sequences

the freedom to allow each element to be any member of some species (i.e., class), for instance, the species of natural numbers—or may elect, from the beginning of the construction or at any point during it, to impose restrictions on his own future choices. An important example is the decision to create a real number. This amounts to the subject self-imposing the restriction that each element shall be a rational number qn of the form m · 2−n−1 , satisfying |qn−1 − qn | ≤ 2−(n−1) if n > 1. Some terminology: a choice sequence governed by an algorithm will be called “lawlike”, whereas one that is not will be called “freely proceeding”. Although this seems like a simple distinction, some further clarification is required. First, the creating subject may impose some restrictions on future choices, but without going so far that there is only one option for each element. Sequences characterized by such restrictions will still be called “freely proceeding” even though that freedom is partial. Second, a choice sequence may be governed by, for example, a law that each element is the sum of the corresponding elements of two freely proceeding sequences. Since this does not qualify as an algorithm, when everything is taken into account, such a sequence will be considered freely proceeding. However, complicated examples like these are not very relevant to my purposes, so the reader is encouraged to keep a simpler stereotype of arbitrary choice in mind when freely proceeding sequences are discussed. Third, the categorization of choice sequences into lawlike and freely proceeding is intended to be exclusive and exhaustive. That exhaustiveness is achieved when one category is defined from the other by negation is not obvious when the negation is intuitionistic. But the categories are meant to be time-relative; i.e., a sequence is considered freely proceeding at a given instant if the subject has not, at that instant, decided to follow an algorithm for the rest of the sequence. Thus, at every instant, each sequence is either lawlike or freely proceeding, although a freely proceeding sequence may become lawlike later. According to the platonist, there are among the abstract mathematical objects real numbers that cannot be defined. While disagreeing with classical mathematics in many other respects, including whether abstract objects exist, Brouwer also claimed to have found a place for undefinable real numbers in the intuitionist ontology, namely, among the freely proceeding choice sequences. This thesis is perhaps presented most clearly in the following passage: [Intuitionism] also allows infinite sequences of pre-constructed elements which proceed in total or partial freedom. After the abandonment of logic one needed this to create all the real numbers which make up the one-dimensional continuum. If only the predeterminate sequences of classical mathematics were available, one could by introspective construction only generate subspecies of an ever-unfinished

3

Intuitionism and Choice Sequences

35

countable species of real numbers which is doomed always to have the measure zero. To introduce a species of real numbers which can represent the continuum and therefore must have positive measure, classical mathematics had to resort to some logical process, starting from anything-but-evident axioms[. . . ]. Of course, this so-called complete system of real numbers has thereby not yet been created; in fact only a logical system was created, not a mathematical one. On these grounds we may say that classical analysis, however suitable for technology and science, has less mathematical reality than intuitionist analysis, which succeeds in structuring the positively-measured continuum from real numbers by admitting the species of freely-proceeding convergent infinite sequences of rational numbers and without the need to resort to language or logic. (Brouwer 1951, 451–452)7 So, Brouwer’s claim is that the free creation of sequences—an arbitrary choice of an element, followed by another arbitrary choice of an element, ad infinitum in potentia—can result in sequences that cannot be defined. Without relying on abstract objects, but just the human potential for free mental construction, the intuitionist has access to the “full” set of real numbers. It is this notion that I want to dispute. There are two slightly different ways to interpret it. The stronger interpretation is that Brouwer does, in one crucial respect, exactly the same thing as the classical mathematician, by finding a non-denumerable totality of points with which to identify the continuum. If so, Brouwer changed his mind in his late work, because in his early years, before he came up with the idea of freely proceeding choice sequences, he was of the opinions that the continuum is a primitive notion; that it cannot be constructed out of entities of any other type; and that, specifically, it cannot be identified with a set of points. His 1907 description of the continuous and the discrete holds that they are complementary and equally basic aspects of the Primordial Intuition, and that points (and numbers) can only be used to analyze a pre-existing continuum by being the endpoints of the subintervals into which it could be decomposed.8 One reason Brouwer gives for why a continuum cannot be a set of points is that the available points are only those that can be identified with rational numbers or definable real numbers, i.e., lawlike sequences, implying that there is only a denumerable infinity of them and hence not enough to exhaust the continuum (Brouwer 1913). However, it is also possible to interpret the above passage in a weaker way: instead of Brouwer’s claim being that his reals make up the one-dimensional intuitive continuum, they just make up the mathematical continuum, i.e., the best possible 7 See

also Brouwer (1930) and (1952). is, of course, a view originating with Aristotle (1930).

8 This

36

3.3 Constitution of Free Choice Sequences

model we can have of the intuitive continuum. That would be consistent with this model falling short of perfection. In this interpretation, Brouwer makes a more modest claim, namely that the freely proceeding sequences add to the model something that the lawlike sequences cannot. However, the subtle differences between these exegetical theses do not affect the critique made below.

3.3

Constitution of Free Choice Sequences

Several claims discussed up to this point have been prefaced with “according to Brouwer”, and for most of those that were not, it was implicit. That ends now, as I will move into a critical mode to seek a more precise answer to the question of what a freely proceeding choice sequence is, independently of Brouwer’s position. I will reach an answer to that question of constitution at the end of this section, and then, in Sect. 3.4, argue on the basis of that answer that Brouwer is mistaken in thinking that freely proceeding choice sequences can contribute something to the analysis of the continuum that lawlike sequences cannot. The project of determining the constitution of freely proceeding choice sequences independently of Brouwer’s position is a little dicey, since, as the inventor of the term, Brouwer has some authority over the meaning of “freely proceeding choice sequence”. So let me clarify the rules of the game. I do not believe that it would be reasonable to say that, if there are no entities with all the properties Brouwer claims for freely proceeding sequences, then there are no freely proceeding sequences. Instead, I think that a basic characterization can serve as a common ground by picking out a certain class of entities, whose more sophisticated properties we can then disagree about. Let the first three paragraphs of the previous section serve as that basic characterization. So what exactly constitutes a freely proceeding choice sequence? As is witnessed by the debate on personal identity, questions of constitution can often be elucidated by first asking related questions of individuation and self-identity over time. So, if I begin a freely proceeding sequence of natural numbers now at t1 by making the first element 4, and then now at t2 add 9 to it as its second element, what is it that makes the sequence at t1 identical to the sequence at t2 ? The strongest possible answer, that they are qualitatively identical, can quickly be ruled out. If they were qualitatively identical they would have exactly the same properties, and so would already at t1 have 9 as its second element. So, by the same token, for each n, at t1 it would be a property of the sequence that there was some

3

Intuitionism and Choice Sequences

37

specific number that was its nth element. But then, the sequence would be actually infinite. Instead of the relevant property being has 9 for its second element, it could be has, at t2 and later, 9 for its second element. But this makes little difference; the problem still arises, mutatis mutandis, because there are still an actual infinity of properties. The fact that some of them are about the future does not make a meaningful difference. Brouwer and those of us who also reject actual infinity cannot accept that what will happen in the future corresponds, in general, to facts in the present—at least not when one assumes the possibility of an infinite future with genuinely random events; yet, Brouwer needs that premise if his choice sequences are to play the rôle of non-definable real numbers. Hence, he is committed to anti-realism with respect to the future. The failure of the foregoing attempt to reach a satisfactory answer teaches us two things: that we must look for some criterion of numerical identity instead, and that such a criterion must allow for the sequence to be genuinely dynamic. This is acknowledged by Brouwer (1955, 114): In intuitionist mathematics a mathematical entity is not necessarily predeterminate, and may, in its state of free growth, at some time acquire a property which it did not possess before. However, commenting on this quote, van Atten (2007, 14) states that a property such as ‘The number n occurs in the choice sequence x’ is constitutive of the identity of x, but is generally undecidable and does not satisfy PEM [the principle of the excluded middle]. If van Atten’s statement were true, the property the number 9 occurs in the choice sequence α would be constitutive of α, but that would imply that the t1 -incarnation of α is not α. Consequently, diachronic self-identity of a choice sequence would be impossible. At most, the property the number n occurs in the choice sequence x being constitutive of the identity of x is the case only from the point in time at which n is added to the sequence. On pain of commitment to actual infinity, it cannot be before that. And from that time onwards, it is decided.9 To avoid actual infinity in both its explicit and implicit forms, do we need to conclude that the temporal instantiation of our freely proceeding sequence at t2 is the object 4, 9 ? 9 Van

Atten has informed me that he only intended to say that if the third element of α has been chosen to be 1, then it is known that a choice sequence β, for which something different from 1 has been chosen as its third element, is not equal to α.

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3.3 Constitution of Free Choice Sequences

No, for that is just an ordered tuple, and a choice sequence is not just that. There is a dynamic aspect to a sequence that is absent from the n-tuple. This difference is, however, not in the past; the tuple has also been created, one element added at a time, in a temporal process. In Brouwer’s universe there are no atemporal mathematical objects; it is just that some of the temporal objects have been completed. That is the difference between the tuple and the sequence: the former has found its final form, while the latter will continue to undergo changes. This is, however, exactly the kind of claim that we must be cautious about interpreting. The notion that it “will continue to undergo changes” must not be understood as an assertion about the actual future of the sequence, for the actual future does not exist. Given the commitment to anti-realism with regard to the future, the only content this claim can have is that the creating subject has an intention to amend the sequence. So, allowing “intention to expand” to be short for “intention to expand in keeping with the restriction . . . ” if there is a restriction, the following is a more promising proposal regarding the constitution of our freely proceeding sequence at t2 : 4, 9, intention to expand Under that proposal, the present product of an ongoing construction is merely what has actually been constructed plus the psychological fact that its creator does not consider it finished. The self-identity of the sequence being created over time does not rely on any objects in the future, but simply on the subject choosing, when he adds a new element, to consider the extended finite sequence a part of the same freely proceeding sequence as the old one.10 I think this is the correct answer, that is, it is the closest thing we can find in “the inventory of the world” to what Brouwer envisions a freely proceeding choice sequence to be. Nevertheless, in the next section it will be useful to contrast this answer with another possible answer, namely that the constitution of our choice sequence at t2 looks like this: 4, 9, x3N , x4N , x5N , . . . Here, xnN is supposed to be an indeterminate element that is restricted to N. That is, at t2 it is true that the third element (e.g.) is a natural number, but neither true nor 10 One

might deny that such a decision to identify really has the force to secure actual identity. But then, we would launch into an even more extensive disagreement with Brouwer, so I will not argue against it here.

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Intuitionism and Choice Sequences

39

false that it is equal to 7. Then, at t3 , the choice sequence may change to 4, 9, 7, x4N , x5N , x6N , . . . , as the next choice determinates the third element, which until then was indeterminate. I think that 4, 9, intention to expand is the correct answer to the question of the constitution of the choice sequence at t2 because it captures all of what seem to be the facts of the situation under a parsimonious ontological analysis thereof. That is, 4 has been chosen as the first element, 9 has been chosen as the second element, and the creating subject has an intention to continue expanding the sequence—that’s it! It is a simple situation and there is no need to invoke the existence of mysterious indeterminate objects to understand it.11 Hence, let us refer to it as the “simple answer” and to the alternative answer as the “indeterminacy answer”. However, I will consider both of these answers to the question of the constitution of freely proceeding choice sequences in the following section.

3.4

Evaluation of Brouwer’s Claim

Let us evaluate Brouwer’s claim that he has succeeded in supplying an adequate ontology for the “full” system of real numbers that includes non-definable sequences of rational numbers, in light of the above analysis of the constitution of a freely proceeding choice sequence. A preliminary point is that, at any given time, only a finite number of choice sequences actually exist, because a choice sequence only exists if someone has created it. Thus, relying only on choice sequences that actually exist will definitely not suffice; rather R must consist of all possible choice sequences that satisfy the definition given above. And that is the idea: where the classical, platonic reals, by virtue of the hierarchical nature of the set-theoretical universe, must all exist for the set of them to do the same, Brouwer only commits himself to the possibility of constructing each 11 I am overstating the simplicity a bit. If you and I each produce a choice sequence and we have so far, by chance, picked the same elements in the same order, and we both intend to expand our respective sequences according to the same restrictions, if any, we have nevertheless produced different sequences. (The two sequences will be equal (so far), but not identical. Brouwer also makes this distinction, for example in his definition of “species” (1952, 142). Troelstra (1977) makes the distinction using the terminology “extensional identity” and “intensional identity”.) That is not captured by “4, 9, intention to expand”; there are also concrete facts about who the creator of the sequence is, when it was started, etc., that belong in a complete analysis of the constitution of a choice sequence. However, this complication is irrelevant to the issue at hand, for there is still no need to invoke the existence of indeterminate objects.

40

3.4 Evaluation of Brouwer’s Claim

of his reals. They do not all have to exist prior to them being collected in the species of all reals. His continuum is the totality of all possible convergent sequences of rationals. As I stated in the Introduction, I also think that the merely possible is acceptable building blocks for mathematics, so I will not take issue with that. Instead, the question to be asked is: which choice sequences are really possible? Assume that a is a platonic real number, i.e., that a is an actually infinite (and converging) sequence of rational numbers a1 , a2 , . . .; and assume further that this sequence is undefinable. If a creating subject attempts to construct the same real number (“same” in a mathematical, but not an ontological sense), it is possible for him to construct a1 , intention to expand, after which it is possible to expand it to a1 , a2 , intention to expand, and then to a1 , a2 , a3 , intention to expand. However, at any given instant, only a finite initial segment of a has been created. Assuming for the moment that they exist, the actually infinite, undefinable sequences do not correspond to possible routes for potentially infinite choice sequences: “possible” means “can be taken”, and the entire route corresponding to a platonic undefinable sequence can never be taken, only initial segments of it. There is a nice metaphor of Posy’s (1976, 98–99) that we can make use of here. He likens choice sequences to the route of a bus traveling on a forking highway. The journey of the bus can be seen from different perspectives. First, there is the perspective of a passenger seated with his back to the bus driver, so that he can only see the route already traversed. Second, there is the perspective of the driver, who in addition to the knowledge possessed by his passenger has an intention regarding where to travel to from his present position. And third, there is the perspective of a helicopter pilot looking down on the bus and road system from above, seeing both the traveled path and the roads ahead. Given the rejection of actual infinity, there is no helicopter perspective. Actually infinite roads are no less actually infinite than completed infinite travels. The only legitimate perspectives are the passenger’s and the driver’s: the former being finitely extensional and the latter both finitely extensional and finitely intensional. For the bus driver or the creating subject, there is an infinity of possibilities in the indefinite future. But one must not conflate an infinity of possibilities with the possibility of infinity. If that was a bit too metaphorical, the same point can also be made more formally, either by a comparison with classical mathematics or by employing tense logic. Assuming classical mathematics, we can say that freely proceeding choice sequences can only deliver the elements of N m) ∀n, m(n, m ∈ Q ⇒ n ≤ m ∨ n ≤ m) ∀n, m(n, m ∈ Q ⇒ n ≥ m ∨ n ≥ m) Proof Like the proofs of Theorems 2 and 5.

Q

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8.5

161

Classicality So Far

Thus far, we have reconstructed the classical theories of natural numbers, integers, and rational numbers. We state this in a classical theorem that generalizes Theorems 2, 5, and 8: Theorem F Let n and m be closed terms s.t. there are closed terms n1 , . . . , nk and m1 , . . . , ml s.t. n1 , . . . , nk , m1 , . . . , ml ∈ N is true, and n is built up syntactically from only n1 , . . . , nk , and , +, and ·, and similarly with m. Further, let nc be the N N result of making the following changes to n: First replace each ni with the canonical natural number that is the element of ni . Then replace all remaining occurrences of + with + and all remaining occurrences of · with ·. Let mc be defined similarly. N N Then n = m is true (false) iff nc = mc is classically true (false). The analogous N statements hold for Z, Q, and Q+ in place of N with the operations that are defined for those number systems; and for , ≤, and ≥ (with appropriate subscript) in place of =. Proof The reader should go through the various relevant rules to convince himor herself that they are direct adaptions of the Peano axioms for arithmetic and the classical methods for constructing the integers and rational numbers from the natural numbers. The use of this theorem below could be avoided if we went to the trouble of stating and proving all the relevant theorems about natural numbers, integers, and rational numbers. However, I trust that the reader can see that the real challenge does not lie with these classically countable number systems, and will appreciate this shortcut to the interesting stuff. In the following developments, similarity to classical mathematics will be an important desideratum, but not the only one, and not one that will be fully satisfied.

8.6

Classes

This section introduces general ways to make classes. The first is by listing all the elements: {x 1 , . . . , x n } term y ∈ {x 1 , . . . , x n } satisfied iff y ≡ x 1 ∨ . . . ∨ y ≡ x n true The next is by comprehension—unrestricted comprehension:

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8.6 Classes

{x | φ} term (binds all free variables of x) y ∈ {x | φ} satisfied iff, letting z1 , . . . , zn be the free variables of x, there are closed terms z1 , . . . , zn such that xJz1 z1 K · · · Jzn zn K ≡ y true and φJz1 z1 K · · · Jzn zn K true By way of example, here is a class of the irreducible canonical rational numbers:6 { n/m | n, m ∈ Z ∧ m = 0 ∧ ¬∃k, ln , lm (| k | = 1 ∧ k · ln = n ∧ k · lm = m)} Z

0,0/ 0,0

The rational number

Z

Z

Z

Z

Z

Z

Z

is an element of this class because ,0

/m Jn 0, 0KJm 0, 0K ≡ 0

n

/0,0

and n, m ∈ Z ∧ m = 0 ∧ ¬∃k, ln , lm (| k | = 1 ∧ k · ln = n ∧ k · lm = m) Z Z Z Z Z Z Z Z Jn 0, 0KJm 0, 0K are both true. The next stipulation allows for classes written with a more compact notation: {x | φ} term (binds all free variables of x) X

y ∈ {x | φ} satisfied iff y ∈ {x | x ∈ X ∧ φ} true X

In {x | φ} and {x | φ}, φ is called the defining condition. X Let me briefly discuss the “pathological” side of the system. This stipulation introduces Russell’s Class: R term x ∈ R satisfied iff x ∈ {y | y ∈ / y} true Obviously, R ∈ R is undefined. Using this class we can now give counterexamples to the existence of theorems similar to Theorems 1, 4, and 6 for N, Z, and Q. A counterexample for N is given by the sentence {n | n ≡ 0 ∧ R ∈ R} ∈ N, which 6 Note the use of articles here. It should not be “the set of irreducible rational numbers”, because in this system there are many different classes of all the irreducible canonical rational numbers—for example, you get one by interchanging two conjuncts in the displayed class—so it has to be “a class”. However, “a class of irreducible canonical rational numbers” would not work, as that might mean a class containing just some of the irreducible canonical rational numbers. So the formulation has to be “a class of the irreducible canonical rational numbers”.

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is undefined. Here is why. It follows from the undefinedness of R ∈ R that it is undefined whether 0 is an element of {n | n ≡ 0 ∧ R ∈ R}, while it is false for any other term that it is an element. Hence the sentence {n | n ≡ 0 ∧ R ∈ R} = 0 is N undefined, while {n | n ≡ 0 ∧ R ∈ R} = n is false for any other canonical natural N number n. From this the claim follows. Similar counterexamples can be constructed for Z and Q. In fact, a counterexample that works for all three simultaneously is provided by this completely indeterminate class: I term x ∈ I satisfied iff x ∈ {y | R ∈ R} true For any closed term x, the sentence x ∈ I is undefined. I is useful in the construction of examples that show the limitations of this system. Because N is determinate, the sentence ∀n (n + 1 ∈ N) is true. On the other hand, N N because N is indeterminate, ∀n (n + 1 ∈ N) is undefined. Truth can be “restored” N N by using the strong conditional; thus, the sentence ∀n(n ∈ N ⇒ n + 1 ∈ N) is true. N This difference has consequences for how we can formulate theorems. In the sequel, almost all theorems will have ⇒ as their main connective within some quantifiers, and the antecedents and consequents of these conditionals will be open sentences that have undefined instances. Because of the restrictions on supervaluation when ⇒ is iterated (see Sect. 7.3.5), those antecedents and consequents cannot be formulated using ⇒. Since that method of “restoring” definite truth values is therefore not available, quantification within an antecedent or a consequent will have to be over determinate classes only. This will require some creativity, and some deviations from the way things are done in classical mathematics. However, that will come later. Unions, intersections, relative complements, and subclasses are standard:7 X ∪ Y term z ∈ X ∪ Y satisfied iff z ∈ X ∨ z ∈ Y true X ∩ Y term z ∈ X ∩ Y satisfied iff z ∈ X ∧ z ∈ Y true X \ Y term z ∈ X \ Y satisfied iff z ∈ X ∧ z ∈ / Y true 7 Absolute

complements are of course also legitimate, but we shall not need it.

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8.6 Classes

X ⊂ Y wff X ⊂ Y satisfied iff ∀x (x ∈ Y ) true X

The final additions of this section are stipulations that allow for taking the minimum and maximum of a class: min X term iff x ≥ y wff and x V term V

V

y ∈ min X satisfied iff ∃x (y ∈ x V ∧ ∀x0 (x0 ≥ x)) true V

X

X

V

max X term iff x ≤ y wff and x V term V

V

y ∈ max X satisfied iff ∃x (y ∈ x V ∧ ∀x0 (x0 ≤ x)) true V

X

X

V

An example will help to bring out the salient features of these stipulations. Consider this term: min{0, 2, 3, 0} Z

The class over which the minimum is taken has three elements. The first two of these are equal natural numbers and the last is an integer, which, intuitively, is larger than the first two. Because the stipulation for minimum contains the term x V instead of just x, and because of the stipulation for ≥ and those that it builds on, the two V natural numbers are in effect “raised” to their integer equivalents before anything else “happens”. So ∀  x0 (x0 ≥ x) is true for both 0 and 2 in place of x. The {0 ,2,3,0} Z elements of 0Z and 2Z are the same, and they are, it follows, also the elements of min{0, 2, 3, 0}, which therefore is equal to integer-2. Z

From this example it can be seen that, when there is more than one minimal, respectively maximal, element, minimum and maximum only work as intended if equal numbers have the same elements (at least after being “raised”). That will not be the case for real numbers, which we will come to shortly; so for them, infimum and supremum must be used instead. Another example is max Z. In classical mathematics, we would say that this is Z not well-defined. Here, however, it is well-defined: it is perfectly legitimate as a term, and it is an empty class. What we can say instead is that it is not a natural number. As of now, minimum and maximum do not work for (unraised) natural numbers because x N is not a term when x is. We need this auxiliary stipulation to remedy that problem:

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nN term m ∈ nN satisfied iff m ∈ n true

8.7

An Example of Applied Mathematics

To apply our mathematics to the empirical world outside of our language, it will be useful to have terms of the language that denote the objects to which we would like to apply the mathematics. The terms of pure mathematics we have introduced so far denote nothing, so we will here make use of denotation for the first time. As an example, let us add some of the stipulations needed if we are to apply arithmetic to the study of demographics. We first add a name for each human being to the language:8 For n a canonical natural number: – hn term – hn denotes the firstborn human currently alive and not denoted by any of h0 , h0 , . . . , hn (if that is ambivalent because two people were born at the same instant, it denotes the one born furthest north; if that is also ambivalent, it denotes the one born furthest east) A meta-convention that should be added to the list in the beginning of this chapter is this one: if one attempts to assign a term of the formal language a denotation by a definite natural-language description that is not satisfied by a unique object, the term does not denote anything. Hence, the term h0···  with 10 billion primes does not denote anything. Just as we earlier imported the conjunction from English, we can now import the common noun “human” and (pretending that there is no philosophical problem of personal identity) the existing criterion for identity between individuals:

8 That is one way to do it, but I don’t think that it is necessary: a language convention that allows

us to communicate about the number of humans could work without any reliance on names for each person, because humans, unlike mathematical objects, are different from and exist independently of their names. But in a book, conventions of language have to be explained using language and it would therefore be a challenge to explain such a language convention without using mathematical vocabulary, which would make it difficult to get the point across that the nominalistic mathematics here developed does not rely on some other, antecedent, kind of mathematics.

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8.7 An Example of Applied Mathematics

H term x ∈ H satisfied iff x denotes a human x = y wff H

x = y satisfied iff x and y denote the same person H

The last thing we need is a (finite) cardinality operator: #X term V

0 ∈ #X satisfied iff ¬∃y (y = y) true V

V

V

For n a canonical natural number: – n ∈ #X satisfied iff ∃y (y = y ∧ n ∈ #{z | z = y}) V

X

V

V

X

V

∧ ¬∃y1 , y2 (y1 = y1 ∧ y2 = y2 ∧ y1 = y2 ∧ n ∈ #{z | z = y1 ∧ z = y2 }) X

V

V

V

V

X

V

V

true With these stipulations, the sentence # H > 7,500,000,000, for instance, comes H N out true. Thus, we have managed to convey a mathematical fact about the empirical world without relying on the existence of abstract mathematical objects. The short version of the indispensability argument discussed earlier goes as follows. Our best scientific theories quantify over mathematical objects, and therefore we are committed to the existence of mathematical objects on pain of intellectual dishonesty. What we have developed in this chapter is a language that, as a matter of syntax, allows for quantification over mathematical objects, but arrives at the right truth conditions even though the mathematical terms do not refer to anything. If we are able to develop this language to the point that it is adequate for all of science, we will have undermined the indispensability argument. I hope that the simple example of application in this section suffices to make it plausible that the areas of science that use no more advanced mathematics than countable number systems are covered. In the next chapter, we will turn to the more challenging case of real analysis. First, though, let me say a little more about the relationship between pure and applied mathematics, again taking Wittgenstein as my interlocutor: 37. Put two apples on a bare table, see that no one comes near them and nothing shakes the table; now put another two apples on the table; now count the apples that are there. You have made an experiment; the result of the counting is probably 4. (We should present the result like this: when, in such-and-such circumstances, one puts first 2 apples and then another 2 on a table, mostly none disappear and none get added.)

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And analogous experiments can be carried out, with the same result, with all kinds of solid bodies.—This is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums. “But shouldn’t we then still have 2 + 2 = 4?”—This sentence would have become unusable. (Wittgenstein 1956) If I am right, there is a sharper distinction between pure mathematics and its application than Wittgenstein claims here. It is conceivable that apples, sticks, fingers, lines, and most other things could behave in such a way that the truth of the sentence 2 + 2 = 4 would be utterly irrelevant to everything outside of the mathematical lanN N guage game, but that we had nevertheless adopted the conventions of this chapter. Then, the sentence would indeed have become unusable for practical purposes, but that would not be the end of all sums. That is not to say, however, that the existence of arithmetic is necessary, as the platonist would have it. If physical objects behaved as imagined by Wittgenstein, humans might not have managed to—or bothered to—institute the conventions that are needed to do arithmetic. That “however” has to be counteracted by another “however”, however: most of the platonist’s intuitions about the modal resilience of arithmetic can be preserved by noting that we, who do have arithmetic, can truly say that if those strange people had had our arithmetical conventions, then 2 + 2 = 4 would have been true in their N N world.

Chapter 9

Real Analysis

I will now turn to mathematics that is classically uncountable.

9.1

Functions

Having functions available is a prerequisite for introducing real numbers. As in classical mathematics, we will let a function be a class of ordered pairs. The stipulation for ordered pairs is simple: (x, y) term Unlike in classical mathematics, an ordered pair has no elements. It does not have to, because its syntax reveals its two coordinates. We then add terms of the form X →  Y , a class of the partial functions from X to Y , and notation for the application of a function to a term: X→  Y term iff x 1 = x 2 and y 1 = y 2 wffs X

Y

f ∈X→  Y satisfied iff ∀w ∃x0 ∃y0 (w ≡ (x0 , y0 ) ∧ ∀x1 , y1 ((x1 , y1 ) ∈ f ∧ x1 = x0 → y1 = y0 )) f

X

Y

X

Y

true f (x) term iff x 1 = x 2 wff X

X

 Y ∧ x = x ∧ v ∈ y) true v ∈ f (x) satisfied iff ∃x, y, Y ((x, y) ∈ f ∈ X → X

X

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C.S. Hansen, Founding Mathematics on Semantic Conventions, Studies in Epistemology, Logic, Methodology, and Philosophy of Science, https://doi.org/10.1007/978-3-030-88534-2_9

169

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9.1 Functions

A simple, yet instructive, example is the function {(0, 0), (2, 1)}. The reader should verify that these two sentences are true: {(0, 0), (2, 1)} ∈ N →  N {(0, 0), (2, 1)}(1 + 1) = 1 N

N

N

As the example shows, these stipulations ensure a high degree of flexibility when working with functions. A function is allowed to contain several different ordered pairs with equal but different first coordinates; but it is not required to contain ordered pairs with all the, say, rational numbers of a given class of equal rational numbers as first coordinate in order for that function to be applicable to all of those equal rational numbers. This flexibility comes at a small price in elegance: a uniform rule for functional application cannot be given. The one given here works for N as a co-domain, but only because any two equal natural numbers have the same elements. That also holds, mutatis mutandis, for Z and Q, but, as already mentioned, that will not be the case for real numbers when they are introduced below. Thus, they will require a separate kind of functional application. Next up is a class of the full functions from X to Y : X → Y term iff x 1 = x 2 and y 1 = y 2 wffs X

Y

f ∈ X → Y satisfied iff f ∈ X →  Y ∧ ∀x0 ∃x ∃y (x = x0 ∧ (x, y) ∈ f ) true X

Y

X

This one only works if the domain is one of those classes with which an equality relation is associated. If it is just a sub-class of such a class, this syntax must be used instead: X → Y term iff z1 = z2 and y 1 = y 2 wffs Z

Z

Y

f ∈ X → Y satisfied iff f ∈ Z →  Y ∧ ∀x0 ∃x ∃y (x = x0 ∧ (x, y) ∈ f ) Z

X

Y

X

∧ ∀w ∃x0 ∃x ∃y (x = x0 ∧ w ≡ (x, y)) true f

X

Z

Y

Z

Continuing with the previous example, this sentence is true: {(0, 0), (2, 1)} ∈ {2} → N N

Finally, we allow a function to be given by means of a domain and a formula, using lambda notation:

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Real Analysis

171

λx .y term (binds x in y and the displayed x) X

(x, yJx xK) ∈ λx .y satisfied iff x ∈ X true X

A simple example is the function λn.2 · n + 3, whose elements are (0, 2 · 0 + 3), N N N N N (0, 2 · 0 + 3), (0, 2 · 0 + 3), . . ., and nothing else. Because the second of these N N N N elements has as first coordinate a term that is equal to 1, and as second coordinate a term that is equal to 5, this sentence is true: (λn.2 · n + 3)(1) = 5. N

9.2

N

N

N

N

Real Numbers

We have seen that it works just fine to have several different terms that correspond to the same classical natural number, integer, or rational number, as long as we have an equality relation that binds them together. It would not have worked well if we had instead decreed that, say, the natural number zero should be an equivalence class of all those classes that are equal to 0 (as that term is actually used), for such a class would be indeterminate; and we do not want more of that than is absolutely necessary. For the same reason, we will let real numbers be Cauchy sequences of rational numbers, rather than equivalence classes of such sequences: R term a ∈ R satisfied iff a ∈ N → Q ∧ ∀ + ∃n ∀k1 , k2 > n (| a(k1 ) − a(k2 )| < ) true Q

N

N

N

Q

N

Q

N

Q

However, we still need the equivalence relation itself. That stipulation is given below, preceded by a prerequisite: namely, the operator that “raises” natural numbers, integers, and rational numbers to real numbers. If x is already a real number, x R simply inherits its elements. If, instead, x is a natural number, integer, or rational number, x R is the constant sequence that has x in each “position” of the sequence: x R term (n, y) ∈ x R satisfied iff n ∈ N ∧ ((n, y) ∈ x ∨ (x ∈ Q ∧ y ≡ x)) true a = b wff R

a = b satisfied iff a R , bR ∈ R ∧ ∀ + ∃n ∀k > n (| a R (k) − bR (k)| < ) satisfied R

Q

N

N N

Q

N

Q

N

Q

The first of many theorems in this section confirms that the equality stipulation is successful:

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9.2 Real Numbers

THEOREM 9: R-EQUALITY IS AN EQUIVALENCE RELATION ∀a (a R ∈ R ⇒ a = a) R

∀a, b (a = b ⇒ b = a) R

R

∀a, b, c (a = b ∧ b = c ⇒ a = c) R

R

R

Proof I prove the first sentence (in great detail) and leave the other two to the reader. Let a be a closed term and assume a R ∈ R. That presupposes a R ∈ N → Q,  Q and (2) ∀n0 ∃n ∃p (n = n0 ∧ (n, p) ∈ a R ). which presupposes (1) a R ∈ N → N Q N With the aim of proving ∀ +∃n∀k > n(| a R (k)− a R (k)| < ), let a closed term  be Q N N N Q N Q N Q given and assume that  ∈ Q+ is not false. By Theorem 7, it follows that  ∈ Q+ is true. For n, 0 will do. Let a closed term k be given and assume that k ∈ N ∧ k > 0 N is not false. Theorem 1 implies that k ∈ N is either true or false, so it is true. The truth of (2) presupposes k ∈ N → ∃n ∃p (n = k ∧ (n, p) ∈ a R ), which Q N in turn presupposes either the falsity of k ∈ N or the truth of ∃n ∃p (n = k ∧ Q N (n, p) ∈ a R ); but the former option is ruled out. The latter presupposes that there are terms n and p such that p ∈ Q, n = k, (n, p) ∈ a R , and—in conjunction N with (1)—that any ordered pair in a R with a first coordinate equal to n has a second  Y ∧ coordinate equal to p. All this grounds that ∃n, p, Y ((n, p) ∈ a R ∈ N → n = k ∧ v ∈ p), and hence v ∈ a R (k), are true for exactly those v that are elements X N of p. This grounds | a R (k) − a R (k)| < . Q N Q N Q Discharging the assumption that k ∈ N ∧ k > 0 is not false, this again N grounds k ∈ N ∧ k > 0 → | a R (k) − a R (k)| < . As k was arbitrary, N Q N Q N Q ∀k > 0 (| a R (k) − a R (k)| < ) is grounded. Together with the basic truth 0 ∈ N, N N Q N Q N Q this in turn grounds 0 ∈ N ∧ ∀k > 0 (| a R (k) − a R (k)| < ). This grounds N N Q N Q N Q ∃n ∀k > 0 (| a R (k) − a R (k)| < ). Discharging the assumption that  ∈ Q+ is not N N N Q N Q N Q false, this again grounds  ∈ Q+ → ∃n ∀k > 0 (| a R (k) − a R (k)| < ). As  was N N N Q N Q N Q arbitrary, ∀ + ∃n ∀k > 0 (| a R (k) − a R (k)| < ) is grounded. Together with the iniQ N N N Q N Q N Q tial assumption that a R ∈ R, this grounds a R , a R ∈ R ∧ ∀ + ∃n ∀k > 0 (| a R (k) − Q N N N Q N Q a R (k)| < ), which grounds a = a. Discharging the assumption that a R ∈ R, N Q R we can infer a R ∈ R ⇒ a = a, as the reasoning has been “down-up”. As a was R arbitrary, ∀a (a R ∈ R ⇒ a = a) can be concluded. R

It is important to note how we moved without effort from “not false” to “true” in the third paragraph of this proof. A little reflection shows that this means that restricted quantification over N, Z, Q, and Q+ does in general function as it does in classical mathematics. We will rely on this observation to make subsequent proofs shorter.

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Our next theorem makes use of the class R, so a comment has to be made about the relationship between R and R. It is unlike the relationship between N and N, etc., and we will not refer to R as the class of canonical real numbers. As mentioned in Sect. 8.3, N, Z, and Q share the nice features of (1) being determinate and (2) containing a counterpart of each element of the corresponding classical sets. R has neither of these features.1 So in particular, it is not feature (1) that distinguishes R from R. The difference is merely that R contains natural numbers, integers, and rational numbers (since, by being “raised”, they are equal to real numbers). Two real numbers are added in the same way as in classical mathematics: a + b term R

(n, a R (n) + bR (n)) ∈ a + b satisfied iff n ∈ N ∧ a, b ∈ R true N

Q

N

R

THEOREM 10: R IS CLOSED UNDER ADDITION ∀a, b (a, b ∈ R ⇒ a + b ∈ R) R

Proof Assume a, b ∈ R. That presupposes a R ∈ N→Q, which presupposes that the elements of a R are, for each canonical natural number n, one or more terms of the form (n, p a,n ), where p a,n ∈ Q; and that if there are more than one, then the p a,n are equal; and that a R contains nothing else. That pa,n ∈ Q presupposes that p a,n contains, for some canonical rational number, all and only the canonical rational numbers that are equal to it. That grounds that a R (n) contains those same canonical N rational numbers; and similarly for b, by analogous reasoning. That grounds, for each n, that a R (n) + bR (n) is equal to some canonical rational number. That, in N Q N turn, grounds that a + b ∈ N → Q. R With the aim of proving ∀ + ∃n ∀k1 , k2 > n (| (a + b)(k1 ) − (a + b)(k2 )| < ), Q N N N Q R N Q R N Q let an  ∈ Q+ be given. Let  half be a canonical rational number such that  half = Q /2. That a ∈ R presupposes that ∃n ∀k1 , k2 > n (| a R (k1 ) − a R (k2 )| <  half ). That Q N N N Q N Q N Q again presupposes that there is a canonical natural number na such that ∀k1 , k2 > na N N (| a R (k1 )− a R (k2 )| <  half ). Let nb be given similarly. Let n be the canonical natural Q N Q N Q number such that n = max{na , nb }. Let k 1 and k 2 be canonical natural numbers N N such that k 1 , k 2 > n. Then | a R (k 1 )−a R (k 2 )| <  half and | bR (k 1 )−bR (k 2 )| <  half . N Q N Q N Q Q N Q N Q It follows from what was shown above that (a + b)(k 1 ) = a R (k 1 ) + bR (k 1 ) and R N Q N Q N similarly with k 2 , so | (a + b)(k 1 ) − (a + b)(k 2 )| = | (a R (k 1 ) + bR (k 1 )) − (a R (k 2 ) + Q R N Q R N Q Q N Q N Q N Q bR (k 2 ))|. Hence, Theorem F ensures that | (a + b)(k 1 ) − (a + b)(k 2 )| ≤ | a R (k 1 ) − N

1 It

Q

R

N

Q

R

N

Q Q

N

Q

will later be possible to formulate this point in a different way, by using the concept of indefinite extensibility; see Sect. 10.1.

174

9.2 Real Numbers

a R (k 2 )| + | bR (k 1 ) − bR (k 2 )|. Ergo, | (a + b)(k 1 ) − (a + b)(k 2 )| < . We can N Q Q N Q N Q R N Q R N Q conclude that the target sentence, ∀ +∃n ∀k1 , k2 > n(| (a + b)(k1 )− (a + b)(k2 )| < ), Q N N N Q R N Q R N Q is grounded; so a + b ∈ R is grounded; so ∀a, b (a, b ∈ R ⇒ a + b ∈ R) is R R true. This proof amounts to going down in the presuppositions of the antecedent, plugging in the classical proof, and then grounding the consequent. Very many proofs are like that. In the sequel, when that is the case, I will not give the proof but just state the theorem without one.2 THEOREM 11: SUMS ARE INVARIANT W.R.T. EQUALITY ∀a, b, c, d (a = b ∧ c = d ⇒ a + c = b + d) R

R

R

R

R

a · b term R

(n, a R (n) · bR (n)) ∈ a · b satisfied iff n ∈ N ∧ a, b ∈ R true N

Q

N

R

LEMMA 12: REAL NUMBERS ARE BOUNDED ∀a (a ∈ R ⇒ ∃p+ ∀n (| a(n)| ≤ p)) Q

N

Q

N

Q

THEOREM 13: R IS CLOSED UNDER MULTIPLICATION ∀a, b (a, b ∈ R ⇒ a · b ∈ R) R

THEOREM 14: PRODUCTS ARE INVARIANT W.R.T. EQUALITY ∀a, b, c, d (a = b ∧ c = d ⇒ a · c = b · d) R

R

R

R

R

THEOREM 15: ASSOCIATIVITY, COMMUTATIVITY, AND DISTRIBUTIVITY ∀a, b (a, b ∈ R ⇒ a + b = b + a) R

R

R

∀a, b, c (a, b, c ∈ R ⇒ a + (b + c) = (a + b) + c) R

R

R

∀a, b (a, b ∈ R ⇒ a · b = b · a) R

R

R

R

R

∀a, b, c (a, b, c ∈ R ⇒ a · (b · c) = (a · b) · c) R

R

R

R

R

R

R

∀a, b, c (a, b, c ∈ R ⇒ a · (b + c) = a · b + a · c) R

2 In

R

R

R

such cases, the classical proof can be found in Sprecher (1970).

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Real Analysis

175

THEOREM 16: ADDITIVE IDENTITY ELEMENT ∀a (a ∈ R ⇒ 0 + a = a) R

R

a − b term R

(n, a R (n) − bR (n)) ∈ a − b satisfied iff n ∈ N ∧ a, b ∈ R true N

Q

N

R

−a term R

(n, −a R (n)) ∈ −a satisfied iff n ∈ N ∧ a ∈ R true Q

N

R

THEOREM 17: ADDITIVE INVERSE ∀a (a ∈ R ⇒ a + (−a) = 0) R

R

R

THEOREM 18: MULTIPLICATIVE IDENTITY ELEMENT ∀a (a ∈ R ⇒ 1 · a = a) R

R

a/b term R

(n, a R (n)/bR (n)) ∈ a/b satisfied iff n ∈ N ∧ a, b ∈ R ∧ bR (n) = 0 ∧ b = 0 N

Q

N

R

N

Q

R

true (n, 0) ∈ a/b satisfied iff n ∈ N ∧ a, b ∈ R ∧ bR (n) = 0 ∧ b = 0 true R

N

Q

R

THEOREM 19: MULTIPLICATIVE INVERSE ∀a (a ∈ R ∧ a = 0 ⇒ a · (1/a) = 1) R

R

R

R

Proof The truth of a ∈ R ∧ a = 0 presupposes that a R is a sequence of rational R numbers (elements of Q) such that from some point onward, they are all different from 0. Since only the tail is relevant for present purposes, it is easy to see that this grounds a · (1/a) = 1. R

R

R

Taken together, the foregoing theorems imply that R is a field. Following the introduction of an ordering on the real numbers, Theorems 20, 22, 23, and 24 will show that it is an ordered field.

176

9.2 Real Numbers

a < b wff R

a < b satisfied iff a R , bR ∈ R ∧ ∃ρ+ ∃n ∀k > n (bR (k) − a R (k) > ρ) satisfied R

Q

N

N N

N

Q

N

Q

Theorem 20 is of particular interest, as it marks an important dividing point between classical mathematics and intuitionism: namely, the proposition that each real number is either negative, equal to zero, or positive. In this respect too, this system is like classical mathematics. (And again the difference between the classical proof and the proof for our system is too trivial to make it worth the effort to write out the latter.) By now it should be clear that, when we are dealing with individual real numbers, not much is different from the classical case. THEOREM 20: TRICHOTOMY FOR REAL NUMBERS ∀a, b (a, b ∈ R ⇒ ((a < b ∧ a = b ∧ a > b) R

R

R

∨ (a < b ∧ a = b ∧ a > b) ∨ (a < b ∧ a = b ∧ a > b)) R

R

R

R

R

R

THEOREM 21: LESS-THAN RELATION IS INVARIANT W.R.T. EQUALITY ∀a, b, c, d (a = b ∧ c = d ∧ a < c ⇒ b < d) R

R

R

R

THEOREM 22: TRANSITIVITY OF LESS-THAN RELATION ∀a, b, c (a < b ∧ b < c ⇒ a < c) R

R

R

THEOREM 23: LESS-THAN RELATION PRESERVED UNDER ADDITION ∀a, b, c (a < b ∧ c ∈ R ⇒ a + c < b + c) R

R

R

R

THEOREM 24: POSITIVE REAL NUMBERS CLOSED UNDER MULTIPLICATION ∀a, b (a, b > 0 ⇒ a · b > 0) R

R

R

We also add a term for the absolute value of a real number: | a | term

R

(n, p) ∈ | a | satisfied iff a ≥ 0 ∧ (n, p) ∈ a R true R

R

(n, −p) ∈ | a | satisfied iff a < 0 ∧ (n, p) ∈ a R true Q

R

R

We complete this section with another anti-intuitionistic theorem, showing that the real numbers divide neatly into rational reals and irrational reals.

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177

Q term R

a ∈ Q satisfied iff ∃p (a = p) true R

Q

R

THEOREM 25: REAL NUMBERS ARE RATIONAL OR IRRATIONAL ∀a (a ∈ R ⇒ a ∈ Q ∨ a ∈ R \ Q) R

R

Proof Assume a ∈ R. That presupposes a is such that, for each canonical natural number k, it is the case that a R (k) = q for some canonical rational number q. That N Q grounds ∀ + ∃n ∀k > n (| a R (k) − p R (k)| < ) as being either true or false for each Q N N N Q N Q N Q canonical rational number p. That, in turn, grounds that a = p is true or false. As R Q is determinate, this again grounds the consequent of the theorem. The formulation of the second sentence of this proof calls for an explanation. It would have been incorrect to write “That presupposes, for each canonical natural number k, that a R (k) = q for some canonical rational number q”, because the N Q sentence a R (k) = q does not appear “under” a ∈ R in the language for the theorem N Q sentence with which the proof is concerned. What does appear under a ∈ R are all the sentences of the form x ∈ a; and the possible combinations of truth values of these sentences are what ground the truth of ∀ + ∃n ∀k > n (| a R (k) − p R (k)| < ). Q N N N Q N Q N Q The easiest way to state what those combinations are is to say that they are those compatible with the proposition following “is such that” in the second sentence of the proof. I will make extensive use of such formulations below.

9.3

Exponentiation

Raising 2 to some natural number power will be used repeatedly below to construct various real numbers, so we need to introduce exponentiation. It is quite simple to formulate a stipulation that allows the base number to be a natural number, an integer, a rational number, or a real number, and outputs a number of the same type. However, we will refrain from generalizing to cases where the exponent is not a natural number. x nN term iff x · x, x V , and V terms V

V

y ∈ x 0N satisfied iff y ∈ 1V ∧ x ∈ V true V

For n a canonical natural number: – y ∈ x n satisfied iff y ∈ x · x Nn true N V

V V

178

9.4 Completeness

For n not a canonical natural number: – y ∈ x nN satisfied iff n ∈ N ∧ ∃i (i ∈ n ∧ y ∈ x iN) true N

V

V

Let me give a simple example of how we are going to use powers of 2 in the construction of real numbers. It concerns how we could give a stipulation for the square root of 3. (It will not be an official stipulation, for a stipulation for roots that is both general and elegant must await the introduction of suprema.) We could stipulate √ √ that 3 is a term, and that (n, m/2nN ) ∈ 3 has its bivalent immediate view-fromQ N nowhere truth conditions satisfied iff n ∈ N ∧ m ≡ max {m | (m/2Nn )2N ≤ 3} is true.3 N N Q N Q Q √ Then 3 would contain exactly these terms:   (0, max {m | (m/20N )2N ≤ 3}/20N ), (0, max {m | (m/20N )N2 ≤ 3}/20N ), . . . N

N

Q N Q

Q

Q N

N

N

Q N

Q

Q

Q N

Since N is determinate, the class over which the maximum is taken is, in each case, determinate. So, the maximum is equal to some natural number, and therefore the quotient is equal to some rational number. Those rational numbers are 1/1, 3/2, Q Q 6/4, 13/8, . . ., that is, better and better approximations to the square root of 3. It is Q Q √ √ thus seen that 3 ∈ N → Q. And it is also true that 3 ∈ R: for a given positive canonical rational number , just let n be a canonical natural number such that 1/2nN < . Q N

Q

9.4

Completeness

As mentioned above, we need a separate way to apply a function when the codomain is R. To illustrate the problem, consider a function f ∈ N →  R that contains (0, a) and (0, b), where a and b are equal real numbers, but a contains (0, 1/2), Q whereas b contains (0, 1/3). Then, f (0) contains both (0, 1/2) and (0, 1/3), and Q N Q Q can therefore not be a real number. A naïve idea for a solution is to let f (0) inherit N only the elements of a and not those of b. This idea is naïve because there is no general method for choosing the real number from which the inheritance shall come. However, there is a general method for constructing a single real number c that is equal to both a and b. Even though a and b are different, their equality ensures that for any rational number p, p ≤ a holds if and only if p ≤ b holds. Therefore, c R R can be the real number that has elements of the form m/2nN , where n is any natural Q N number and m is chosen in a way that is similar to the square root example. Square brackets will indicate that this alternative form of functional application is in play: 3 An

alternative and more “direct” √ way to give essentially the same stipulation would be: “(n, max {m | (m/2nN )2N ≤ 3}/2nN ) ∈ 3 satisfied iff n ∈ N true”. That, however, is less perspicN N Q N Q Q Q N uous.

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179

f x ! term iff x 1 = x 2 wff X

X

(n, m/2nN ) ∈ f x ! satisfied iff Q N

X

∃x, a ((x, a) ∈ f ∈ X →  R ∧ x = x ∧ m ≡ max {m | m/2Nn ≤ a}) true Z

X

Z

Q N

R

With that, we can introduce an equality relation on the sequences of real numbers, and can name a sub-class of the class of those sequences containing those that are Cauchy: α = β wff N→R

α = β satisfied iff α, β ∈ N → R ∧ ∀n (αn! = βn!) satisfied N→R

N

N

R

N

c

N → R term c

α ∈ N → R satisfied iff α ∈ N → R ∧ ∀ + ∃n ∀k1 , k2 > n∃m ∀l > m(| αk1 !(l) − αk2 !(l)| < ) true Q

N

N

N

N

N N

Q

N

N

Q

N

N

Q

The classical theorem stating that whether a sequence of real numbers is Cauchy is independent of representative, takes the following form here: C

THEOREM 26: MEMBERSHIP OF N → R IS INVARIANT W.R.T. N → R-EQUALITY c

c

∀α, β (α = β ⇒ α ∈ N → R ↔ β ∈ N → R) N→R

Again, the essence of the proof is just the classical proof, so I will not give it. Instead, we will proceed to treating limits of sequences of real numbers. If we wanted to introduce a class of the limits of a given sequence α, then we could transfer the classical criterion directly, by stipulating that a is an element of that class iff α ∈ N → R ∧ a ∈ R ∧ ∀ + ∃n ∀k > n∃r ∀m > r (| αk !(m) − a(m)| < ) is true. But Q N N N N N N Q N N Q N Q we need a limit that is a real number, not a class of real numbers. Therefore, we will instead introduce the limit like this: lim αn! term (binds the displayed occurrences of n) N

n
∞

(n, αn!(i)) ∈ lim αn! satisfied iff α ∈ N → R ∧ n ∈ N N

N

n
∞

N

∧ i ≡ min {i | ∀k1 , k2 > i (| αn!(k1 ) − αn!(k2 )| < 1/n)} true N

N

N

Q

N

N

Q

N

N

Q

Q

With this stipulation, we are shifting some things around compared to classical mathematics. It makes use of an idea that, in classical mathematics, does not enter into the definition of limit, but instead shows up in the subsequent existence

180

9.4 Completeness

theorem. Conversely, some of the material that in classical mathematics is to be found in the definition here appears instead in the formulation of the theorem that corresponds to that existence theorem. In this case, that existence theorem is the Completeness Theorem: THEOREM 27: COMPLETENESS THEOREM c

∀α (α ∈ N → R ⇒ lim αn! ∈ R ∧ ∀ + ∃n ∀k > n ∃m ∀l > m(| αk !(l) − lim αn!(l)| < )) N

n
∞

Q

N

N N

N

N N

Q

N

N

Q

N

n
∞

N

Q

c

Proof Assume (1) α ∈ N → R. That presupposes (2) that α is such that αn! ∈ R N for each canonical natural number n.4 For each canonical natural number n, it is a presupposition of (2) that: (3) there is a smallest canonical natural number i n such that ∀k1 , k2 > i n (| αn!(k1 )− αn!(k2 )| < 1/n).5 In part 1, below, the first conjunct N N Q N N Q N N Q Q of the consequent of the theorem will be proved; and in part 2, the second. Part 1: We prove ∀ + ∃n ∀k1 , k2 > n (| lim αn!(k1 ) − lim αn!(k2 )| < ). Let a Q N N N Q n
∞ N N Q n
∞ N N N positive canonical rational number  be given. Let n be the canonical maximum6 of (4a) the smallest canonical natural number n1 s.t. ∀k1 , k2 > n1 ∃m∀l > m(| αk1 !(l)− N N N N N Q N N Q αk2 !(l)| < /3) (existence is a presupposition of (1)) and (4b) the smallest canonN N Q Q ical natural number n2 s.t. 1/n2 < /3. Let canonical natural numbers k 1 and k 2 be Q Q Q given such that k 1 , k 2 > n. Let m be the canonical maximum of (5a) the smallest N canonical natural number m1 s.t. ∀l> m1 (| αk 1 !(l) − αk 2 !(l)| < /3) (existence N Q N N Q N N Q Q is a presupposition of (4a)), (5b) i k 1 and (5c) i k 2 . Let a canonical natural number l be given such that l > m. N Because of (4b) and (5b), it is a presupposition for (3) that | αk 1 !(i k 1 ) − Q N N Q αk 1 !(l)| < /3. It is a presupposition for (5a) that | αk 1 !(l) − αk 2 !(l)| < N N Q Q Q N N Q N N Q /3. Because of (4b) and (5c), it is a presupposition for (3) that | αk 2 !(l) − Q Q N N Q αk 2 !(i k2 )| < /3. This presupposes that the relevant terms are such that N

N

Q

Q

| αk 1 !(i k 1 ) − αk 2 !(i k2 )|

Q

N

N

Q

N

N

≤ | αk 1 !(i k 1 ) − αk 1 !(l)| + | αk 1 !(l) − αk 2 !(l)| + | αk 2 !(l) − αk 2 !(i k 2 )| Q Q

N

N

Q

N

N

Q Q

N

N

Q

N

N

Q Q

N

N

Q

N

N

< /3 + /3 + /3 =  . Q

4 This

Q

Q

Q

Q

Q

Q

kind of formulation was explained in the final paragraph of Sect. 9.2. 5 Because the stipulation for membership of R only quantifies over canonical rational numbers in the position to the right of the i n (| αn!(k1 ) − αn!(k2 ) | < p), where p is a canonical rational N N Q N N Q N N Q number such that p = 1/n”. (b) “| αn!(k 1 ) − αn!(k 2 ) | is such that it is smaller than 1/n for all Q Q Q N N Q N N Q canonical natural numbers k 1 and k 2 larger than n”. 6 That is, the canonical natural number that is equal to the maximum of what follows.

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181

By the stipulation for the limit, this grounds that | lim αn!(k 1 )−lim αn!(k 2 )| < . Q n
∞ N N Q n
∞ N N Q It is thus grounded that lim αn! ∈ R. n
∞ N Part 2: We prove ∀ + ∃n ∀k > n ∃m ∀l > m (| αk !(l) − lim αn!(l)| < ). Let a Q N N N N N N Q N N Q n
∞ N N Q positive canonical rational number  be given, and let n be defined as above. Let a canonical natural number k be given such that k > n. Let m be the canonical maxiN mum of (6a) the smallest canonical natural number m1 s.t. ∀l1 , l2 > m1 (| αk !(l1 ) − N Q N N Q αk !(l2 )| < /3) (existence is a presupposition of (2)) and (6b) n. Let a canonical N N Q Q natural number l be given such that l > m. Let (7) o be the smallest canonical natN ural number s.t. ∀j > o (| αk !(j ) − αl !(j )| < /3) (existence is a presupposition N N Q N N Q N N Q Q of (4a)). And let j be the canonical natural number such that j = max{m, o, i l }+ 1. N N N It is a presupposition for (6a) that | αk !(l) − αk !(j )| < /3. It is a presupQ N N Q N N Q Q position for (7) that | αk !(j ) − αl !(j )| < /3. Because of (4b) and (6b), it is Q N N Q N N Q Q a presupposition for (3) that | αl !(j ) − αl !(i l )| < 1/j < 1/n < /3. This Q N N Q N N Q Q Q Q Q Q presupposes that the relevant terms are such that | αk !(l) − αl !(i l )|

Q

N

N

Q

N

N

≤ | αk !(l) − αk !(j )| + | αk !(j ) − αl !(j )| + | αl !(j ) − αl !(i l )| Q Q

N

N

Q

N

N

Q Q

N

N

Q

N

N

Q Q

N

N

Q

N

N

< /3 + /3 + /3 =  . Q

Q

Q

Q

Q

Q

Q

This grounds that | αk !(l) − lim αn!(l)| ≤ .7 Q

N

N

Q

n
∞

N

N

Q

This is a very important result. One might think that, to avoid “gaps” in the number line, uncountably many numbers are needed. However, there are no gaps in this number system—that is, any convergent sequence of real numbers converges to some real number—and yet there are only countably many real numbers (because there are only countably many terms in total). Classically, the Completeness Theorem, which is an existence theorem, is accompanied by a uniqueness theorem. That theorem takes the following form in this setting: THEOREM 28: LIMITS ARE INVARIANT W.R.T. N → R-EQUALITY ∀α, β (α = β ⇒ lim αn! = lim βn!) N→R

9.5

n
∞

N

R

n
∞

N

Suprema, Infima, and Roots

In the introduction of suprema we will again “shift things around” compared to classical mathematics, in the way explained just before Theorem 27: 7 This

proof is fashioned after a classical proof kindly provided to me by Alain Genestier.

182

9.5 Suprema, Infima, and Roots

sup A term (n, m/2nN ) ∈ sup A satisfied iff Q N

n ∈ N ∧ A ⊂ R ∧ m ≡ max {m | ∃a (a ≥ m/2Nn )} true Z

Z

A

R

Q N

In words, the sequence sup A contains as its n’th element the largest fraction with denominator 2nN and integer numerator that is not larger than all the elements of the N class of real numbers A. That is, when A has an upper bound, the n’th element differs from the intended position in the number line by at most 1/2nN ; and if A has Q N no upper bound, then sup A is not a real number. This should make it clear that the following theorem, which corresponds to the classical existence theorem for suprema, holds: THEOREM 29: SUPREMUM PROPERTY ∀A, b (∃a (a ∈ A) ∧ ∀x (x ∈ A ∨ x ∈ / A) ∧ ∀a (a ≤ b) ⇒ b ≥ sup A) R

A

R

It is important to note that the consequent implies that sup A is a real number. So, the theorem says that any determinate non-empty class of real numbers with an upper bound has a supremum. The requirement that the class be determinate could be weakened slightly. For instance, a class A such that 0 ∈ A is true, 1 ∈ A is undefined, and x ∈ A is false for all other terms also has a supremum (i.e., sup A is a real number with the intended properties). From Theorem 27 it follows that bounded sequences of real numbers have upper bounds; and there, it is similarly a requirement that the sequence is “almost-determinate”. Thus, unlike in Weyl’s predicativism, taking a supremum over an arbitrary class and taking a supremum over a sequence work similarly—at least in theory. In praxis, it does not make much of a difference either to the strength of the theorems we can prove or to the amount of classical mathematics that can be reconstructed. This is because sub-classes of R defined using comprehension will typically be indeterminate, while determinate sequences of real numbers are much easier to come by. The real advantage of this system over Weyl’s is different, as explained in Sects. 4.4 and 10.1. Obviously, the dual case of infima works similarly: inf A term (n, m/2nN ) ∈ inf A satisfied iff Q N

n ∈ N ∧ A ⊂ R ∧ m ≡ min {m | ∃a (a ≤ m/2Nn )} true Z

Z

A

R

Q N

9

Real Analysis

183

THEOREM 30: INFIMUM PROPERTY ∀A, b (∃a (a ∈ A) ∧ ∀x (x ∈ A ∨ x ∈ / A) ∧ ∀a (a ≥ b) ⇒ b ≤ inf A) R

A

R

Suprema offer us the means to introduce roots in the general and elegant way previously promised. We will restrict ourselves to natural number degrees, and adopt the usual preference for positive results when possible. And we will stay within the real numbers rather than introduce complex numbers. √ n

a term √ w ∈ n a satisfied iff n > 0 ∧ a ∈ R ∧ w ∈ sup {p | pNn ≤ a} true NR

NR

N

Q

Q

R

To see that this works as intended, three cases have to be considered separately. In the case of an even degree and a non-negative radicand, the class over which the supremum is taken is a bounded interval centered around 0, so the supremum will pick out the non-negative root. If the degree is even but the radicand is negative, the class will be empty, so the supremum will be, too. And, in the case of an odd degree, the class is an interval that is unbounded below and bounded above, no matter the sign of the radicand; so the supremum will give the unique root. As part of the motivation for a move we will make in the next section, it is instructive to note that because of the indeterminacy of R, we could not have used sup {b | bnN ≤ a} instead of sup {p | p Nn ≤ a} in this stipulation. To see why, recall the R R R Q Q R completely indeterminate class I from Sect. 8.6, and let us have a look at the exam√ ple 2 3. The sentence I ∈ R ∧ I2N ≤ 3 is undefined, so therefore this sentence is too, NR R R for all canonical natural numbers n and canonical integers m: I ∈ {b | b2N ≤ 3} ∧ R R R I ≥ m/2nN . Therefore, it would not be have been true to say of any term that it is an R Q N √ element of 2 3. NR Fortunately, we can use Q instead of R, as it combines two important features: it is a determinate class, and it is dense in R. This is essential to the present reconstruction of real analysis.

9.6

Continuity

Consider a theorem stating that all functions from the real numbers to the real numbers that have some property also have some other property. (Example: the classical Intermediate Value Theorem says that all such functions that have the property of being continuous also have the Intermediate Value Property.) If the property in, say, the antecedent is formulated using universal quantification over the real numbers

184

9.6 Continuity

(as continuity is), then we have a problem. Setting aside certain extremely trivial examples of such properties, I and similar classes will prevent intuitively true antecedents from coming out true. This is due to the fact that we cannot use ⇒ in the antecedent when the wide-scope conditional is also formulated using ⇒; and the fact that we do not have a determinate class of canonical real numbers. Therefore, we must make do with quantification over the rational numbers. In the strategy for overcoming the challenges this creates the most important element is that we will focus on functions that are in a primary sense functions from Q (or a subclass thereof) to R, and only in a secondary sense functions from R (or a subclass thereof containing irrational numbers) to R. The simplest approach to specifying such functions is to first specify the function values for rational values of the independent variable, and then to let the function value (in the secondary sense) of an irrational value be the limit of the function values for the independent variable approaching that irrational value. We will restrict ourselves to this simple approach for the time being, and postpone generalizations of the idea until Sect. 9.11. Above, we introduced limits of sequences, i.e., limits of functions with the natural numbers as their domain, but not limits of functions that have rational or real numbers as their domain. To serve the purpose just described, we allow the limit point to be a real number (a below); and to avoid the problems just described, we only allow the function to have rational numbers in its domain: lim f q ! term (binds the displayed occurrences of q) Q

q
a

 R∧m≡ (n, m/2nN ) ∈ lim f q ! satisfied iff n ∈ N ∧ a ∈ R ∧ f ∈ Q → Q N

Q

q
a

min {m | ∃δ+ ∀q (f q ! ∈ R ∧ 0 < | q − a | < δ → | f q ! − m/2Nn | < 1/2nN )} Z

Z

Q

Q

Q

R R

R

R

R

Q

R

Q N

R

Q N

true Before proceeding, we prove a theorem that states that a limit has the property that, in classical mathematics, would be the defining property: THEOREM 31: LIMITS BEHAVE AS EXPECTED ∀a, f (lim f q ! ∈ N → Q ⇒ lim f q ! ∈ R Q

q
a

q
a

Q

∧ ∀ + ∃δ+ ∀q (f q ! ∈ R ∧ 0 < | q − a | < δ → | lim f q ! − f q !| < ) Q

Q

Q

Q

R R

R

R

R q
a

Q

R

Q

R

Proof Assume lim f q ! ∈ N→Q. That presupposes that for each canonical natural q
a Q number n, min {m | ∃δ+ ∀q (f q ! ∈ R ∧ 0 < | q − a | < δ → | f q ! − m/2Nn | < 1/2nN )} Z

Z

Q

Q

Q

R R

R

R

R

Q

R

Q N

R

Q N

9

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185

is equal to some canonical integer. That again presupposes that (1) there is a smallest (up to equality) canonical integer mn for which the defining condition of the class is true, while it is false for all smaller canonical integers; let l n be one of them. Part 1: We first show that it is a presupposition that there are canonical rational numbers q arbitrarily close to a such that f q ! is a real number.8 For this purQ pose, let a positive canonical rational number δ be given. The falsity of the defining condition for l n (for any n) presupposes the falsity of ∀q (f q ! ∈ R ∧ 0 < | q − a | < δ → | f q ! − l n /2nN | < 1/2nN ) . Q

Q

R R

R

R

R

Q

R

Q N

R

Q N

That falsity presupposes a false instance: f q ! ∈ R ∧ 0 < | q − a | < δ → | f q ! − l n /2nN | < 1/2nN Q

R R

R

R

R

Q

R

Q N

R

Q N

That presupposes the truth of the antecedent: f q ! ∈ R ∧ 0 < | q − a | < δ Q

R R

R

R

Part 2: We then prove the first conjunct of the consequent of the theorem; i.e., we need to ground ∀ + ∃n ∀k1 , k2 > n (| lim f q !(k1 ) − lim f q !(k2 )| < ) . Q

N

N

N

Q q
a

Q

N

Q

Q

q
a

N

Q

Let a positive canonical rational number  be given. Let n be the smallest canonical natural number s.t. 2/2nN < . Let canonical natural numbers k 1 and k 2 larger than Q N Q n be given. Let δ 1 be a positive canonical rational number such that ∀q (f q ! ∈ R ∧ 0 < | q − a | < δ 1 → | f q ! − mk 1 /2Nk1 | < 1/2kN 1 ) , Q

Q

R R

R

R

R

Q

R

Q N

R

Q N

and similarly for δ 2 (existence is a presupposition of (1)). Let δ be the smaller of the two (or, if they are equal, let δ be δ 1 ). Let q be a rational number that satisfies f q ! ∈ R ∧ 0 < | q − a | < δ, cf. part 1. The truth of these sentences presupposes Q R R R R that the relevant terms are such that | lim f q !(k 1 ) − lim f q !(k 2 )| = | mk 1 /2kN 1 − mk 2 /2kN 2 |

Q q
a

Q

N

Q

q
a

Q

N

Q Q

Q N

Q

Q N

≤ | f q ! − mk 1 /2kN 1 | + | f q ! − mk 2 /2kN 2 | < 1/2kN 1 + 1/2kN 2 ≤ 2/2nN <  , Q Q

Q

Q

Q N

Q Q

Q

Q

Q N

Q

Q N

Q

Q N

Q

Q N

Q

which grounds the target sentence. Part 3: Grounding the second conjunct is simple. I leave that to the reader. 8 The expression “f  q ! is a real number” corresponds to “q is in the domain of f ” in classical Q mathematics.

186

9.6 Continuity

Using this concept of limits, we can specify the secondary sense in which a function defined only on rational numbers can be applied to irrational numbers. We introduce yet another notion of functional application, this one with the notation “f a ”. If a is a rational real q, f a is made equal to f q !; and if a is irrational, Q f a is made equal to the limit of f for the independent variable approaching a, if this limit is a real number. The formal stipulation looks like this: f a term w ∈ f a satisfied iff f ∈Q→  R ∧ (∃q (a = q ∧ w ∈ f q !) ∨ (a ∈ R \ Q ∧ w ∈ lim f q !)) Q

R

Q

R

q
a

Q

true

√  With this stipulation, sentences such as λp.p 2N 2 3 = 3 come out true; and this Q Q NR R type of functional application gives us, in effect, what might be called the continuous completification of the function in question. And continuity is what we will now focus on. In particular, we will introduce three different notions of continuity. The first is the simple, local type of continuity, normally named “continuity” simpliciter. We will use the notation Cs @p for a class of those functions that are continuous at the rational number p, and the notation Cs (P ) for a class of the functions that are continuous on the class of rational numbers P : Cs @p term f ∈ Cs @p satisfied iff f ∈ Q →  R ∧ f p ! ∈ R Q

∧ ∀ + ∃δ+ ∀q (f q ! ∈ R ∧ | p − q | < δ → | f p ! − f q !| < ) true Q

Q

Q

Q

Q

Q

Q

R

Q

R

Q

R

Cs (P ) term f ∈ Cs (P ) satisfied iff P ⊂ Q ∧ ∀p (f ∈ Cs @p) true P

That the square function is continuous can thus be stated, truly, with the sentence λp.p N2 ∈ Cs (Q). However, because we only check the rational numbers, this notion Q Q of continuity is weak—too weak, in fact, to capture the intuitive idea of continuity, √ as this example shows: λp .1/(p − 2 3) ∈ Cs (Q). This is a true sentence, expressing Q R√ R NR that the function λp .1/(p − 2 3) is continuous on all of Q in spite of the fact that Q R R NR √ the function value goes towards plus infinity for p approaching 2 3 from the left, NR √ and towards minus infinity for p approaching 2 3 from the right (see Fig. 9.1). In NR classical mathematics, this has an impact on what can be said about the continuity properties of that function (i.e., the classical function with the same formula and

9

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187

√ 2 3 NR

Fig. 9.1 Graph of λp .1/(p Q

R

− R

√ 2 3) NR

the largest possible subset of R as its domain): it is only continuous on the set √ R \ { 3}, not all of R. We would like to be able to capture the same phenomenon. However, as quantification over the real numbers creates problems in the current setting, as explained at the end of the previous section, we must take another route. The first step along that route is to introduce the second of the three notions of continuity: uniform continuity. As uniform continuity is not defined point-wise like simple continuity is, this only requires one stipulation: Cu (P ) term f ∈ Cu (P ) satisfied iff f ∈ Q →R  ∧ P ⊂ Q ∧ ∀ +∃δ+∀p, q (| p − q | < δ → | f p !− f q !| < ) Q

Q

P

Q

Q

Q

R

Q

R

Q

R

true Uniform continuity will prove very useful to us, because it is, on the one hand, stronger than simple continuity and not fooled by discontinuities at irrational values; and on the other, it is classically equivalent to simple continuity in the case of closed, bounded intervals,9 which for many applications is what is relevant (think for instance of the Intermediate Value Theorem). However, if the goal were to come up with a stronger notion of continuity that √ could give us a characterization of λp .1/(p − 2 3) reflecting the intuitive verdict Q R R NR √ that this function is continuous everywhere except at 2 3 (like classical simple conNR √ tinuity), then we have now overshot our target. For it is false that λp .1/(p − 2 3) is Q

9 See

Sprecher (1970, Theorem 27.4).

R

R

NR

188

9.6 Continuity

√ uniformly continuous on a class of the canonical rational numbers smaller than 2 3, NR and false that it is uniformly continuous on a class of the canonical rational numbers √ larger than 2 3. Uniform continuity is too strong for many purposes. NR Fortunately, there is a “compromise notion” of continuity. We shall call it “piecewise uniform continuity”. A function is piecewise uniformly continuous on a subclass P of the class of rational numbers if the function is uniformly continuous on every closed interval with rational endpoints that is included in P . Before we can formally introduce this notion, we must add some notation for intervals to our language. An interval will be a class of canonical rational numbers only, but of course, that does not prevent it from being characterized by endpoints that are real numbers: a, b!, !a, b, a, b, !a, b!, !−∞, b!, !−∞, b, a, ∞, !a, ∞, and I terms p ∈ a, b! satisfied iff p ∈ Q ∧ a ≤ p ≤ b true R

R

R

R

R

R

R

R

p ∈ !a, b satisfied iff p ∈ Q ∧ a < p < b true p ∈ a, b satisfied iff p ∈ Q ∧ a ≤ p < b true p ∈ !a, b! satisfied iff p ∈ Q ∧ a < p ≤ b true p ∈ !−∞, b! satisfied iff p ∈ Q ∧ p ≤ b true R

p ∈ !−∞, b satisfied iff p ∈ Q ∧ p < b true R

p ∈ a, ∞ satisfied iff p ∈ Q ∧ a ≤ p true R

p ∈ !a, ∞ satisfied iff p ∈ Q ∧ a < p true R

P ∈ I satisfied iff P ⊂ Q ∧ ∀p, q, r (p, r ∈ P ∧ p ≤ q ≤ r → q ∈ P ) true Q

Q

It is easy to see that the various interval notations really do give intervals. That is, when a and b are real numbers, a, b! ∈ I, and similarly for the other types of interval notation. And this lemma shows that intervals behave nicely: LEMMA 32: INTERVALS ARE DETERMINATE ∀P (P ∈ I ⇒ ∀x (x ∈ P ∨ x ∈ / P )) Proof Assume P ∈ I. That presupposes P ⊂ Q. Since Q is determinate, there are only two cases: elements of Q and non-elements of Q. P ⊂ Q presupposes that non-elements of Q are non-elements of P . Let s be a canonical rational number, and consider the instantiation of the second conjunct of the truth condition for P ∈ I in which p, q, and r are all replaced with s. That would be undefined if s ∈ P were undefined; ergo, it is not.

9

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189

We now have what we need to introduce a class of the functions that are piecewise uniformly continuous on a given class of rational numbers: Cp (P ) term f ∈ Cp (P ) satisfied iff P ⊂ Q ∧ ∀p, q (p, q ! ⊂ P → f ∈ Cu (p, q !)) true Q

Returning to our previous example of the function λp .1/(p − Q R R able to express what we intended: for the sentence λp.1/(p − Q

R

R

√ 2 NR

√ 2 NR

3), we are now

√ √ √ 2 2 2 3) ∈ Cp (!−∞, 3) ∧ λp.1/(p − 3) ∈ Cp (! 3, ∞) NR

Q

R

R

NR

NR

is true, while λp.1/(p − Q

R

R

√ 2 3) ∈ Cp (Q) NR

is false. Since simple continuity does not fare well when transferred from the classical setting to this one, this concept of continuity is a useful replacement, especially because it is closely connected with simple continuity in classical mathematics. To be precise, this theorem holds classically: for any classical interval I and real function f on it, f is classically piecewise uniformly continuous on I if and only if f is classically simply continuous on I , where classical piecewise uniform continuity is defined in the obvious way by quantification over real numbers instead of just rational numbers.10 In the next few sections, only continuity on closed, bounded intervals will be relevant to the theorems we will formulate and prove. We will therefore for simplicity’s sake, use uniform continuity. Piecewise uniform continuity will become relevant in Sect. 9.10. The first theorem to be proved that makes use of a notion of continuity is one stating that f b is “well-defined”: i.e., is a real number when f is uniformly continuous on the intersection of a neighborhood of b and the domain of f , and b is a limit point of that domain. We therefore first introduce a class of the limit points of the domain of f : 10 Classical proof: Let I and f be given and assume that f is piecewise uniformly continuous on I . Let x ∈ I be given. There is a closed, bounded interval I  such that x ∈ I  ⊆ I ; and from the assumption it follows that f is uniformly continuous on I  , from which it follows that f is simply continuous on I  and hence at x. By the arbitrariness of x, it follows that f is simply continuous on I . Now, assume instead that f is simply continuous on I . Let I  be a closed, bounded sub-interval of I . The assumption implies that f is simply continuous on I  , and from Sprecher’s (1970) Theorem 27.4, it follows that f is uniformly continuous on I  . By the arbitrariness of I  it follows that f is piecewise uniformly continuous on I .

190

9.6 Continuity

L(f ) term a ∈ L(f ) satisfied iff f ∈ Q →  R ∧ ∀δ+ ∃q (| a − q | < δ ∧ f q ! ∈ R) true Q

Q

R

R

R

Q

THEOREM 33: COMPLETIFIED FUNCTIONS WELL-DEFINED ON LIMIT POINTS OF UNIFORMLY CONTINUOUS FUNCTIONS

∀a, b, c, f (b ∈ R \ Q ∧ a < b < c R

R

R

∧ f ∈ Cu ({ p | f p ! ∈ R}) ∧ b ∈ L(f ) ⇒ f b ∈ R) a,c!

Q

Proof Consider the defining property of the class over which the minimum is taken in the stipulation for limits, on which the stipulation for the secondary notion of functional application depends. The assumption that b is a limit point ensures that there is no δ so small that all canonical integers m would vacuously satisfy the universally quantified formula. On the other hand, the assumption that f is uniformly continuous around b ensures that some m do satisfy the defining property. The class of those m must be bounded and hence have a minimum. This implies that the limit is a sequence of rational numbers, which by Theorem 31 implies that f b ∈ R. Next up is the Intermediate Value Theorem mentioned earlier. To recapitulate: in classical mathematics it is formulated using simple continuity, which would not work here; but it could just as well have been formulated using uniform continuity— as it is the continuity of a closed interval that is in question—and that works for us too. THEOREM 34: INTERMEDIATE VALUE THEOREM ∀f, a, b, d (a < b ∧ f ∈ Cu (a, b!) ∧ (f a < d < f b R

R

R

∨ f a > d > f b ) ⇒ ∃c (a < c < b ∧ f c = d)) R

R

R

R

R

Proof Assume a < b ∧ f ∈ Cu (a, b!) ∧ (f a < d < f b ∨ f a > d > R R R R R f b ). The last conjunct presupposes one of its disjuncts. Say that the former is true. Let c abbreviate sup { p | f p ! ≤ d)}. In part 1 we prove the first conjunct of R a,b! Q the consequent, and in part 2, the second. Part 1: The class { p | f p ! ≤ d)} is non-empty, as it contains real numbers that a,b! Q R are larger than a. That is grounded in presuppositions of f ∈ Cu (a, b!) and in the fact that f a is a real number, as seen by letting the of Theorem 31 be a positive canonical number smaller than d − f a . As the interval a, b! is bounded above R

9

Real Analysis

191

and determinate, it is grounded—cf. Theorem 29—that c is a real number. As the class contains real numbers larger than a, and is bounded above by real numbers that are smaller than b, it is grounded that a < c < b is true. R R Part 2: According to Theorem 33, f c is a real number. Let  ∈ Q+ be given. We need to prove that f c is within a distance of  to d. Let δ be given according to the truth condition for uniform continuity. Let p ∈ Q be smaller then c by a distance of less than δ/2, and such that f p ! is less than or equal to d. Similarly, let q ∈ Q Q Q be larger than c by a distance of less than δ/2, and such that f p ! is larger than d. Q Q All function values of values between p and q are within a distance of  of d. As  was arbitrary, this shows that f c = d. R

Another of the important continuity theorems is the Extreme Value Theorem, which we, for the same reason as with the Intermediate Value Theorem, will formulate using uniform continuity without thereby weakening it: THEOREM 35: EXTREME VALUE THEOREM ∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Cu (a, b!) R

⇒ ∃c (a ≤ c ≤ b ∧ f c = sup {f p ! | p ∈ a, b!})) R

R

R

Q

∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Cu (a, b!) R

⇒ ∃c (a ≤ c ≤ b ∧ f c = inf {f p ! | p ∈ a, b!})) R

R

R

Q

Proof I will not prove this in detail, but just mention which class does the job of c. In the case of the first sentence, either a does or sup { q | sup {f p ! | p ∈ a, q !} < sup {f p ! | p ∈ a, b!}} a,b!

Q

R

Q

does. For the second sentence, it is either a or sup { q | inf {f p ! | p ∈ a, q !} < inf {f p ! | p ∈ a, b!}} . a,b!

9.7

Q

R

Q

Operations on Functions

As preparation for the treatment of differentiation and integration, we introduce algebraic operations on functions that have the real numbers as their co-domain: f + g, f · g, f − g, f / g terms iff X →  R term X

→  R

X

→  R

X

→  R

X

→  R

(x, f x ! + gx !) ∈ f + g satisfied iff x ∈ X ∧ f x !, gx ! ∈ R true Q

R

Q

X

→  R

Q

Q

192

9.7 Operations on Functions

(x, f x ! · gx !) ∈ f · g satisfied iff x ∈ X ∧ f x !, gx ! ∈ R true Q

R

Q

X

→  R

Q

Q

(x, f x ! − gx !) ∈ f − g satisfied iff x ∈ X ∧ f x !, gx ! ∈ R true Q

R

Q

X

→  R

Q

Q

(x, f x !/gx !) ∈ f / g satisfied iff x ∈ X ∧ f x !, gx ! ∈ R ∧ gx ! = 0 Q

R

Q

X

→  R

Q

Q

Q

R

true This kind of composition is only appropriate when X is a class for which the primary sense of functional application is appropriate. If the domain and the functions that go into the algebraic operation are determinate, then the resulting function is as well: THEOREM 36: DETERMINACY OF COMBINED FUNCTIONS ∀f, g, X (∀x (x ∈ f ∨ x ∈ / f ) ∧ ∀x (x ∈ g ∨ x ∈ / g) ∧ ∀x (x ∈ X ∨ x ∈ / X) / f + g)) ⇒ ∀x (x ∈ f + g ∨ x ∈ X

→  R

X

→  R

∀f, g, X (∀x (x ∈ f ∨ x ∈ / f ) ∧ ∀x (x ∈ g ∨ x ∈ / g) ∧ ∀x (x ∈ X ∨ x ∈ / X) ⇒ ∀x (x ∈ f · g ∨ x ∈ / f · g)) X

→  R

X

→  R

∀f, g, X (∀x (x ∈ f ∨ x ∈ / f ) ∧ ∀x (x ∈ g ∨ x ∈ / g) ∧ ∀x (x ∈ X ∨ x ∈ / X) / f − g)) ⇒ ∀x (x ∈ f − g ∨ x ∈ X

→  R

X

→  R

∀f, g, X (∀x (x ∈ f ∨ x ∈ / f ) ∧ ∀x (x ∈ g ∨ x ∈ / g) ∧ ∀x (x ∈ X ∨ x ∈ / X) ⇒ ∀x (x ∈ f / g ∨ x ∈ / f / g)) X

→  R

X

→  R

Proof If f , g, and X are determinate, then x ∈ X ∧ f x !, gx ! ∈ R is true or Q Q false for all terms x, so (x, f x ! + gx !) ∈ f + g is true or false. If x is of a Q R Q X→  R different form than (x, f x ! + gx !), then x ∈ f + g is false. The argument is Q R Q X→  R of course the same for f · g, f − g, and f / g. X

→  R

X

→  R

X

→  R

Limits of the combined functions can be calculated in the obvious way, and the classical proof for the theorem stating as much transfers directly. It makes no difference that we are only quantifying over rational numbers, since they are dense in the real numbers. THEOREM 37: LIMITS OF COMBINED FUNCTIONS ∀f, g, a (a ∈ L(f + g) ∧ lim f q !, lim gq ! ∈ R →  R

Q

Q

q
a

q
a

Q

⇒ lim(f + g)q ! = lim f q ! + lim gq !) q
a

→  R

Q

Q

R

q
a

Q

R

Q

q
a

∀f, g, a (a ∈ L(f · g) ∧ lim f q !, lim gq ! ∈ R →  R

Q

Q

q
a

q
a

Q

⇒ lim(f · g)q ! = lim f q ! · lim gq !) q
a

→  R

Q

Q

R

q
a

Q

R q
a

Q

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Real Analysis

193

∀f, g, a (a ∈ L(f − g) ∧ lim f q !, lim gq ! ∈ R →  R

Q

Q

q
a

Q

q
a

⇒ lim(f − g)q ! = lim f q ! − lim gq !) q
a

→  R

Q

Q

R

Q

q
a

R

Q

q
a

∀f, g, a (a ∈ L(f / g) ∧ lim f q !, lim gq ! ∈ R ∧ lim gq ! = 0 →  R

Q

Q

q
a

Q

q
a

q
a

⇒ lim(f / g)q ! = lim f q !/ lim gq !) q
a

→  R

Q

Q

R

Q

q
a

R q
a

Q

R

Q

It is also obvious (at least if one knows the corresponding classical theorem) that simple continuity is preserved under these operations: THEOREM 38: SIMPLE CONTINUITY PRESERVED UNDER ALGEBRAIC OPERATIONS

∀f, g, p (f, g ∈ Cs @p ⇒ f + g ∈ Cs @p) →  R

Q

∀f, g, p (f, g ∈ Cs @p ⇒ f · g ∈ Cs @p) →  R

Q

∀f, g, p (f, g ∈ Cs @p ⇒ f − g ∈ Cs @p) →  R

Q

∀f, g, p (f, g ∈ Cs @p ∧ gp ! = 0 ⇒ f / g ∈ Cs @p) Q

R

→  R

Q

In addition to being obvious, this theorem is also uninteresting, since simple continuity is not a very useful concept—uninteresting, that is, except when comparing antecedents with the corresponding theorems for those types of continuity that are useful. For addition and subtraction there is no difference: uniform continuity and piecewise uniform continuity are preserved given any class of rational numbers as the domain, just as in the case of simple continuity. But this is not the case for uniform continuity and multiplication and division. A simple counterexample is the identity function, which is uniformly continuous on Q but produces a non-uniformly continuous function when multiplied by itself. However, the extra assumption needed for preservation to go through is that the domain is a closed, bounded interval, which means that the need for the assumption disappears again when we come to piecewise uniform continuity. For simple continuity, the extra assumption needed in the case of division is just that the divisor function does not assume the value zero on any of the rational numbers in the domain. The weakness of this assumption is what accounts for the failure of the concept of simple continuity to give the intuitively correct verdict in √ the previously mentioned example: λp.1/(p − 2 3). For uniform continuity, the Q R R NR assumption is instead that the values do not come arbitrarily close to zero: a terribly strong assumption. Think for example of a function that is unbounded above and approaches zero for the independent variable tending to infinity. Again piecewise uniform continuity is the golden mean. The requirement here is that for any closed, bounded interval with rational endpoints contained in the domain, the values do

194

9.7 Operations on Functions

not come arbitrarily close to zero. That requirement is of just the right strength, because if they do come arbitrarily close to zero in some such interval, then there is a real number “in” that interval at which the function assumes the value zero, cf. the Extreme Value Theorem and the Intermediate Value Theorem. All this is made precise in the next two theorems.

THEOREM 39: UNIFORM CONTINUITY PRESERVED UNDER ALGEBRAIC OPERATIONS

∀f, g, P (f, g ∈ Cu (P ) ⇒ f + g ∈ Cu (P )) →  R

Q

∀f, g, a, b (f, g ∈ Cu (a, b!) ⇒ f · g ∈ Cu (a, b!)) →  R

Q

∀f, g, P (f, g ∈ Cu (P ) ⇒ f − g ∈ Cu (P )) →  R

Q

∀f, g, a, b (f, g ∈ Cu (a, b!) ∧ inf {| gp !| | p ∈ a, b!} = 0 R

⇒ f / g ∈ Cu (a, b!))

Q

R

→  R

Q

Proof We first prove the sentence concerning addition. Assume f , g ∈ Cu (P ). Let  ∈ Q+ be given. Let δ f be a positive canonical rational number such that ∀p, q (| p − q | < δ f → | f p ! − f q !| < /2), and let δ g be given similarly. Let P Q Q Q R Q R Q R Q δ be the canonical minimum of δ f and δ g , and p and q be elements of P such that | p − q | < δ. It is then simple to show that | (f + g)p ! − (f + g)q !| < :

Q

Q

Q

→  R

R

Q

Q

→  R

R

Q

Q

R

| (f + g)p ! − (f + g)q !| = | (f p! + gp !) − (f q ! + gq !)|

R

→  R

Q

Q

→  R

R

Q

Q

R R

Q

R

Q

R

Q

R

Q

= | (f p! − f q !) + (gp ! − gq !)| R R

Q

R

Q

R

Q

R

Q

≤ | f p ! − f q !| + | gp ! − gq !| R R

Q

R

Q

R R

Q

R

Q

< /2 + /2 R

Q

R

Q

= R

The sentence for subtraction is proved similarly. We then turn to the sentence about multiplication, so assume f , g ∈ Cu (a, b!). Let  ∈ Q+ be given. It follows from the Extreme Value Theorem that f and g are bounded, i.e., there are positive rational numbers R f and R g such that ∀ p a,b! (| f p !| < R f ) and ∀ p (| gp !| < R g ). Let δ f and δ g be positive rational numbers R Q R a,b! R Q R such that ∀p, q (| p − q | < δ f → | f p ! − f q !| < /(2 · R g )) a,b!

Q

Q

Q

R

Q

R

Q

R

Q

Q

9

Real Analysis

195

and ∀p, q (| p − q | < δ g → | gp ! − gq !| < /(2 · R f )) . a,b!

Q

Q

Q

R

Q

R

Q

R

Q

Q

Let δ be defined, and p and q given, the same way as above. We then have the inequality we need: | (f · g)p ! − (f · g)q !| →  R

R

Q

Q

→  R

R

Q

Q

= | f p ! · gp ! − f q ! · gq !| R R

Q

R

Q

R

Q

R

Q

= | f p ! · gp ! − f p ! · gq ! + f p ! · gq ! − f q ! · gq !| R R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

≤ | f p !| · | gp ! − gq !| + | gq !| · | f p! − f q !| R R

Q

RR

Q

R

Q

R R

Q

RR

Q

R

Q

< R f · /(2 · R f ) + R g · /(2 · R g ) R

R

Q

Q

R

R

Q

Q

= R

We can deal with division by first proving that the reciprocal to g is uniformly continuous on a, b!, and then applying the sentence just proved. Proving the former can be done by rewriting | 1/gp ! − 1/gq !| as | 1/(gp ! · gq !)| · | gp ! − gq !|, R R Q R R Q R R Q R Q RR Q R Q and noting that the assumption inf {| gp !| | p ∈ a, b!} = 0 implies that there is an R Q R upper bound to | 1/(gp ! · gq !)| over a, b!. R

R

Q

R

Q

THEOREM 40: PIECEWISE UNIFORM CONTINUITY PRESERVED UNDER ALGEBRAIC OPERATIONS

∀f, g, P (f, g ∈ Cp (P ) ⇒ f + g ∈ Cp (P )) →  R

Q

∀f, g, P (f, g ∈ Cp (P ) ⇒ f · g ∈ Cp (P )) →  R

Q

∀f, g, P (f, g ∈ Cp (P ) ⇒ f − g ∈ Cp (P )) Q

→  R

∀f, g, P (f, g ∈ Cp (P ) ∧ ∀q, r (q, r ! ⊂ P → inf {| gp !| | p ∈ q, r !} = 0) ⇒ f / g ∈ Cp (P ))

Q

R

Q

R

→  R

Q

Proof This follows easily from the previous theorem. In preparation for the treatment of differentiation, we also introduce composition of functions, and prove that composition preserves uniform continuity and piecewise uniform continuity under suitable assumptions: g ◦ f term →  R

Q

(p, gf p ! ) ∈ g ◦ f satisfied iff p ∈ Q ∧ gf p ! ∈ R true Q

→  R

Q

Q

196

9.7 Operations on Functions

THEOREM 41: UNIFORM CONTINUITY PRESERVED UNDER COMPOSITION ∀f, g, a, b, c, d (f ∈ Cu (a, b!) ∧ g ∈ Cu (c, d !) ∧ ∀ p (c < f p ! < d) a,b!

⇒ g ◦ f ∈ Cu (a, b!)) Q

R

Q

R

→  R

Proof Let  ∈ Q+ be given. Let δ g ∈ Q+ be such that ∀p, q (| p − q | < δ g → c,d ! Q Q Q | gp ! − gq !| < ). Let δ f ∈ Q+ be such that ∀p, q (| p − q | < δ f → | f p ! − R Q R Q R a,b! Q Q Q R Q R f q !| < δ g ). Let p, q ∈ a, b! such that | p − q | < δ f be given. Then, | f p ! − Q R Q Q Q R Q R f q !| < δ g . From the third conjunct of the antecedent it follows that both f p! Q R Q and f q ! lie between c and d. Hence, since g is uniformly continuous, it follows Q that | gf p ! − gf q ! | < . R

Q

R

Q

R

THEOREM 42: PIECEWISE UNIFORM CONTINUITY PRESERVED UNDER COMPOSITION

∀f, g, P , Q(P , Q ∈ I ∧ f ∈ Cp (P ) ∧ g ∈ Cp (Q) ∧ ∃q, r ∀p (q < f p ! < r) Q

⇒ g ◦ f ∈ Cp (P ))

P

R

Q

R

→  R

Q

Proof Let p and s be elements of P . Then f is uniformly continuous on p, s !. It follows from the last conjunct of the antecedent that there are q, r ∈ Q such that f applied to any rational number in p, s ! gives a value between q and r. Since g is uniformly continuous on q, r !, it follows from the previous theorem that g ◦ f ∈ Cu (p, s !). We can conclude that g ◦ f ∈ Cp (P ). →  R

→  R

Q

Q

We conclude this section with a stipulation that will help to simplify notation. The stipulation for limits at the beginning of Sect. 9.6 takes a function as its “input”. In practice we are often interested in the limit of something that formally is an expression rather than a function. For instance, we say that the limit of q + 2 for Q q approaching 0 is 2, so we would like to write lim(q + 2) = 2. With the current q
0 Q R stipulation, however, we have to convert it to a function and write the cumbersome lim(λq .q + 2)q ! = 2. This next stipulation allows us to write the former: q
0

Q

Q

Q

R

lim x term (binds q in x and the displayed q) q
a

w ∈ lim x satisfied iff w ∈ lim(λ q .x)q ! true q
a

q
a

{q |x∈R} Q

Q

9

Real Analysis

9.8

197

Differentiation

We now have the necessary pieces in place for the culmination of our development of mathematical analysis, namely differentiation (this section) and integration (next section). The differential quotient is introduced using the familiar formula, but with the independent variable restricted to rational numbers: f  term  R ∧ p ∈ Q ∧ f p ! ∈ R (p, a) ∈ f  satisfied iff f ∈ Q → ∧ a ≡ lim((f p ! − f q !)/(p − q)) ∈ R true Q

q
p

R

Q

R

Q

Q

This section comprises three subsections. The first contains the familiar theorems about how to calculate derivatives when functions are combined. The middle subsection is about the specific challenges that arise here because we do not have all the resources of classical mathematics available. The last contains other well-known theorems: Rolle’s, the Mean Value Theorem, and the theorem that connects the sign of the derivative with monotonicity.

9.8.1

Calculating Derivatives

Because of the division between the primary and secondary senses of functional application, the familiar theorem about the derivatives of combined functions comes in two variants. We first state a lemma concerned with derivative functions applied in the primary sense. Then, a theorem with the “same” formulas, but for the secondary sense of application, is derived therefrom. The question about how to interpret a term of the form f  a —whether it can be “trusted”—is postponed to the next subsection. LEMMA 43: DIFFERENTIATION OF COMBINED FUNCTIONS ∀f, g, p (p ∈ L(f + g) ∧ f  p !, g  p ! ∈ R →  R

Q

Q

Q

⇒ (f + g) p ! = f  p ! + g  p !) →  R

Q

Q

R

Q

R

Q

∀f, g, p (p ∈ L(f · g) ∧ f  p !, g  p ! ∈ R →  R

Q

Q

Q

⇒ (f · g) p ! = f  p ! · gp ! + f p ! · g  p !) →  R

Q

Q

R

Q

R

Q

R

Q

∀f, g, p (p ∈ L(f − g) ∧ f  p !, g  p ! ∈ R →  R

Q

Q

R

Q

Q

⇒ (f − g) p ! = f  p ! − g  p !) →  R

Q

Q

R

Q

R

Q

∀f, g, p (p ∈ L(f / g) ∧ f  p !, g  p ! ∈ R ∧ gp ! = 0 →  R

Q

Q

Q

Q

R

Q

R

⇒ (f / g) p ! = (gp ! · f  p ! − g  p ! · f p !)/(gp !)2N ) →  R

Q

Q

R

Q

R

Q

R

Q

R

Q

R

198

9.8 Differentiation

Proof Like the classical theorem.

THEOREM 44: DIFFERENTIATION OF COMBINED FUNCTIONS ∀f, g, a (a ∈ L(f + g) ∧ f  a , g  a ∈ R →  R

Q

⇒ (f + g) a = f  a + g  a ) →  R

Q

R

R

∀f, g, a (a ∈ L(f · g) ∧ f  a , g  a ∈ R →  R

Q

⇒ (f · g) a = f  a · ga + f a · g  a ) →  R

Q

R

R

R

∀f, g, a (a ∈ L(f − g) ∧ f  a , g  a ∈ R

R

→  R

Q

⇒ (f − g) a = f  a − g  a ) →  R

Q

R

R

∀f, g, a (a ∈ L(f / g) ∧ f  a , g  a ∈ R ∧ ga = 0 →  R

Q

R

⇒ (f / g) a = (ga · f  a − g  a · f a )/(ga )2N ) →  R

Q

R

R

R

R

R

R

Proof I only prove the first sentence. If a is a rational real, this sentence follows directly from the lemma and from the stipulation of the secondary sense of functional application. If a is irrational, the antecedent of the first sentence presupposes that the relevant terms are such that (f + g) a = lim(f + g) q ! = lim(f  + g  )q ! →  R

Q

R

→  R

Q

q
a

Q

R

→  R

Q

q
a

Q

= lim f  q ! + lim g  q ! = f  a + g  a . R

Q

q
a

R

q
a

Q

R

R

The first and last equality follow from the stipulation of the secondary sense of functional application, while the second equality follows from Lemma 43, and the third from Theorem 37. The splitting of a classical theorem into a lemma that can be proved essentially like the classical theorem, and a theorem that follows from it, is repeated in the case of the Chain Rule: LEMMA 45: CHAIN RULE ∀f, g, p (f  p !, g  f p ! ∈ R ⇒ (g ◦ f ) p ! = g  f p ! · f  p !) Q

→  R

Q

Q

Q

R

Q

R

Q

THEOREM 46: CHAIN RULE ∀f, g, a (f  a , g  f a

∈ R ⇒ (g ◦ f ) a = g  f a →  R

Q

R

· f  a )

R

9

Real Analysis

199

Proof This is proved from Lemma 45 in much the same way that Theorem 44 was proved from Lemma 43. In this case, the relevant string of equalities is as follows: (g ◦ f ) a = lim(g ◦ f ) q ! = lim g  f q ! · f  q ! →  R

Q

9.8.2

R

q
a

R

q
a

→  R

Q

Q

R

Q

q
a

R

= lim g  f q ! · lim f  q ! = g  f a Q

R q
a

Q

R

Q

· f  a

R

Uniform Differentiability

Our approach to reconstructing analysis without recourse to a classically uncountable set of numbers has been to build real functions on a skeleton of rational numbers, and to let the function values of irrational numbers be determined indirectly, much as the position of the webbing between an aquatic bird’s toes is determined indirectly by the position of the toes. When we do that for derivative functions, it raises the question of whether the webbing can be trusted. The value of f  a is only useful if it actually represents the derivative at a, i.e., the limit of (f a − f q !)/(a − q). Prima facie it is just the limit of the derivative for rational R Q R R values, and these two can come apart. To get a sense of the challenge, we will first have a look at two examples from classical mathematics where they do come apart. The standard example of a continuous function with a discontinuous derivative is f : R → R given by ⎧   ⎨x 2 sin 1 if x = 0 x f (x) = ⎩0 if x = 0 , with derivative ⎧     ⎨2x sin 1 − cos 1 if x =  0 x x f  (x) = ⎩0 if x = 0 . Their graphs are shown in Fig. 9.2. If we consider a Q→R-function g in our system that is like f except that everything is shifted sideways so that the discontinuity √ √ lands on an irrational number, say 2 3, then we have a problem: g   2 3 is not a real NR NR √ √ √ number, but the limit of g 2 3 − gq !)/( 2 3 − q) for q approaching 2 3 is 0. R Q R NR R NR NR The second example is Minkowski’s Question Mark Function on the unit interval. For rational numbers, the function can be defined recursively. The base case of the recursion is stage 1, where ?( 01 ) = 01 and ?( 11 ) = 11 are stipulated. At stage n for n > 1, all irreducible fractions with denominator n are assigned a function value.

200

9.8 Differentiation

Fig. 9.2 Graph of f (top) and f  (bottom). One the one hand, f is squeezed in between ±x 2 , forc-

ing differentiablity at 0, and on the other, the graph crosses the x-axis with a slope of approximately ±1 within any neighborhood of 0, resulting in f  being discontinuous.

The function value of m n is the average of (1) the function values of the largest m− fraction n− that is smaller than m n and has been assigned a function value at a previous stage, and (2) and the smallest fraction also already been assigned a function value:

?

m n

 =

?

m− n−



+? 2

m+ n+

that is larger than





m+ n+

m n

and has

9

Real Analysis

201

1 4

1 3

1 2

2 3

3 4

1 1

Fig. 9.3 Graph of Minkowski’s Question Mark Function

So stages 2, 3, and 4 see these function values added: ? 1 ? = 2     ? 0 +? 1 1 2 1 1 = = ? 3 2 4     ? 0 +? 1 1 3 1 1 = ? = 4 2 8

  0 1

+? 2

  1 1

=

1 2

    ? 1 +? 1 2 1 2 3 ? = = 3 2 4     ? 2 +? 1 3 1 7 3 = ? = 4 2 8

Obviously, all rational numbers are assigned a function value in this process. The resulting function on Q ∩ 0, 1! is continuous, so the function values of irrational numbers can then be added as the limits of the function values for rational numbers.11 The function’s graph, which is shown in Fig. 9.3, has a fractal structure. The flattening out that happens around 12 is repeated on a smaller scale around 13 and 23 , and on an even smaller scale around 14 and 34 , etc., ad infinitum. This has the astonishing effect that the derivative for any rational point is 0, even though the function is monotonically increasing.12 But, of course, the derivative is not 0 for all values: it is 11 The function was first described by Minkowski (1904). However, the recursive definition is due to Denjoy (1938). 12 To prove this, we need the following fact: In each case, m can be computed as the “freshman n −

+

−

+

sum” of its two “Farey parents”, m and m : m = mn− +m . Let an irreducible fraction m  n ∈ !0, 1 n− n+ n +n+ m be given. We prove that the right-derivative of n is 0; the proof for the left-derivative is similar. Let

202

9.8 Differentiation

in fact 0 for some irrational values but undefined for others.13 This means that for all irrational values a we have ? a = 0 (for ? suitably stipulated in our system), but it R is not necessarily the case that the limit of (?a − ?q !)/(a − q) for q approaching R Q R R a is 0. The upshot of these examples is that it is both possible for f  a to fail to have a value when it “ought to” and, conversely, have a value when it “ought not to”. That is, it can be untrustworthy in two different ways. And even though we have the derivative for any given irrational number a represented by the term lim((f a − f q !)/(a − q)) , R

q
a

Q

R

R

we need to be able to trust f  a so that the answers to questions such as “Is f  continuous?” and “What is f  ?” will not to be misleading. We will do two things to handle this challenge. First, we will identify a condition under which f  a can be trusted. That is the purpose of the rest of this section. The condition will be used in the antecedents of several of the following theorems about differentiation and integration. It is weak enough to allow us to state and prove a version of the Fundamental Theorem of Calculus that is as general as the version usually found in textbooks of classical mathematics. Second, in Sect. 9.11, we will consider changes that can be made to the way real functions are handled, so that f  a can be trusted even if the condition fails to obtain. The solution to this problem, in the form of an extra condition, is analogous to the solution to the problem concerning simple continuity we discovered in Sect. 9.6: go uniform! If f is differentiable at p then ∀ + ∃δ+ ∀q (0 < | p − q | < δ → | (f p ! − f q !)/(p − q) − f  p!| < ) Q

mk nk

Q

P

Q Q

Q

Q

R

Q

R

Q

be the k’th fraction that is added as a Farey child of

iteration. Let

m0 n0

be the Farey parent of

m n

some constant K. We also have that | m n −

m n

R

=

1 n·nk ;

R

Q

R

m n itself is added in m 1 k N0 , |?( n )−?( m nk )| = 2K+k

to its right after

to its right. For all k ∈ mk nk |

Q

the for

this is a general fact about Farey neighbors

1 which is easy to prove from the above fact by induction when | 01 − 11 | = 1·1 is used as the base m+mk−1 mk case. From that fact, we also know that nk = n+nk−1 , so it follows by induction that nk = kn+n0 . mk mk−1 m1 Let x ∈ ! m n , n1 ! be given. There is a k ∈ N such that x ∈ ! nk , nk−1 !. This k goes towards ∞ as m x approaches n from the right. Thus   mk−1   m   m 1  ?( n )−?(x)  ?( n )−?( nk−1 ) kn2 + n · n0 2K+k−1  ≤   = = →0 m   1 m mk  − x 2K+k−1 n  n − nk  n·(kn+n0 ) + for x → m n . 13 See Paradís et al. (2001).

9

Real Analysis

203

(where P is the domain of f ). The δ is allowed to depend on p, just as it is in the case of simple continuity. Demanding that the δ works for all p is what defines uniform continuity. Adding the same demand here gives us uniform differentiability: Du (P ) term  R∧P ⊂Q f ∈ Du (P ) satisfied iff f ∈ Q → ∧ ∀ +∃δ+∀p, q (0 < | p− q | < δ → | (f p !− f q !)/(p− q)− f  p !| < ) Q

Q

P

Q Q

Q

Q

R

Q

R

Q

R

Q

R

Q

R

true Note that this condition implies that f is differentiable in P (because otherwise there would be p ∈ P such that | (f p! − f q !)/(p − q) − f  p!| would not be R Q R Q R Q R Q a real number and could not be smaller than ). Classically, being uniformly differentiable on a closed, bounded interval is equivalent to being differentiable with a continuous derivative in that interval.14 Thus, we could introduce a notion of piecewise uniform differentiability analogous to piecewise uniform continuity, and we would then have a classical equivalence between that and the continuity of the derivative. However, I will not do so formally, as uniform differentiability on a closed, bounded interval is sufficient for the theorems I will state and prove below. That uniform differentiability suffices to make f  a trustworthy is established in Theorem 49, but we must first prove two simple lemmas. LEMMA 47: UNIFORMLY DIFFERENTIABLE FUNCTIONS HAVE UNIFORMLY CONTINUOUS DERIVATIVES

∀f, P (P ⊂ Q ∧ f ∈ Du (P ) ⇒ f  ∈ Cu (P ))

14 Classical proof: On a closed, bounded interval, continuity of the derivative is equivalent to uniform continuity of the derivative; see, again, Sprecher (1970, Theorem 27.4). Uniform continuity of the derivative, i.e.,

∀ > 0 ∃δ > 0 ∀x1 , x2 ∈R : |x1 − x2 | < δ → |f  (x1 ) − f  (x2 )| < implies uniform derivability, i.e.,    f (x1 ) − f (x2 )  ∀ > 0 ∃δ > 0 ∀x1 , x2 ∈R : 0 < |x1 − x2 | < δ →  − f  (x1 ) < x −x 1

f (x1 )−f (x2 ) x1 −x2

can be substituted with f  (x3 ) for some x3

2

because between x1 and x2 , according to the Mean Value Theorem. The opposite direction can be proved in classical mathematics essentially in the same way that Lemma 47 is proved below.

204

9.8 Differentiation

Proof Let  ∈ Q+ be given. The second assumption presupposes that there is a positive canonical rational number δ such that ∀p, q (0 < | p − q | < δ → | (f p ! − f q !)/(p − q) − f  p !| < /2) . Q Q

P

Q

R

Q

R

Q

R

Q

R

Q

Q

Let canonical rational numbers p and q be given such that | p − q | < δ. If p and q Q Q Q are equal, it is obvious that | f  p ! − f  q !| < . If not, then this inequality also R Q R Q R holds, because | f  p ! − f  q !| ≤ | (f p ! − f q !)/(p − q) − f  p !|

R

Q

R

Q

R R

Q

R

Q

R

Q

R

Q

+ | (f p ! − f q !)/(p − q) − f  q !| < /2 + /2 =  . R R

Q

R

Q

R

Q

R

Q

R

Q

Q

Q

R

LEMMA 48: UNIFORMLY DIFFERENTIABLE FUNCTIONS ARE UNIFORMLY CONTINUOUS

∀f, a, b (a < b ∧ f ∈ Du (a, b!) ⇒ f ∈ Cu (a, b!)) R

Proof Let  ∈ Q+ be given. It follows from the previous lemma together with the Extreme Value Theorem that f  is bounded on a, b!: there is a positive rational number R such that ∀ p (| f  p !| < R). Let a positive canonical rational number a,b! R Q R δ 1 be given such that ∀p, q (0 < | p − q | < δ 1 → | (f p ! − f q !)/(p − q) − a,b! Q Q Q Q R Q R Q R Q R f  p !| < R). Let δ be the canonical minimum of /(2 · R) and δ 1 . Let canonical Q R Q Q rational numbers p, q ∈ a, b! be given such that | p − q | < δ. Again, the case Q Q Q where p and q are equal is trivial, so we can assume that they are not: | f p ! − f q !| = | (p − q) · (f p ! − f q !)/(p − q)|

R

Q

R

Q

R R

Q

R

Q

R

Q

R

Q

< /(2 · R) · | (f p ! − f q !)/(p − q)| < /(2 · R) · (| f  p!| + R) R

Q

Q

RR

Q

R

Q

R

Q

R

Q

Q

R

R

Q

R

< /(2 · R) · (R + R) =  R

Q

Q

R

R

R

We are now ready to prove that f  c can be trusted to represent what you would think it represents when c belongs to an interval on which f is uniformly differentiable: THEOREM 49: DIFFERENTIATION BEHAVES AS EXPECTED FOR IRRATIONAL VALUES

∀f, a, b, c (a < c < b ∧ f ∈ Du (a, b!) R

R

⇒ f  c = lim((f c − f q !)/(c − q))) R

q
c

R

Q

R

R

9

Real Analysis

205

Proof We need to prove ∀ + ∃δ+ ∀ q (0 < | c − q | < δ → | (f c − f q !)/(c − q) − f  c | < ) . Q

Q

a,b!

R R

R

R

R

R

Q

R

R

R

R

It should first be noted that f c and f  c are both real numbers, since both f and f  are uniformly continuous on a, b!. Let  ∈ Q+ be given. Let δ 1 be a positive canonical rational number such that ∀q (| c − q | < δ 1 → | f  c − f  q !| < /4) Q R R R R R Q R Q (Lemma 47), and let δ 2 be a positive canonical rational number such that ∀p, q (0 < a,b! Q | p − q | < δ 2 → | (f p ! − f q !)/(p − q) − f  p !| < /4). Let δ be the canonical Q Q Q R Q R Q R Q R Q R Q minimum of δ 1 and δ 2 . Let a canonical rational number q in a, b! be given such that 0 < | c − q | < δ. To show that q satisfies the desired -inequality, we need to R R R R show that there is a rational number p that is non-equal to q and that satisfies the following: (a) (b) (c) (d)

| (f c − f q !)/(c − q) − (f p! − f q !)/(c − q)| < /4 R Q R R R Q R Q R R R Q | (f p ! − f q !)/(c − q) − (f p! − f q !)/(p − q)| < /4 R Q R Q R R R Q R Q R R R Q | p − q | < δ, because then | (f p ! − f q !)/(p − q) − f  p !| < /4 R Q R R Q R Q R Q R Q R Q | p − c | < δ, because then | f  p ! − f  c | < /4 R

R

R

R

R

Q

R

R

Q

To satisfy (c) and (d) we simply restrict ourselves to rational numbers between q and c. To also satisfy (a), we restrict ourselves further to within a distance of c wherein all canonical rational numbers r are such that | f c −f r !| < | c−q |·/4. R

R

Q

R R

R

R

Q

Lastly, we need to satisfy (b). Since f is uniformly continuous (Lemma 48), it follows from the Extreme Value Theorem that sup {| f r ! − f q !| | r ∈ c, q !} is R Q R Q a real number (in the case where c is smaller than q; otherwise replace c, q ! with q, c!). If this supremum is equal to 0, then | (f p! − f q !)/(c − q) − (f p ! − f q !)/(p − q)| = 0 < /4 .

R

Q

R

Q

R

R

R

Q

R

Q

R

R

R

R

Q

If not, then restrict further to within a distance from c of /4 · (q − c)2N / sup {| f r ! − f q !| | r ∈ c, q !} Q

R

R

R

R

R

Q

R

Q

in which case we have | (f p! − f q !)/(c − q) − (f p! − f q !)/(p − q)|

R

Q

R

Q

R

R

R

Q

R

Q

R

R

= | f p! − f q !| · | p − c |/| (c − q) · (p − q)| R R

Q

R

Q

RR

R

RR

R

R

R

< sup {| f r ! − f q !| | r ∈ c, q !} · | p − c |/(c − q)2N < /4 . R

R

Q

R

Q

RR

R

R

R

R

R

Q

206

9.8 Differentiation

We will add one more theorem about uniform differentiability. It states that the class of uniformly differentiable functions is closed under addition, subtraction, multiplication, and division. In the cases of the first two, it holds for any domain, but for the other two, the theorem is restricted to closed, bounded domains. But again: turning to piecewise uniform differentiability would lift this restriction. THEOREM 50: UNIFORM DIFFERENTIABILITY PRESERVED UNDER ALGEBRAIC OPERATIONS

∀f, g, P (f, g ∈ Du (P ) ⇒ f + g ∈ Du (P )) →  R

Q

∀f, g, a, b (f, g ∈ Du (a, b!) ⇒ f · g ∈ Du (a, b!)) →  R

Q

∀f, g, P (f, g ∈ Du (P ) ⇒ f − g ∈ Du (P )) →  R

Q

∀f, g, a, b (f, g ∈ Du (a, b!) ∧ inf {| gp !| | p ∈ a, b!} = 0 R

⇒ f / g ∈ Du (a, b!))

Q

R

→  R

Q

Proof I will just prove the sentence about multiplication. Let  ∈ Q+ be given. Let R 1 , R 2 , and R 3 be positive rational numbers such that ∀ p (| f p !| < R 1 ), a,b! R Q R ∀ p (| gp !| < R 2 ), and ∀ p (| f  p !| < R 3 ). Let δ 1 , δ 2 , and δ 3 be positive rational a,b! R Q R a,b! R Q R numbers such that ∀p, q (0 < | p − q | < δ 1 → | (gp ! − gq !)/(p − q) − g  p !| < /(3 · R 1 )) , a,b!

Q Q

Q

Q

R

Q

R

Q

R

Q

R

Q

R

Q

Q

and ∀p, q (0 < | p − q | < δ 2 → | (f p ! − f q !)/(p − q) − f  p !| < /(3 · R 2 )) , a,b!

Q Q

Q

Q

R

Q

R

Q

R

Q

R

Q

R

Q

Q

and, using Lemma 48, ∀p, q (| p − q | < δ 3 → | gp ! − gq !| < /(3 · R 3 )) . a,b!

Q

Q

Q

R

Q

R

Q

R

Q

Q

Let δ be the canonical minimum of δ 1 , δ 2 , and δ 3 , and let non-equal canonical rational numbers p and q be given. Through the trick of adding and subtracting the same number twice, as well as appealing to Lemma 43, we can verify that the desired inequality holds: | ((f · g)p! − (f · g)q !)/(p − q) − (f · g) p !|

R

→  R

Q

Q

→  R

R

Q

Q

R

Q

R

= | (f p ! · gp ! − f q ! · gq !)/(p − q) R R

Q

R

Q

R

Q

R

Q

R

− f  p! · gp ! − f p ! · g  p !| R

Q

R

Q

R

Q

R

Q

Q

→  R

Q

Q

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= | (f p ! · gp ! + f p ! · gq ! − f p ! · gq ! − f q ! · gq !)/(p − q) R R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

− f  p! · gp ! − f p ! · g  p ! − f  p ! · (gq ! − gp !) R

Q

R

R

Q

R

Q

R

Q

R

+ f  p! · (gq ! − gp !)| Q

R

Q

R

Q

R

Q

R

R

Q

Q

Q

= | f p ! · ((gp ! − gq !)/(p − q) − g  p !) R R

Q

R

Q

R

Q

R

Q

R

Q

+ gq ! · ((f p ! − f q !)/(p − q) − f  p !) R

Q

R

Q

R

Q

R

Q

R

Q

+ f  p! · (gq ! − gp !)| R

Q

R

R

Q

Q

< R 1 · /(3 · R 1 ) + R 2 · /(3 · R 2 ) + R 3 · /(3 · R 3 ) R

R

Q

Q

R

R

Q

Q

R

R

Q

Q

= R

9.8.3

Mean Value Theorem and Monotonicity

We now return to theorems that one would find in standard textbooks on classical analysis. This section will conclude with the theorem that gives the relationship between the sign of the derivative of a function and the function’s monotonicity properties. The buildup to this theorem requires three other theorems, starting with Rolle’s. These three theorems will expand on the by now somewhat familiar point that, even though we cannot rely on the platonic assumption of the existence of uncountably many undescribable reals, we nevertheless often have the real numbers that are needed as truth makers of classical theorems that assert the existence of certain specific reals. THEOREM 51: ROLLE’S THEOREM ∀f, a, b (a < b ∧ f ∈ Du (a, b!) ∧ f a = f b = 0 R

R

⇒ ∃c (a < c < b ∧ f  c = 0)) R

R

R

R

Proof If f is identically 0 on a, b!, the consequent is trivially true. Otherwise, it follows from Theorem 35 that f attains a non-zero maximum or minimum in the inner of that interval—say, a maximum at c. For all canonical rational numbers q not equal to c in a neighborhood of c contained in a, b!, f c − f q ! is therefore R Q non-negative. Hence, for such q that are smaller than c, we have (f c − f q !)/(c − q) ≥ 0 , R

Q

R

R

R

and for q larger than c we have (f c − f q !)/(c − q) ≤ 0 . R

Q

R

R

R

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9.8 Differentiation

Using Theorem 49 we can therefore conclude that f  c = lim((f c − f q !)/(c − q)) = 0 . R

R

q
c

Q

R

R

R

THEOREM 52: GENERALIZED MEAN VALUE THEOREM ∀f, g, a, b (a < b ∧ f a , f b , ga , gb ∈ R ∧ f, g ∈ Du (a, b!) R

⇒ ∃c (a < c < b ∧ (f b − f a ) · g  c = (gb − ga ) · f  c )) R

R

R

R

R

R

R

Proof Consider the function λ

p

.((f b − f a ) · (gp ! − ga ) − (gb − ga ) · (f p ! − f a )) .

{p|f p !,gp !∈R} Q

Q

R

Q

R

Q

R

R

R

R

Q

R

It assumes the value 0 at a and b, so it follows from Rolle’s Theorem that there is a value in the inner of the interval where the derivative function assumes the value 0. With that value, the desired equation is satisfied. THEOREM 53: MEAN VALUE THEOREM ∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) R

⇒ ∃c (a < c < b ∧ f b − f a = (b − a) · f  c )) R

R

R

R

R

R

Proof Let g in the previous theorem be the identity function. THEOREM 54: DERIVATIVES AND MONOTONICITY ∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) ∧ ∀ p (f  p ! ≥ 0) R

⇒ f a ≤ f b )

a,b!

Q

R

R

∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) ∧ ∀ p (f  p ! ≤ 0) R

⇒ f a ≥ f b )

a,b!

Q

R

R

∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) ∧ ∀ p (f  p ! = 0) R

⇒ f a = f b )

a,b!

Q

R

R

∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) ∧ ∀ p (f  p ! > 0) R

⇒ f a < f b )

a,b!

Q

R

R

∀f, a, b (a < b ∧ f a , f b ∈ R ∧ f ∈ Du (a, b!) ∧ ∀ p (f  p ! < 0) R

⇒ f a > f b ) R

a,b!

Q

R

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Proof Let f , a, and b be given, and assume the antecedent of the first sentence. It follows by Theorem 53 that there exists a c between a and b such that f b − f a = (b − a) · f  c . Since f  c is equal to a limit of non-negative R R R R numbers, it is itself non-negative. As b − a is positive, this implies that f b − f a R R is non-negative, and the consequent follows. A similar argument works for sentences two and three. As the limit of negative (positive) numbers is not necessarily negative (positive) itself, another argument is needed regarding sentences four and five. Let p be a canonical rational number in !a, b. From f  p ! > 0, it follows that there Q R is a positive rational number δ smaller than the difference between p and b such that f p! < f p + δ !. From sentence one, it follows that f a ≤ f p ! and Q R Q Q R Q f p + δ ! ≤ f b . Ergo f a < f b . Q

9.9

Q

R

R

Integration

Let us now turn to reconstructing the Darboux integral.15 To do so, we first have to overcome a minor challenge. The Darboux integral is defined using quantification over all partitions of the interval over which the given function is to be integrated, which essentially amounts to quantification over all finite sets of real numbers in that interval. We, however, need to just quantify over all finite sets of rational numbers. Fortunately, in classical mathematics, amending the definition in that way gives an equivalent result. Intuitively, a partition of a given interval is a finite class/set of non-overlapping intervals, the union of which is co-extensional with the given interval. Technically, we can identify a partition with just a finite set of numbers. Then, the first subinterval in the intuitive understanding is the interval between the left endpoint of the given interval and the smallest of the numbers in the set; the second sub-interval is between the smallest number in the set and the second-smallest number in the set; . . .; and the last sub-interval is between the largest number in the set and the right endpoint of the given interval. Let us agree to call a partition (in the technical sense just explained) plus the two endpoints of the given interval an “extended partition” (for lack of a better term) of that interval. We will not forbid the extended partition from containing real numbers—i.e., the real numbers corresponding to the endpoints of the given interval. We cannot, for that would prevent us from calculating the integral over 15 The Darboux integral is equivalent to the better-known Riemann integral, and slightly easier to reconstruct.

210

9.9 Integration

an interval with irrational endpoints; and we do not have to. Only the partitions (simpliciter) need to be so restricted. I prove in a footnote that, classically, the normal definition of a Darboux integral is equivalent to a definition that only quantifies over partitions with rational numbers only.16 The first step in the formal development of integration is to introduce a class of the partitions of any given bounded interval. We need such a class to be determinate, and therefore we limit it to classes of one specific form of canonical rational numbers in the interval: a term of the form {q 1 , . . . , q n }, where q 1 , . . . , q n are canonical rational numbers in an interval a, b!, is called a canonical partition of a, b!. That is reflected in the formal language as follows: Pa,b! term {q 1 , . . . , q n } ∈ Pa,b! satisfied iff q 1 , . . . , q n ∈ a, b! true The upper sum of a function f given an extended partition σ is denoted (f ; σ )+ . We give a stipulation for membership of this class that, in effect, is a recursive definition. The recursion is on the number of non-equal partition points. The base case of the recursion is the case where all elements of σ are equal; that is, where the interval is degenerate. Then, of course, (f ; σ )+ is made equal to 0. In the recursion step, the length of the left-most subinterval, i.e., inf σ − inf {a | a = inf σ } , R

σ

R

multiplied by the supremum of the function values in that interval, i.e., sup {f q ! | q ∈ inf σ , inf {a | a = inf σ }!} , Q

16 Call

σ

R

a partition consisting of only rational numbers a “rational partition”, and rename what is normally called just a “partition” a “real partition”. Let f be a bounded function on a bounded interval I ; let P be a real partition of I ; and let U be the upper sum of f over I given P . Further, let > 0 be given. Let b be a real number such that |f (x)| < b for all x ∈ I . Let P  be the result of replacing each irrational number x in P with two rational numbers x1 and x2 such that

, where n is the number of elements in P . Also, let U  be the upper x1 < x < x2 and x2 − x1 < 2bn

sum of f over I given the rational partition P  . U  can at most be n · 2bn · 2b = larger than U . That is, given any upper sum for a real partition, there is an upper sum for a rational partition arbitrarily close to it, which means that the infimum over the upper sums of the one kind is equal to the infimum over the upper sums of the other kind. The same holds for lower sums. Ergo, f is integrable under the one definition iff it is under the other, and the values of the two integrals, if they exist, are equal. The final step in the proof is to note that the equivalence extends to improper integrals because they are then limits of equal integrals.

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is added to the upper sum of the rest of the given interval, namely (f ; {a | a = inf σ })+ . R

σ

The full stipulation looks like this: (f ; σ )+ term w ∈ (f ; σ )+ satisfied iff (∀a, b (a = b) ∧ w ∈ 0) ∨ (∃a, b (a = b) R

σ

σ

∧ w ∈ (inf σ − inf {a | a = inf σ }) · R

R

σ

R

R

sup {f q ! | q ∈ inf σ , inf {a | a = inf σ }!} Q

R

σ

+ (f ; {a | a = inf σ })+ ) true R

σ

R

The stipulation for the lower sum is analogous: (f ; σ )− term w ∈ (f ; σ )− satisfied iff (∀a, b (a = b) ∧ w ∈ 0) ∨ (∃a, b (a = b) R

σ

σ

∧ w ∈ (inf σ − inf {a | a = inf σ }) · R

R

σ

R

R

inf {f q ! | q ∈ inf σ , inf {a | a = inf σ }!} Q

σ

+ (f ; {a | a = inf σ })− ) true R

σ

R

R

The upper (lower) integral of f from a to b is made equal to the infimum (supremum) of the upper (lower) sums of f , given extended partitions of the form σ ∪ {a, b}, where σ is a canonical partition of a, b!—or, that with a minus in front, if a is larger than b: b

f q ! dq term (binds the displayed occurrences of q) Q  w ∈ ab f q ! dq satisfied iff a

Q

∀ q (f q ! ∈ R) ∧ ((a ≤ b ∧ w ∈ inf {(f ; σ ∪ {a, b})+ | σ ∈ Pa,b! }) a,b!

Q

R

∨ (a > b ∧ w ∈ − inf {(f ; σ ∪ {a, b})+ | σ ∈ Pb,a ! })) true R

R

b

f q ! dq term (binds the displayed occurrences of q) Q w ∈ ab f q ! dq satisfied iff a

Q

∀ q (f q ! ∈ R) ∧ ((a ≤ b ∧ w ∈ sup {(f ; σ ∪ {a, b})− | σ ∈ Pa,b! }) a,b!

Q

R

∨ (a > b ∧ w ∈ − sup {(f ; σ ∪ {a, b})− | σ ∈ Pb,a ! })) true R

R

If the upper and lower integrals are equal, then the integral is also made equal to them. To be precise, the integral inherits the elements of the upper integral:

212

9.9 Integration

b

f q ! dq term (binds the displayed occurrences of q) Q   w ∈ ab f q ! dq satisfied iff w ∈ ab f q ! dq = ab f q ! dq true a

Q

Q

R

Q

The classical theorem stating that integration is additive on intervals carries over; however, the proof is a little more complicated: THEOREM 55: ADDITIVITY OF INTEGRATION ON INTERVALS ∀a, b, c, f (a < c < b ∧ ∃p ∀ q (| f q !| < p) R R R   c Q a,b! R Q ⇒ ab f q ! dq = a f q ! dq + cb f q ! dq) Q

R

Q

R

Q

Q

R

Q

R

Q

∀a, b, c, f (a < c < b ∧ ∃p ∀ q (| f q !| < p) R R R  c Q a,b! R Q  ⇒ ab f q ! dq = a f q ! dq + cb f q ! dq)

Proof We prove the first sentence. Let a, b, c, and f be given, and assume the antecedent. Let p satisfy the condition of the second conjunct thereof. And let a positive canonical rational number  be given. In part 1, we show that c

q ! dq af Q

+ R

b c

f q ! dq ≤ Q

R

q ! dq af Q

+

b a

f q ! dq +  , Q

R

and in part 2, that b a

f q ! dq ≤ Q

R

c

R

b c

f q ! dq +  , Q

R

from which the consequent follows. Part 1: Let {r 1 , . . . , r n } ∈ Pa,b! be given such that (f ; {a, r 1 , . . . , r n , b})+ < R

b a

f q ! dq +  . Q

R

Assume without loss of generality that the elements of {r 1 , . . . , r n } are ordered by size and are all non-equal; and let r i be the largest of them that is smaller than c. Then, {r 1 , . . . , r i } ∈ Pa,c! and {r i+1 , . . . , r n } ∈ Pc,b! . This gives us the inequality we need: c

q ! dq af Q

+ R

b c

f q ! dq ≤ (f ; {a, r 1 , . . . , r i , c})+ + (f ; {c, r i+1 , . . . , r n , b})+ Q

R

≤ (f ; {a, r 1 , . . . , r n , b})+ R  ≤ ab f q ! dq +  R

Q

R

R

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Part 2: Let {r 1 , . . . , r n } ∈ Pa,c! be given such that (f ; {a, r 1 , . . . , r n , c})+ < R

c

q ! dq af Q

+ /4 . R

Q

Similarly, let {s 1 , . . . , s m } ∈ Pc,b! be given such that (f ; {c, s 1 , . . . , s m , b})+ < R

b c

f q ! dq + /4 . Q

R

Q

Since c may be irrational and therefore not equal to a number that is an element of any partition of a, b!, we pick two rational numbers close to it: let r c ∈ a, c! and s c ∈ c, b! be such that (c − r c ) · 2 · p < /4 and (s c − c) · 2 · p < /4. We then R R R R Q R R R R Q have, as desired, that b a

f q ! dq Q

≤ (f ; {a, r 1 , . . . , r n , r c , s c , s 1 , . . . , s m , b})+ R

≤ (f ; {a, r 1 , . . . , r n , r c , c})+ + /4 + (f ; {c, s c , s 1 , . . . , s m , b})+ + /4 R

R

Q

R

≤ (f ; {a, r 1 , . . . , r n , c})+ + /4 + (f ; {c, s 1 , . . . , s m , b})+ + /4 R R R Q c Q R ≤ a f q ! dq + /4 + /4 + cb f q ! dq + /4 + /4 . R

Q

R

Q

R

Q

R

Q

R

Q

R

R

Q

Q

A classical theorem states that anti-derivatives are unique, up to the addition of a constant. That is not true here. A counterexample is a zero function, which not only has constant functions as anti-derivatives, but also—for example—a function which assumes the value 0 on values less than the square root of 3, and 1 on values higher than the square root of 3. Another counterexample is Minkowski’s Question Mark Function from Sect. 9.8.2. To achieve uniqueness up to the addition of a constant, we have to restrict ourselves to the uniformly differentiable functions (or, in the case of unbounded domains, piecewise uniformly differentiable functions; see again Sect. 9.8.2). The modified theorem looks like this: THEOREM 56: UNIFORMLY CONTINUOUS ANTI-DERIVATIVES UNIQUE UP TO CONSTANT

∀a, b, F, G(a < b ∧ F, G ∈ Du (a, b!) ∧ ∀ q (F  q ! = G q !) R

⇒ ∃c ∀ q (F q ! = Gq ! + c)) R

a,b!

Q

R

Q

a,b!

Q

R

Q

R

Proof Let q be an element of a, b!. From Lemma 43 it follows that (F − G) q ! Q→  R Q is equal to F  q ! − G q !, which is again equal to 0 according to the third conjunct Q R Q of the antecedent. Theorems 39 and 50 tell us that F − G is uniformly continuous →  R

Q

214

9.9 Integration

and uniformly differentiable on a, b!. Hence, using the third sentence of Theorem 54, we can infer that F − G is constant, which implies the consequent. →  R

Q

We finally come to the Fundamental Theorem of Calculus. Both classically and here, the proof of this theorem builds on the uniqueness of anti-derivatives. But even though our version of that theorem requires an extra assumption, the Fundamental Theorem itself holds, in a form that is essentially the same as in classical mathematics: THEOREM 57: FUNDAMENTAL THEOREM OF CALCULUS p ∀a, b, c, f (a < c < b ∧ f ∈ Cu (a, b!) ⇒ (λ p . a f q ! dq) c = f c ) R

R

a,b!

Q

R

Proof Let a, b, c, and f be given, and assume the antecedent. We prove that for p all p ∈ a, b!, it is the case that (λ p . a f q ! dq) p ! = f p !, from which the a,b! Q Q R Q consequent follows. Part 1: Let p and r be non-equal elements of a, b!. If r > p, then Q r inf {f q ! | q ∈ p, r !} · (r − p) ≤ p f q ! dq ≤ sup {f q ! | q ∈ p, r !} · (r − p) , Q

R

Q

R

Q

R

Q

R

Q

while if p > r, then Q

inf {f q ! | q ∈ r, p !} · (p − r) ≤ Q

R

Q

R

p r

f q ! dq ≤ sup {f q ! | q ∈ r, p !} · (p − r) . Q

R

Q

R

Q

Hence, either way, it is true that: inf {f q ! | q ∈ p, r ! ∪ r, p !} ≤ Q

p r

R

f q ! dq/(p − r) Q

R

Q

≤ sup {f q ! | q ∈ p, r ! ∪ r, p !} . R

Q

Taking the limit for r approaching p, both the left-hand side and the right-hand side become equal to f p ! because of the uniform continuity of f . Hence, the Q same holds for the middle. That justifies the third of these equalities, whereas the first is given by the stipulation for differentiation, and the second is sanctioned by Theorem 55: p p r (λ p . a f q ! dq) p ! = lim(( a f q ! dq − a f q ! dq)/(p − r)) a,b! Q Q R r
p Q R R Q p Q = lim( r f q ! dq/(p − r)) = f p ! . R

Q

r
p

In the same way it can be shown that p (λ p . a f q ! dq) p! = f p ! . a,b!

Q

Q

R

Q

R

Q

R

Q

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215

p Part 2: Next we prove that λ p . a f q ! dq is uniformly differentiable; i.e., that a,b!

Q

p r ∀ +∃δ+∀p, r (0 < | p−r | < δ → | ( a f q ! dq− a f q ! dq)/(p−r)−f p !| < ) . Q

Q

a,b!

Q Q

Q

Q

R

Q

R

Q

R

Q

R

Q

R

Let a positive canonical rational number  be given. From the uniform continuity of f , it follows that there is a positive canonical rational number δ s.t. ∀p, r (| p − r | < a,b! Q Q Q δ → | f p ! − f r !| < /2). Let canonical rational numbers p and r from a, b! R Q R Q R Q be given. Assume without loss of generality that r < p. Let c (d) be the value at Q which f attains its maximum (minimum) in p, r !; and let s c and s d be canonical rational numbers from the same interval that satisfy | f s c ! − f c | < /2 and R Q R R Q | f s d ! − f d | < /2. With this we can verify the inequality:

R

Q

R

R

Q

p r | ( a f q ! dq − a f q ! dq)/(p − r) − f p !| R R Q R Q R Q  Qp = | r f q ! dq/(p − r) − f p!| R R

Q

R

Q

R

Q

≤ max{| f c · (p − r)/(p − r) − f p !|, | f d · (p − r)/(p − r) − f p!|} R

R

R

R

R

R

R

R

Q

R

R

R

R

R

R

Q

= max{| f c − f p !|, | f d − f p !|} R

R

R

R

Q

R

R

Q

< max{| f s c ! − f p!| + /2, | f s d ! − f p !| + /2} R

R

R

Q

R

Q

R

Q

R

Q

R

Q

R

Q

< max{/2 + /2, /2 + /2} R

R

Q

R

Q

Q

R

Q

= R

p It can of course be demonstrated analogously that λ p . a f q ! dq is uniformly difa,b! Q ferentiable. Part 3: Using Theorem 56, we can infer from parts 1 and 2 that there is a conp p stant d such that a f q ! dq = a f q ! dq + d for all p ∈ a, b!. Since both Q R Q R p p q ! dq and a f q ! dq have 0 as their limit when p goes toward a from the a f Q Q p p right, d is equal to 0, and we can simplify the equation to a f q ! dq = a f q ! dq. Q R Q p It follows, by the stipulation for integration, that a f q ! dq has the same elements Q p p as a f q ! dq, and therefore that (λ p . a f q ! dq) p ! = f p !. Q

a,b!

Q

Q

R

Q

That concludes the treatment of the reconstructed Darboux integral. Let me briefly comment on the Lebesgue integral.17 There are cases where physicists use Lebesgue’s theory to calculate integrals of functions that are not Darbouxintegrable. Does that mean that it would be a problem for semantic conventionalism if it cannot be reconstructed? That depends on whether the use of the Lebesgue integral is essential to physics. Some applications are merely due to the convenience of idealization. One example of this is the integration of the Dirac delta function 17 One relatively accessible introduction to the Lebesgue integral and how it differs from the Darboux and Riemann integrals is Bear (2001).

216

9.10 Unbounded Intervals and Piecewise Continuity

Dirac (1958, §15) when it is convenient to pretend that momentum is transferred instantaneously from one object to another—and that is clearly not enough to show that a foundation that cannot support Lebesgue integration is inadequate. Similarly, if an application is made in the context of an incorrect theory, then that application would not be a philosophical problem for me, no matter how good an approximation the incorrect theory is to physical reality. It is therefore a difficult question to answer, and I am, unfortunately, not qualified to undertake the detailed investigation into physical theory (in particular quantum mechanics) required to do so. Can it be reconstructed? Well, the standard way of defining it would not work in this setting, because it involves taking the infimum over the class of lengths of a certain class of open subsets, which—as an indefinite class—would not satisfy the antecedent of Theorem 30. However, we could partially reconstruct the integral by exploiting the fact that, classically, the Lebesgue integral for a function f is equal to the limit of the integrals (defined as a finite sum) of a sequence (f1 , f2 , f3 , . . .) of step functions that has f as its limit (Riesz and Sz.-Nagy 1956). We can exploit that by simply defining the Lebesgue integral as that limit of integrals (and, if necessary, define some of the step functions in the way explained in Sect. 9.11 below). I say that the reconstruction is only partial because we can only define the integral for functions that are syntactically in the form of the limit of a sequence of step functions. Thus, to use this stronger integral for the complicated functions that cannot be handled simply by the Darboux integral, we would have to define functional application to irrational values directly (instead of the via the “skeleton” of rational numbers). At that point, equality of functions cannot be made to come out true without the use of the strong conditional in ways that leads to iterations thereof; so the integral cannot be applied to functions that are not syntactically the limit of a sequence of step functions. If there is an essential need for the Lebesgue integral, is this partial reconstruction then sufficient to satisfy that need? Again I must plead ignorance and leave that question for others.

9.10

Unbounded Intervals and Piecewise Continuity

We shall not go further “upwards” in the reconstruction of analysis; the four remaining sections of this chapter have different goals. This one provides some generalizations of previous theorems to functions on unbounded domains where the concept of piecewise uniform continuity has to take over from the concept of uniform continuity. Section 9.11 is concerned with a different way of basing functional application

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217

in the secondary sense on determinate functions. Section 9.12 revisits applied mathematics. And the final section explores diagonalization and the status of the class of real numbers. Many theorems in the preceding sections have been concerned with the behavior of a function on an interval a, b!. In this section I will—in a brief and unsystematic way—illustrate how to generalize our version of analysis to unbounded intervals. We will just deal with the example of the Intermediate Value Theorem on intervals of the form a, ∞, that is, the theorem stating that if f is a piecewise uniformly continuous function on a, ∞, and d is a value between the value assumed by f on a and the limit of f for the independent variable tending to infinity, then f assumes the value d somewhere to the right of a. To formulate this theorem, we first have to introduce this new kind of limit: lim f q ! term (binds the displayed occurrences of q) q
∞

Q

(n, m/2nN ) ∈ lim f q ! satisfied iff n ∈ N ∧ f ∈ Q →  R Q N

Q

q
∞

∧ m ≡ min {m | ∃N ∀q (f q ! ∈ R ∧ q > N → | f q ! − m/2nN | < 1/2nN )} Z

Z

N

Q

Q

Q

R

Q

R

Q N

R

Q N

true THEOREM 58: INTERMEDIATE VALUE THEOREM FOR UNBOUNDED INTERVALS

∀f, a, d (f ∈ Cp (a, ∞) ∧ (f a < d < lim f q ! ∨ f a > d > lim f q !) ⇒ ∃c (a < c ∧ f c = d)) R

R

R

Q

q
∞

R

R

q
∞

Q

R

Proof Assume for the sake of argument that f a < d < lim f q !. From the R R q
∞ Q stipulation for the secondary sense of functional application, it follows that there is a p ∈ a, ∞ such that f p ! < d. And from the newest stipulation, it folQ R lows that there is a canonical rational number r such that p < r and d < f r !. R R Q From the assumption that f ∈ Cp (a, ∞), it can be deduced that f ∈ Cu (p, r). Hence, the existence of a c with the desired properties can be concluded from Theorem 34. As an aside, I should mention that the new type of limit can also be used to introduce improper integrals. I will just state the stipulation without building any theory on top of it: ∞

f q ! dq term (binds the displayed occurrences of q) p  ∞Q w ∈ a f q ! dq satisfied iff w ∈ lim λ p . a f q ! dqp ! ∈ R true a

Q

p
∞

a,∞

Q

Q

218

9.11

9.11 Completifications of Functions Generalized

Completifications of Functions Generalized

So far, we have worked with just one method for applying functions to irrational values: namely, letting the result of such an application be the limit of the values for rational numbers, insofar as that limit exists. For the vast majority of empirical applications of real analysis, this will suffice. However, we are not limited to this specific method. It is not a theorem of semantic conventionalism that there is no √ function that, say, assumes the value 0 for values less than 2 3, and the value 1 for NR √ values equal to or larger than 2 3, unlike in intuitionism where it is a theorem that NR there are no discontinuous functions on the real numbers. Recall the challenge that prompted the use of this method. The class R is indeterminate, so a real function f cannot be a real function by virtue of having as an element an ordered pair with a as first coordinate for every a ∈ R. Instead, we have used Q as a skeleton, to return to a previous metaphor. The essential properties of Q that allow it to fill that rôle are that it is determinate, and that it is dense in R. But it is not essential that exactly Q be used; any other class with those properties could be employed instead, and there are of course classes that contain some irrational numbers that enjoy those properties. We could, for instance, use the class that contains √ (p, 0) for all canonical rational numbers p less than 2 3; (p, 1) for all canonical NR √ √ rational numbers p larger than 2 3; and ( 2 3, 1) as a function. This function would NR NR then map any number equal to one of the terms that appear as a first coordinate to the corresponding second coordinate, and any number that is not, to the limit of the function values of those that are. That would give us a function that is defined, but √ discontinuous, at 2 3. NR

Making this change would involve changing or adding quite a few stipulations, including some of high complexity. That is why I have worked with the inflexible method of using a fixed class of numbers for the skeleton, and am only now mentioning the alternative. But it can be done without departing in any radical fashion from the approach we have adopted. Let me connect this discussion with the issue of differentiation. Recall the challenge considered in Sect. 9.8.2: we saw an example of a function f and an irrational number a such that f  a is not a real number but “ought to be”. That problem can now be solved. The remedy is simply to add a to the class of numbers to which f  is applicable in the primary sense. How do we add a to that class? We modify the stipulation for the derivative to “detect” values where f is not uniformly differentiable, and then include, for each of those values, an ordered pair with that value as first coordinate in f  . Assuming we implemented the changes alluded to above, the revised truth condition could look

9

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219

like this: (a, b) ∈ f  has its bivalent immediate view-from-nowhere truth conditions satisfied iff f ∈R→  R ∧ (a ∈ Dom(f ) ∨ ∃ c (a ≡ sup {q | f ∈ Du (c, q !)} ∈ R ∨ a ≡ inf {q | f ∈ Du (q, c!)} ∈ R)) Dom (f )

Q

Q

∧ b ≡ lim((f a − f d !)/(a − d)) ∈ R d
a

R

R

R

R

is true, where Dom(f ) is introduced by the stipulation that x ∈ Dom f has its bivalent immediate view-from-nowhere truth conditions satisfied iff ∃y((x, y) ∈ f ) is true. With that improvement, terms of the form f  a can be trusted whenever the points that are problematic and need special treatment in this way are all isolated.

9.12

Another Example of Applied Mathematics

I will now turn to Putnam’s challenge of how to achieve “numericalization” in nominalistic mathematics, discussed in Sect. 6.1. With the aim of assigning reasonable truth conditions to sentences saying that there is a certain number of meters between two physical points, we first import a number of things from natural language: namely, the common noun “point”, the identity relation for points, and terms for the two end-points of the Paris standard meter: P term x ∈ P satisfied iff x denotes a point x = y wff P

x = y satisfied iff x and y denote the same point P

Parisw , Parise terms Parisw denotes the westernmost point occupied by the Paris standard meter the moment it was declared the standard Parise denotes the easternmost point occupied by the Paris standard meter the moment it was declared the standard We then construct a standard half-meter (from Parisw to what is called Paris0 ), a standard quarter-meter (from Parisw to Paris0 ), etc. using the natural language congruence and “between” relations that Putnam grants that we have available:

220

9.12 Another Example of Applied Mathematics

For n a canonical natural number: – Parisn term – Paris0 denotes the point denoted by Parise – Parisn denotes the point which is between Parisw and Parisn and is such that the interval from Parisw to that point is congruent with the interval from that point to Parisn As an example, let us say that we have two points, x 0 and y 0 , that are 2.7 meters from each other (as we would like to be able to say). By virtue of the next stipulation, 0 0 → y 0 will denote the point 1 meter from x 0 in the direction of y 0 and x 0 − →y x0− 0  0 will denote the point 2 meters from x 0 in the direction of y 0 (and x 0 − →y 0 and so on will denote nothing). For n a canonical natural number: n → y term – x− 0 → y denotes the point denoted by x – x− n n → y denotes the point which is between x − → y and y and is such that the – x− n → y to interval from Parisw to Parise is congruent with the interval from x −

that point The following stipulation does two things. It first gives the name x − → 0 y to the point that represents the “integer part of the journey from x to y”, that is, in the 0 → →y example at hand, x 0 − 0 denotes. Second, it 0 y 0 denotes the same point as x 0 − gives names to points that approximate y, using the standard half-meter, quarter→ meter, etc. In the example, x 0 − 0 y 0 will denote the point 2.5 meters from x 0 in the direction of y 0 ; since 2.75 meters would be too go to far, x 0 − →y 0 will also denote 0 the point 2.5 meters from x 0 in the direction of y 0 ; x 0 −  0 will denote the point 0→y 2.625 meters from x 0 in the direction of y 0 ; and so on. For n a canonical natural number: → – x− n y term n – x− → → y where n is such that 0 y denotes the point denoted by the term x − m → y ∈ P} satisfied n = max {m | x − N

N

N

– x− → → n y denotes the point between x − n y and y such that the interval from → Parisw to Parisn is congruent with the interval from x − n y to that point

9

Real Analysis

221

Using the approximation points we can now, as desired, represent each distance between two points by a real number:

d(x, y) term For n a canonical natural number: m – (0, q) ∈ d(x, y) satisfied iff q ≡ max {m | x − → y ∈ P} true N

N

→ → – (n, q) ∈ d(x, y) satisfied iff ((n, q) ∈ d(x, y) ∧ x − n y = x − n y) P

∨ (∃p ((n, p) ∈ d(x, y) ∧ q = p + 1/2n ) ∧ x− → → n y = x − n y) true N Q

Q

Q N

P

In the example, the sentence d(x 0 , y 0 ) = 27/10 then comes out true. R

Q

This collection of stipulations would seem to show definitively that Putnam’s indispensability claim is wrong. However, the dialectical situation is a little more complicated than that. He explicitly assumes that there are abstract mathematical objects, claiming that that assumption is necessary. He also implicitly assumes that the geometry of the physical universe is continuous, i.e., that there are infinitely many physical points. For the purpose of this section, I have worked under the same assumption (also implicitly, until now), even though I suspect that it is false. If the assumption is false, the extension of the language stipulated here will not work as intended, just as Putnam’s definition won’t. Strictly speaking, this means that the question of whether the existence of abstract objects is indispensable to the numericalization of physical distances is not settled. We may not have demonstrated that they are not, given the actual structure of physical points and physical distances. However, Putnam’s argument as it stands (i.e., as based on the assumption) has been undermined; and it seems plausible that a discrete universe can also be handled in this framework if a continuous one can. However, that is a complicated question, since it is not obvious exactly which geometry is correct for the physical universe if it is discrete; see Hagar (2014). Lastly, we have only considered two examples of applications of mathematics among many. One might be skeptical that all of them can be treated on the model of these two examples. In that connection it should be noted that it is not clear that identifying a counterexample to that (i.e., that they can be treated on that model) would prove Putnam’s anti-nominalism correct. For, even though we have managed here to break down our language conventions concerning distances into the small bits contained in the gray boxes in this section, there might very well be other applied-mathematics language conventions that cannot be similarly reduced— simply because not all language conventions can be explained in terms of simpler language. But on the other hand, that does not mean that any such counterexample

222

9.13 Diagonalization

can be dismissed out of hand. Again, the situation is complicated and any proposed counterexample would have to be given individual consideration.

9.13

Diagonalization

One very important subject remains to be investigated formally: namely, the status of the class of real numbers. I claimed back in Sect. 2.3 that the proof of Cantor’s Diagonal Theorem goes through in our setting, and it is time to verify that. This theorem will be the point of departure for the discussion in the final chapter. We introduce the term Diag(f ), to be read as “the diagonalization of f ”. The point of a diagonalization of a sequence of real numbers is to produce a real number that is not in that sequence. That can be done in many different ways; as will be made clear in the proof below, this is one of them: Diag(f ) term (0, 0) ∈ Diag(f ) satisfied iff f ∈ N → R true (n, m/3n ) ∈ Diag(f ) satisfied iff N Q N

f ∈ N → R ∧ n ∈ N ∧ ∃p ((n, p) ∈ Diag(f ) Q

∧ m ≡ min {m | m/3n ≥ q ∧ ¬(m/3Nn ≤ f n! ≤ m/3Nn )}) true N N

N

Q N

Q

Q N

R

N

R

Q N

THEOREM 59: DIAGONAL THEOREM ∀f (f ∈ N → R ⇒ Diag(f ) ∈ R ∧ ∀n (Diag(f ) = f n!)) N

R

N

Proof Each “step in the creation” of the sequence Diag(f ) can be illustrated like this:  Option 1: Option 2: Option 3:

  

9

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The top-most star indicates the position of Diag(f )(n) (i.e., the nth rational numN ber in the real number Diag(f )) for some natural number n on a horizontal number line. If n is equal to 0, the star is placed at 0—the content of the remaining pairs of parentheses in this paragraph continues this example. The dashed line to its right indicates the interval to which the remaining numbers in the sequence are restricted (the interval 0, 1!). There are then three different options. Option 1 is that the value of f n! is less than Diag(f )(n) or higher than Diag(f )(n) + 1/3n N N N Q Q NN (the value of f 0! is less than 0 or higher than 1/3). Then, Diag(f )(n) is made N Q N equal to Diag(f )(n), and the remaining numbers in the sequence are restricted to N the first third of the interval to which they where previously restricted (0, 1/3!). Q Option 2 is that the value of f n! is equal to or higher than Diag(f )(n) but less N N (larger than 0 but smaller than 1/3). Then, Diag(f )(n) is than Diag(f )(n) + 1/3n N Q Q NN Q N , and the remaining numbers in the sequence are made equal to Diag(f )(n) + 1/3n N N Q Q N restricted to the second third of the interval to which they were previously restricted (1/3, 2/3!). Option 3 is that the value of f n! is equal to Diag(f )(n) + 1/3n Q Q N N Q Q NN (1/3). Then, Diag(f )(n) is made equal to Diag(f )(n) + 2/3n , and the remaining Q N N Q Q NN numbers in the sequence are restricted to the last third of the interval to which they were previously restricted (2/3, 1!). Q It is therefore easy to see that the two conjuncts of the consequent of the theorem are true. Diag(f ) is a sequence of rational numbers, the tail of which is restricted to intervals the length of which goes towards 0. Hence, it is a real number. And this real number is non-equal to each element in the sequence f , because for each of those elements f (n), there is a stage in the “creation” of Diag(f ) at which the rest of that sequence is restricted to a closed interval not containing f (n).

Chapter 10

Possibility

10.1

All Possible Real Numbers

In the last three chapters I developed a language convention that both stays within the philosophical bounds that I suspect limit mathematics, and does a good job of satisfying the kinds of needs that were identified in Sect. 6.1. But it should be emphasized again that it is merely one possible such convention among many, and I would not be surprised if it turns out that there is one that is better at satisfying those needs than what my creativity has sufficed for. Hence, this monograph might just be the first step in a wider research program. I would like to offer a few tentative remarks about the roads traveled in comparison to the roads not traveled—and one of the latter in particular. Doing so must be combined with a renewed discussion of the status of the real numbers, reaching back to some of the issues treated in the early chapters of this book, and Sect. 2.3 in particular. The content of the Diagonal Theorem is that the class of real numbers is “uncountable”; that is, there is no bijection with the natural numbers. Thus formulated, we can agree with classical mathematics. However, for us, a bijection is, like any other function, a linguistic entity. So what the Theorem really says is just that it is impossible to describe a sequence that contains all the reals of the language and only those. Any sequence that purports to do so will, through diagonalization, give rise to a new real number that is not included in that sequence. We might say that the class of real numbers is indefinitely extensible. The phrase “indefinitely extensible” has been defined in different ways, and there is disagreement about what kind of phenomenon it refers to. Russell’s (1907, 36) definition of what he calls “self-reproductive”, is often taken to be equivalent to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C.S. Hansen, Founding Mathematics on Semantic Conventions, Studies in Epistemology, Logic, Methodology, and Philosophy of Science, https://doi.org/10.1007/978-3-030-88534-2_10

225

226

10.1 All Possible Real Numbers

Dummett’s (1993, 441) definition of “indefinitely extensible”, so it interesting that we have here something that satisfies the latter definition but does not satisfy (the precise wording of) the former. Russell writes: [T]here are what we may call self-reproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Given R, we cannot define a new term also having the property of being a real number while being non-equal with each element of the class R, for R does not in itself give us a sequence of reals to which diagonalization can be applied. To obtain a definition that captures what we are dealing with here, we need to add the crucial word “definite”, as Dummett does: An indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under that concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it. Much confusion has surrounded Dummett’s use of the word “definite”, for in this context it is not clear that its meaning is definite. We, however, are in a position to make it precise, namely as meaning “can be ordered in a sequence”. Thus understood, the reals are indefinitely extensible. That is, if a collection, all of whose members fall under the concept of real number, can be ordered in a sequence, we can by reference to that totality characterize a larger collection—the reals of the sequence plus its diagonalization—all of whose members fall under the concept of real number.1 It is not a particularly strong form of indefinite extensibility. It does not have— to borrow a Cantorian (1883, §11) phrase—“the capacity to break through every obstacle”. On the contrary, the indefinitely extensible class R is a subclass of classes that are not indefinitely extensible: namely, universal classes for which the defining condition is empty, for instance {x | x ≡ x}: R ⊂ {x | x ≡ x} is a true sentence. However, all of the foregoing remarks are one-sided. Our analysis so far could only count as complete if we were restricted to what can be expressed in the formal language, and to just one static version thereof. Let us extend the analysis to what 1 Under the modified Dummettian definition, the natural numbers are not indefinitely extensible, for they can be ordered in a sequence. However, a more liberal notion of indefinite extensibility could be added according to which they are, namely by using the stricter interpretation of “definite collection”: as meaning “all elements of the collection actually exist”. Then the natural numbers are indefinitely extensible with the “diagonalizer” being the operation of taking the maximal element of the collection and applying the successor function to it.

10

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227

can be seen when we compare one version of the language (say the version consisting of all and only the stipulations in Chaps. 8 and 9) both against extensions of that language and against other possible languages—when we look at a given language from the outside instead of from the inside. Seen from the outside, it is clear that the Diagonal Theorem does not imply there are more real numbers in the formal language than there are natural numbers. It is easy to see that there are not: the vocabulary of the language is countable, which implies that there are only countably many finite combinations of the elements of that vocabulary, and the terms are among those combinations. That the proof of the Diagonal Theorem can go through in a setting where there are not, seen from an “external point of view”, uncountably many reals, is not in itself something new. That is also the point of Skolem’s paradox,2 so a comparison in order. This so-called paradox consists in the fact that countable first-order axiomatizations of classical set theory have non-standard models in which the set of real numbers is countable, even though the formal sentence that says (or would normally be interpreted as saying) that it is uncountable, is true in those models. That is possible because the formal sentence can be true merely due to the absence of a witness to countability. The formal sentence asserts the non-existence of a bijection between the natural numbers and the reals. Such a correspondence is itself a set, and this set—the witness to the countability—is what is absent from the models in question. The theorem that the reals are uncountable is of course provable in classical set theory. So Skolem’s paradox shows, just as well as the present result, that diagonalization does not on its own suffice to demonstrate that there are more reals than natural numbers. However, in the classical case, Skolem’s result seems like little more than an anomaly: an indication that first-order axiomatizations are imperfect.3 Classical set theory is a theory about all possible sets—i.e., about all the sets that could exist. So if there is an external perspective from which it can be seen that there could be a bijection between the reals of the model and the natural numbers, then this correspondence ought to be in the model. Hence, if it is not, it is deemed a “non-standard” model.4 2 The

primary source is Skolem (1922), but see also Benacerraf and Wright (1985). is the central point of Shapiro (1991). 4 The classical mathematician may try to avoid this problem by denying that she is restricted to first-order logic, or by denying that she is restricted in her expressive power to what is invariant in all models of some set of axioms. See Sect. 2.3. 3 That

228

10.1 All Possible Real Numbers

How are we to understand this phenomenon in our context? At face value: diagonalization shows that there is no (description of a) bijection between the two classes. That is all. Diagonalization does not show that there are more real numbers than natural numbers. However, there is something else that can be seen from the outside, complicating the picture. In addition to the in-language diagonalization of Theorem 59, we can diagonalize on any one version of the formal language in natural language: the closed terms x such that x ∈ R is a true sentence can be ordered in a sequence by a description of English; and by virtue of another description of English that diagonalizes on that sequence, we have a real number that is not in the formal language. In that light, the part of the one-sided analysis above that concerned Russell now seems wrong: given R we can define a new term also having the property of being a real number while being non-equal to each element of the class R. This stronger form of indefinite extensibility, which shows that we can never include all possible real numbers in any one version of the formal language, is what our open-ended language was designed to handle. Any possible real number accessible to humans, i.e., any real number that can be formulated in some possible language that we can understand, can be imported into our formal language; and by limiting what we call a “theorem” to what can be proved to be true under any possible extension of the language, we have ensured that theorems about R are, in a sense, about all those possible real numbers. Does the stronger form of indefinite extensibility show that there are more possible real numbers than natural numbers? I see no basis for claiming that. When the possibility of diagonalization on the real numbers in a system is consistent with there being the same number of real numbers in that system as natural numbers, diagonalization loses its alleged force as a prover of larger-than-N-size. It is a quintessentially platonic belief that whenever two classes are of the same size (in an intuitive sense of “same size”), there must be a bijection between them, consisting—if everything else fails—of arbitrary ordered pairs of one element from each of the two classes. For a nominalist, there is no reason to think that the absence of the latter, in the case of two infinite classes, tells us anything that can reasonably be described as a “difference in size”. It is another phenomenon altogether: the impossibility of capturing all possible real numbers in any one (static) language. Does it nevertheless make sense to talk about all possible real numbers?5 In general, I think there are objective facts about, and sharp boundaries to, what is possible (more on this in the next section); so in the absence of any evidence to the contrary, 5 I raise this question because indefinite extensibility has been used to argue that there are limits to how general quantification can be; see, e.g., Parsons (1974) and Glanzberg (2004).

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Possibility

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I assume that it does. Semantic conventionalism does not force us to conclude that the phenomenon of indefinite extensibility implies some kind of mystical fuzziness at the edge of the space of possibility, or at the edge of our concept thereof. Diagonalization does not provide evidence to the contrary. If one could diagonalize out of the class of all possible real numbers, that would be a paradox that would undermine the assumption. But that is not possible, for (in spite of there not being more of them than there are natural numbers) there is no way to order them all in a sequence. And in that light we must once more revise our judgment about the indefinite extensibility of the reals in Russell’s sense: given the class of all possible real numbers, we cannot define a new term also having the property of being a real number while being non-equal with each element of the class. There might be some mismatches between truth conditions and the satisfaction of view-from-nowhere truth conditions arising from some sentences that make reference to the class of all possible real numbers. Let’s say that we define a classical semantics for a language about real analysis by using the class of all possible real numbers as our domain of quantification (the road not traveled). Can we trust that all the true sentences of this language have their view-from-nowhere truth conditions satisfied, and that all the false sentences do not? There is self-reference that casts some doubt on that: enumerate the sentences of that language and define the real  −n number r := ∞ n=1 tn · 2 , where tn equals 1 if the nth sentence is true and 0 if it is not. The definition of r is sensitive to each truth value of the language, while on the other hand, the truth value of many sentences of the language depend on the properties of r. The first thing to say about this is of course that such mismatches concerning the real numbers do not constitute a real paradox any more than the mismatches concerning the liar do. Hence, even if they exist, they do not provide a reason to think that there is no such thing as all possible real numbers, or that there aren’t many things we can say about them without mismatches. But even so, the (epistemic) possibility of such mismatches must be taken into account when considering whether there are better alternatives to the language convention described above in Chaps. 7, 8, and 9. Setting aside such possible mismatches, there is another challenge for someone who would want to argue for a replacement of my trivalent language that fails to make reference to all possible real numbers with a classical language that does. An important advantage of my system is that it gives us a solid grasp of the interrelationships between the real numbers. We know, for instance, that the real numbers are closed under addition, multiplication, and exponentiation, and that a convergent sequence of real numbers converges towards a real number. Further, the fact that when

230

10.1 All Possible Real Numbers

we introduce a new form of term that takes other terms as “input” to produce yet another term, we commit ourselves to also letting that generator of terms be applicable to any other terms that may be introduced into the language later, thus ensuring that those closure properties will hold under any expansion of the language. In contrast, what do we know about the structure of the class of all possible real numbers? Can we achieve the kind of epistemic access to such facts that would be required to found on that basis a mathematics that has a sufficiently high degree of rigor? I do not rule out that it might be done; but I don’t see how. Hence the more cautious approach of this book.6 A critic might say that the concerns about what I haven’t done also apply to what I have done, and I will conclude this section by explaining and addressing that worry. In Chap. 7, we gave the semantics for the strong conditional in a language for a sentence by quantifying over all simple variants of that language. If the language has infinitely many cut-off sentences,7 then the class of all possible simple variants of that language has strong similarities to the class of all possible real numbers: in both cases, it is possible to diagonalize out of any sequence of elements. Let us consider, in reverse order, the two worries that have just been discussed for real numbers in the context of simple variants. We have as little grasp of the interrelationships of the simple variants as we have of the interrelationships of all possible real numbers. But in this case, it does not matter. Facts about whether there is a simple variant that relates in such-and-such ways to some class or sequence of other simple variants has played no rôle in the construction of our system. We have only had to look at one (generic) simple variant at a time and determine whether its making of an antecedent of a given strong conditional true implies that it itself also made the consequent true. There is nothing corresponding to, e.g., the importance of there being a real number that is the supremum of some class or sequence of other real numbers. Are there mismatches in sentences about all the simple variants of some languages? I can’t rule it out. For what it’s worth, if there are, I find it very unlikely that they affect any of the theorems of Chaps. 8 and 9. But again, I can’t com6 Is

every possible real number equal to a real number in a possible human language? Or are there possible alien languages with real numbers that have no counterpart in any possible human language? I am not sure. If the former, then theorems about R in our system are, in a sense, about all possible real numbers, and not just about the possible real numbers that can be “captured” by humans. And then, what I am expressing doubt about here (as also earlier, namely when discussing Feferman’s interpretation of Weyl’s predicativism in Sect. 4.4), namely whether the class of all possible real numbers has nice closure properties, is in fact the case. But that is then in virtue of open-ended languages like the one I have constructed. It is not something that we could reasonably just have assumed without it. Such a language framework can bring order to what one might otherwise fear to be an amorphous cloud of possibility. 7 This term was defined in Sect. 7.3.1.

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pletely rule it out. This means that there is, unfortunately, some uncertainty about the foundations of the kind of mathematics that has been developed here; one source of uncertainty is that we do not have nominalistic proofs of the meta-theorems of Sect. 7.5, and we have just identified another. I hope that these issues will be cleared up at some point in the future.

10.2

Modal Metaphysics

The merely possible does a job in my account of mathematics that is comparable to what abstract objects do in a platonic account. Even though my commitment to what is possible is smaller than the platonic classical mathematician’s commitment to what is actual, I nevertheless need an infinity of possible objects, and one may therefore doubt whether the desired ontological parsimony has really been achieved. According to Lewis’s (1986) modal realism, every possible object exists. For a world to be possible is for that world to exist in the same concrete way that the actual world exists—different possible worlds are merely causally isolated from each other. There is nothing special about the actual world: “actual” is an indexical like “here” and refers to the possible world inhabited by the utterer of that word. It is not the label of something that is more real than the non-actual. Therefore, if it is possible that an object with a certain property exists, then an object with that property exists. If Lewis is right, then my commitment to potential infinity is, ipso facto, a commitment to actual infinity (in the Aristotelian sense of “actual” used in conjunction with “infinity”, as opposed to the Lewisian sense). Fortunately, there are other positions on the market that avoid that commitment. According to Fine’s (1994) essentialism, for example, modal facts are facts about the essences of things. For instance, Kit Fine is necessarily human (if he exists) because humanity is part of the essence of Kit Fine. Therefore, at least some modal facts are facts about actual things, so they do not require an enlarged ontology. Also, according to dispositionalism,8 modal facts are facts about the dispositions of things. For instance, it is possible for a glass to break because glass has a disposition to break. Hence again, modal facts are facts about actually existing things (assuming that we grant dispositions’ existence). I do not believe that there are such things as essences, so I cannot support my case on Fine’s theory. I think that dispositionalism is closer to the mark. However, that theory is best suited to accounting for what is possible for specific things, like 8 See

Wang (2013) and references therein.

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a glass, and is therefore not relevant to our present concerns. That is, dispositionalism does not seem able to explain in virtue of what global modal claims, such as for instance the claim that there can only be finitely many things, are true or false (Cameron 2008, 273; Wang 2013, 113). I will therefore outline some views on modal metaphysics that, while related to dispositionalism in spirit, do not fall under that heading. This brief discussion will deal separately with three different types of modality, the last of which will be absolute, or metaphysical, modality and the second of which will be the nomological kind of modality concerned with what is possible within the confines of existing laws of nature. Connecting back to a previous discussion, the first type I will discuss is the (also nomological) kind of modality concerned with what it is possible to assert appropriately within the confines of existing language conventions. Just as the possibility of the glass shattering is a fact about the actual constitution, shape, and structure of the glass, rather than a fact about a non-actual already-shattered counterpart of the glass, it seems reasonable to me to claim that the, say, truth of a sentence of the form ∀nφ is a fact about the actually existing N language conventions that are relevant to that sentence, rather than a fact about non-actual counterparts of φJn 0K, φJn 0K, φJn 0K, . . . existing in another possible world. All the information needed to determine the truth value has to be, in some sense, contained in the actually existing convention, because if we have to rely on other possible worlds, then there are also possible worlds where some of the sentences φJn 0K, φJn 0K, φJn 0K, . . . are false according to the conventions in those worlds. As discussed in Sect. 6.4 regarding rule-following, everything needed to make our language work has to be located in our world. Back then, my specific point was that, even if there is a platonic realm of mathematical objects, our concrete conventions have to be rich enough to decide between the platonic + and the platonic ⊕; but that transfers directly to the point that our actual conventions have to be rich enough to decide the truth values of the sentences of our language, whether or not there are alternative existing possible worlds. And as I mentioned in Sect. 6.4, I believe that human beings’ intensional understanding of rules does have the required richness. Understanding a rule is not a disposition (as demonstrated by Wittgenstein and Kripke); but understanding and dispositions both, independently, play rôles in the constitution of modality. What has been said so far does, however, not suffice for sentences that contain the strong conditional or depend on such sentences. That is because the semantics of ⇒ is given by quantification over possible languages, i.e., languages that are typically constituted by non-actual conventions. We cannot locate facts about them in facts about actual conventions, but have to move to a different kind of modality and

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the facts that constitute that kind of modality facts. It might be the case that metaphysical possibility would work for this purpose, but I am not sufficiently certain about what metaphysical modality is to be willing to base any positive claims on it (though I will return to metaphysical modality below, with another aim). Instead, I make use of the type of modality that has to do with what is possible according to the laws of nature. Let me first clarify what I mean by “laws of nature” before explaining the job I have in mind for them. If everything, including minds, is ultimately physical, then “laws of nature” can just be understood in the usual sense. But I want to stay neutral on that issue. If not, then there are presumably a separate set of laws that govern the processes that lead to the creation of minds and individual thoughts in those minds. In that case, let me stipulate that those laws are also to be counted among the laws of nature, even if that makes for a non-standard use of that term. I am quite certain that the laws of nature exist in some reasonable sense of that word. I do not know in what form they exist, i.e., whether a law is a single entity or whether it is multiply instantiated as tropes or something else. But that is irrelevant for present purposes, as long as they exist. Just like our actual understanding of existing conventions gives rise to modal facts about what is possible within the bounds of those conventions, I believe that the laws of nature give rise to a type of modal facts that require no further ontological support. (And I suspect that facts about dispositions are just a special case of this kind of fact, so dispostionalism is partly correct but not a sufficiently general theory.) Thus, a language is possible if it is a fact, about the laws of nature, that a subject could be created who could commit to the conventions that constitute that language. The term “possible language” in Chap. 7 should be (re)interpreted in this way.9 (Again, maybe metaphysical possibility could be used instead, but I feel more confident about this nomological type of modality, and believe that it suffices for the job at hand in the sense that the claims I have made about possible languages come out true under this precification of “possible”.) Finally, let me discuss metaphysical possibility in order to come full circle and relate to one of the themes with which this monograph opened. There is hardly anything in connection with metaphysical modality that I would be willing to take a stand on. But I can indulge in a little speculation. 9 Reliance on merely possible languages is a point of similarity to a competing nominalistic proposal, namely that of Chihara (1990, 2004). One important difference is that I do not have to assume that there are more possible linguistic entities than there are natural numbers, like he does (1990, Sect. 3.3).

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Since the actual world exists (a baffling fact!), it is natural to suppose that there is some kind of mechanism that is responsible for the creation of it. That mechanism might be deterministic, so that there is an explanation for why the world was created in exactly the way that it was. Or it might be indeterministic, so that there was some randomness involved. A third option is that there is no mechanism. After all, the assumption of a mechanism seems to come with a looming threat of an infinite regress, and one might want to cut off that regress at the origin by declaring that it is a brute fact that, say, the Big Bang had the parameters that it did. I don’t know which of these options is correct. But each of them is consistent with the range of possibilities for what could have happened instead being narrow. In the first case, the alternative causes that might have created the world differently might not have been capable of being significantly different from the actual causes. In the second case, it might be that the randomness was contained to a delimited outcome space of possibilities. And even if the creation of the actual world is a brute fact, it is not clear that such a fact could have been replaced with any other brute fact that one might imagine (or that is consistent). If we cannot rule out that that range is narrow—and I do not see how we could— then we cannot rule out that, for every way the world could have been created, it would have been created with only finitely many entities in it. Here I have been assuming that something is possible simpliciter if and only if a world could have been created in which that something would have been the case. Or, to take indeterminism into account: something is possible simpliciter iff a world could have been created in which it is possible according to the laws of that world that that something would have been the case. That seems correct to me. But even if there is some sense in which something could be possible, even though there is no possible world history that leads to it, it is not clear that this would involve worlds containing an actual infinity of entities. A common intuition is that everything that is not inconsistent is possible (and that actual infinity is not inconsistent); but I am not convinced that the formal underivability of a contradiction from a given sentence is sufficient for that sentence to describe something possible.10 The possibility of actual infinity is highly speculative, and that is my point: while I do not think that classical mathematics is dependent on an actually existing platonic realm of mathematical entities, but could be justified on the mere possibility of actual infinity, I think that even this seemingly modest requirement is too speculative to assume satisfied. If we want our mathematics, which has often been thought absolutely cer10 In

Hansen (2017) I provide examples involving reverse supertasks in which the line between the consistent and the inconsistent seems to be too arbitrary to be the line between the possible and the impossible.

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tain, to have a foundation that actually lives up to just a reasonably high standard of certainty, we need to base it on just potential infinity. That has been the motivation for this study.

10.3

Conclusion

In this monograph, I have developed semantic conventionalism based on four major premises. The first is that the “universe” has not been so kind as to make special, sui generis, resources available to the human project of describing and explaining that same universe through mathematics; there is no convenient platonic hierarchy of sets. The second is that there aren’t, and cannot be, any actual infinities. These first two premises are both negative, and make the project of providing a sufficiently strong foundation for mathematics difficult. The remaining two, however, are positive. The third premise is that there are facts about the merely possible, and that an infinity of such facts can be implicitly contained in something finite: specifically, in finite semantic conventions, and in the laws that determine which semantic conventions are possible. The fourth is the conventionalist solution to the semantic paradoxes explained in Chap. 5, which implies an exceptionally high degree of freedom when it comes to designing semantic conventions. Those premises led to the conclusion that classical mathematics is illegitimate (except if understood purely formally), and that the best way to reconstruct mathematics is as a language with stipulated truth conditions. In addition, in Chaps. 7–9, I developed the most useful mathematical-conventional system that I was able to come up with. In summary, the main features of that system are as follows: • The language is trivalent; its true and false sentences are trustworthy; and a strong conditional ensures that more can be expressed with true sentences than in a purely Kripkean convention. • One can infer to and from a sentence saying that something is an element of a given class exactly as one would expect from the membership criterion of that class; modus ponens is valid; and conditional proofs are valid subject to a mild restriction. • Arithmetic and other parts of mathematics that are classically countable are fully reconstructed. • Individual real numbers function essentially as in classical mathematics; and even though there are not more of them than there are natural numbers, we will

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•

•

•

10.3 Conclusion

typically find what we need, because, roughly speaking, we can describe what we need and describing it makes it exist. Some attempts at quantifying over all real numbers work out well: for instance, when they are used to express that any two real numbers’ sum is also a real number. Other attempts fail because of terms whose real-number status is undefined. This affects the use of the concept of continuity. However, in such cases, classical mathematics can be closely approximated through the use of the concept of piecewise uniform continuity and of “skeletons” of determinate sets that are dense in the real numbers. There is a great deal of flexibility with regard to which such determinate sets to employ. Differentiation can be performed as in classical mathematics, and the result can be “trusted” for values that are either in the inner of an interval for which the function is uniformly differentiable, or included in the determinate “skeleton” set. The Darboux/Riemann integral is reconstructed as in classical mathematics, and the Fundamental Theorem of Calculus holds. The Lebesgue integral can only be partially reconstructed. It is possible to diagonalize on a sequence of real numbers both in the language and externally on the whole language. The incompleteness of fixed languages that is a consequence of the latter possibility is accommodated by the use of an open-ended language, and by only considering a proved truth to be a theorem if its truth is resilient to any extension of the language.

Can this approach be considered a success? I would tentatively say that a necessary condition for success is that it delivers a mathematics sufficient for all the needs of empirical scientists. I suspect that if you count all of the mathematics that can be reconstructed by following my example, and not just the mathematics I have actually reconstructed, this condition is satisfied—but I am not sure. Being sure about that would require an overview of all the mathematics-consuming empirical sciences (and all the mathematics they consume) that I simply do not have. What I do have is the privilege of being part of an academic community that values and incentivizes dissent. So hopefully, if there is a counterexample to my optimistic belief, my putting this proposal in the public record will eventually result in someone else finding one. If an apparent counterexample is found, it should of course be subjected to some scrutiny. First, one should investigate whether the piece of classical mathematics that seems to resist reconstruction cannot, after all, be shoehorned into my system with sufficient creativity. There are several examples above where things had to be done differently from how they are done in classical mathematics, in order to end up

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with something that was nevertheless essentially like classical mathematics. That an idea for how to similarly handle other difficult parts of classical mathematics does not immediately present itself does not necessarily indicate that it cannot be done. However, if it really cannot, the second step should be to consider whether the piece of classical mathematics that cannot be reconstructed is really needed by empirical scientists. That they actually use a given piece of mathematics does not definitively settle that question, because empirical scientists are not in the habit of ensuring that they use the simplest and least metaphysically demanding piece of mathematics that suffices for their purpose. Why would they, when all the classical mathematics ever developed seems to be available for free? However, I would care, and I would very carefully check whether the fancy mathematics they are using could be replaced with a cheaper model, perhaps at the cost of some elegance and convenience. If that should also fail, the third step would be to consider alternatives to the framework of Chap. 7. That framework is not an integral part of the philosophical position of semantic conventionalism, but merely one possible convention; and although it is the best convention I have been able to come up with, there might well be better ones. In fact, I strongly suspect that there are, and hope that at some point, someone else will formulate one. Thus, an essential piece of classical mathematics that cannot be reconstructed on top of the specific convention that I have focused on in the last half of this book is not necessarily a counterexample to semantic conventionalism. If all of that fails, and only then, would I be willing to reconsider the two negative premises mentioned above: that there are no platonic mathematical objects, and that there can be no actual infinities.

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Index of Symbols

·, 115 ∗, 122  , 197, 219 , 151 ¬, 114, 147  , 150 \, 163 , 154 ?, 199 #, 166 0, 151 , 115 ⊥, 115 ∃, 114, 148–150 ∀, 148–150 ∈, 147, 151 λ., 171 ω, 13 ℵ, 20 →, 115, 148 →, 170 ↔, 115, 148 →,  169 0 − → , 220 − → 0 , 220 c N → R, 179 ⇒, 126–135

≡, 149 =, 96, 149, 153 =, 166 H =, 153 N = , 179 N→R =, 160 Q =, 171 R =, 158 Z

∨, 114, 148 ∧, 100, 115, 148 ∪, 163 ∩, 163 ⊂, 164