Ways of Proof Theory 9783110324907, 9783110324525

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Ralf Schindler (Ed.) Ways of Proof Theory

ontos mathematical logic edited by Wolfram Pohlers, Thomas Scanlon, Ernest Schimmerling Ralf Schindler, Helmut Schwichtenberg Volume 2

Ralf Schindler (Ed.)

Ways of Proof Theory

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Wolfram Pohlers

Preface

Wolfram Pohlers is one of the leading researchers in the proof theory of ordinal analysis. On the occasion of his retirement the Institut f¨ur Mathematische Logik und Grundlagenforschung of the University of M¨unster organized a colloquium and a workshop which took place July 17 – 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The organizer of the colloquium and workshop gratefully acknowledges financial support from the University of M¨unster, the DVMLG (the German Logic Society), and Springer–Verlag. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory. We thank Martina Pfeifer and Jan–Carl Stegert for helping us organize the colloquium and workshop and produce this volume. We dedicate this volume to Wolfram Pohlers, who has always been an inspiring mathematician, an extraordinary colleague, and a great friend. This book is an attempt to tell him that we are well aware of how much we owe him. M¨unster, June 01, 2010

Ralf Schindler

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III Wolfram Pohlers — Life and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Justus Diller The Proof Theory of Classical and Constructive Inductive Definitions. A Forty Year Saga, 1968 – 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solomon Feferman A New Approach to Predicative Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . Arnon Avron Characterising Definable Search Problems in Bounded Arithmetic via Proof Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arnold Beckmann and Samuel R. Buss

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On Topological Models of GLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Lev Beklemishev, Guram Bezhanishvili, and Thomas Icard Program Extraction via Typed Realisability for Induction and Coinduction . . 157 Ulrich Berger and Monika Seisenberger Another Reduction of Classical IDν to Constructive IDiν . . . . . . . . . . . . . . . . 183 Wilfried Buchholz Elementary Constructive Operational Set Theory . . . . . . . . . . . . . . . . . . . . . . 199 Andrea Cantini and Laura Crosilla Functional Interpretations of Classical Systems . . . . . . . . . . . . . . . . . . . . . . . . 241 Justus Diller Towards a Formal Theory of Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Simon Huber, Basil A. Kar´adais, and Helmut Schwichtenberg Σ11 Choice in a Theory of Sets and Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Gerhard J¨ager and J¨urg Kr¨ahenb¨uhl

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An Extended Predicative Definition of the Mahlo Universe . . . . . . . . . . . . . . 315 Reinhard Kahle and Anton Setzer ITTMs with Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Robert S. Lubarsky Logspace without Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Isabel Oitavem Investigations of Subsystems of Second Order Arithmetic and Set Theory in Strength between Π11 -CA and Δ12 -CA + BI: Part I . . . . . . . . . . . . . . . . . . 363 Michael Rathjen Weak Theories of Operations and Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Thomas Strahm Computing Bounds from Arithmetical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . 469 Stanley S. Wainer

Wolfram Pohlers — Life and Work Justus Diller Institut f¨ur Mathematische Logik und Grundlagenforschung Westf¨alische Wilhelms-Universit¨at M¨unster Einsteinstraße 62, 48149 M¨unster, Germany [email protected]

Coming from Munich in 1985, Wolfram Pohlers followed a call to the department of mathematics of the University of M¨unster. From that time to his retirement in the summer of 2008, he occupied the chair of Heinrich Scholz, the first chair in mathematical logic and foundational research in German speaking Central Europe. During this period, he represented the Bavarian element in our department, by accent as well as by temper. For somebody born in Leipzig to a Saxonian father and a Norwegian mother, that may seem somewhat surprising. He proved his character by energetic engagement in many fields, by stubbornness — which is said to be characteristic for Westfalians, too — showing a definite conservative tendency over the years, and by an ability to compromise resulting out of his respect for his partners in negotiations — which is not a matter of course among mathematicians. We review his activities in administration, scientific organization, and science in due brevity. Wolfram Pohlers served on several committees of our department. He was our dean for two years, from 1990 to 1992, and he was our first dean to hold that job for the full period of two consecutive years. Until then, we had not requested of each other to carry the dean’s burden for so long. In his period of office, he, among other things, resuscitated the deans’ conference of our alma mater. Since then, he represented his colleagues in the senate of our university for about 14 years, and he was speaker of professors in the senate for a considerable number of years. Such a position naturally brings about a close, but also time consuming cooperation with the university administration. It was obviously a consequence of that positive cooperation that the rector of the university of M¨unster, Prof. Ursula Nelles, seized the opportunity to address the conference in honor of Pohlers’ retirement in person. Membership in the senate brought with it tasks of which most of us have never heard, for example in university sports. The central unit university sport is a large organization which moves considerable amounts of money. The steering of this unit lies in the hands of a steering committee, and Wolfram Pohlers presided over this committee for many years. The last meeting over which he presided must have been a moving farewell party. On the other hand, it seems almost a matter

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of course that for many years he also worked on the university’s central IV– (information processing–) committee. Even more important tasks come up, when the university looks for a new chancellor or a new rector. Such a person is not found spontaneously; rather, she or he is looked for by a search committee. If we add to these the committee for the delicate preselection of candidates for the university council — Hochschulrat, a product of recent legislation of our state — we have the impression that within his last five to eight years on our senate, Wolfram Pohlers has been on every such committee of our university. Even a science with a small faculty like mathematical logic has its own national scientific union. Wolfram Pohlers has served many years on the board of the DVMLG, the German logic union. He is also a member of the editorial staff of the two journals on mathematical logic that appear in Germany, the Mathematical Logic Quarterly and the Archive for Mathematical Logic. Of the latter he was editor–in–chief until 2008 when he passed that job on to our colleague Ralf Schindler. In recent years, he is also active as scientific area editor for the Journal of Applied Logic which, following trends of our time, appears in Singapore. A noticeable event was the European summer meeting 2002 of the ASL, the association of symbolic logic, which Pohlers organized in M¨unster, together with the president of the DVMLG, Professor Koepke from Bonn. With more than 200 participants from all over the world, this summer meeting was a big conference, considering the standards of our department in those days. Our Institute as a whole was for quite a while pretty busy with the preparation and implementation of the conference. With such determined engagement of manpower and resources, it proved to be an advantage that once upon a time Wolfram Pohlers had been officer of our federal army. Smaller workshops on proof theory were also organized under his supervision. To a large part, these fruitful activities would not have taken place in M¨unster, if, in 1995, he had followed a call to the university of Vienna. After some inner struggle he turned down this honourable offer. The “old” logic institute of the 1990’s maintains deeply felt gratitude to him for his staying in M¨unster. All this is only the outer framework for his central activity, which is research and teaching of mathematical logic. Wolfram Pohlers graduated from high school in Munich in 1964 where, after two years of military service, he began his studies of mathematics at the Ludwig-Maximilians University. He married his wife Renate in 1970, and passed his diploma in mathematics in March 1971. The day after he had completed his diploma, he started work as scientific assistant with Kurt Sch¨utte with whom he completed his dissertation in mathematical logic in 1973. The area of research from which the topic of his dissertation was taken was to become his research field for all of his career: it is the proof–theoretic field of ordinal analysis, a central topic in the foundations of mathematics. We cast a quick glance at what

Wolfram Pohlers — Life and Work

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ordinal analysis is about, and what Wolfram Pohlers has contributed to this field in the last 39 years. G¨odel’s second incompleteness theorem of 1931 showed that the original goal of Hilbert’s program was unattainable: a mathematically relevant theory like Peano Arithmetic (elementary number theory) PA cannot prove its own consistency. Already in 1936, however, Gerhard Gentzen found a way out of this dilemma. He proved the consistency of PA employing a transfinite induction up to the ordinal number 0 in an otherwise finitary, completely combinatorial proof. Here, 0 is the limit of iterated ω–powers, the first fixed point of the function ω α , i.e. the smallest ordinal α such that ω α = α. (ω designates the first transfinite ordinal.) By his proof, he had isolated transfinite induction up to 0 as the transfinite feature which transcends the means of PA. He had thus proved the consistency of PA by constructive, though not finitary methods. In short: he had shown 0 to be the proof–theoretic ordinal of PA. By this proof, Gentzen had started a revised version of Hilbert’s program which until today plays a central role in the foundations of mathematics. In the 1950’s and 1960’s, Kurt Sch¨utte, Gaisi Takeuti, and Solomon Feferman began to tackle stronger mathematical systems. Feferman and Sch¨utte worked in particular on predicative analysis which allows quantification over sets of natural numbers, however, only in a strictly constructive, so–called predicative way. They proved the first strongly critical ordinal Γ0 , the first common fixed point of the Veblen hierarchy, to be the proof–theoretic ordinal of predicative analysis. After this success, impredicative systems of classical mathematics moved into focus. These were, on the one hand, theories of inductive definitions. In this area, Howard 1972 proved the so–called Howard–Bachmann ordinal to be the proof– theoretic ordinal for one inductive definition, and Pohlers 1978 made an ordinal analysis of iterated inductive definitions. On the other hand, there were subsystems of classical analysis, i.e. of second order number theory. To the ordinal analysis of these, Pohlers’ dissertation of 1973 made an important contribution. The study of both of these areas of research and their interrelations came to some completion, when, in 1980, the authors Buchholz, Feferman, Pohlers, and Sieg published the volume “Iterated inductive definitions and subsystems of analysis: Recent proof–theoretical studies” in the Lecture notes in mathematics. It was coordinated by Solomon Feferman and based on the habilitation theses of Pohlers and Bucholz and the PhD thesis of Sieg. In this volume, Pohlers developed his method of local predicativity which presented an essential progress in the ordinal analysis of stronger and stronger impredicative systems. As Gentzen succeeded in isolating the transfinite element in first order number theory, Pohlers’ method of local predicativity allows the isolation of the impredicative elements of strong theories. His method simplifies the still troublesome computations of the corresponding proof theoretic ordinals considerably.

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Given fresh impetus by this new method, proof theorists now also attacked systems of set theory. While in 1950, Heinz Bachmann was the first to make use of an uncountable number, i.e. ℵ1 , to denote countable ordinals, since 1979 hardly any large cardinal was safe from the grip of ordinal analysts. The hunt was to a large part led by students of Sch¨utte and Pohlers. Gerhard J¨ager in Munich, now Bern, was the first to make use of inaccessible cardinals, and around 1990, Michael Rathjen in M¨unster, now Leeds, proceeded to Mahlo and other large cardinals. Rathjen finally succeeded in an ordinal analysis of Π12 –analysis, a theory much stronger than methods used by classical mathematicians outside measure theory. The ambition of the M¨unster school of proof theory — that is an established term meanwhile — does not only go to still larger, still more complex systems. It also aims at restructuring the already analyzed terrain, at including constructive theories, at applications to other areas of mathematics, computer science, and logic. Michael Rathjen has meanwhile included constructive systems like Martin–L¨of theory and constructive set theory in his proof theoretic analysis. Andreas Weiermann, now in Gent, discovered deep connections between ordinal analysis and pure mathematics. For instance, he proved stunning results in proof theory by methods of analytic number theory. Also lower complexities were analyzed. Theories relevant in bounded arithmetic satisfy the conditions of G¨odel’s second incompleteness theorem only in a restricted sense, and their classical proof theoretic ordinal is ω 2 in all relevant cases. Arnold Beckmann, now in Swansea, developed so–called dynamic ordinals which allow to distinguish between the proof theoretic strengths of some of these theories. They are not ordinals in the classical sense, they may be viewed as cloudy objects assembled around ω. It would be a triumph for proof theory, if they could be used to separate the right theories of bounded arithmetic according to their proof theoretic strength. For that would shed light on the P/NP–problem, the fundamental problem of theoretical computer science, and it would yield the desired answer P = N P . Finally, Michael M¨ollerfeld developed a recursion theory of Π12 -analysis. Thus, after set theory, constructivism, and complexity theory, one more field of mathematical logic could be brought into contact with the proof theoretic subject of ordinal analysis. Wolfram Pohlers accompanied this drive to new frontiers in many ways, in recent years in particular with systematizing publications. These include his “Proof Theory, An Introduction”, but even more so his thorough survey chapter “Subsystems of set theory and second order number theory” in the Handbook of Proof Theory and his recent book “Proof Theory: The First Step into Impredicativity”. Adding up, Wolfram Pohlers as a researcher has contributed substantially to the field of ordinal analysis; as an academic teacher he has founded the M¨unster school of proof theory. To keep this group and other friends in contact, the Pohlers cou-

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ple regularly arranges a summer party at their home in Nienberge. Only on rare occasions at these parties, an old double bass comes into appearance which many years ago helped Wolfram Pohlers finance his student days in Munich and which, during a rafting tour down the Isar, is said to have gone down the river part of the way on its own. Well, out of his school of proof theory, there emerged a number of scientists who work in several countries and have produced remarkable results. Ordinal analysis has become more complex under Pohlers’ influence, but it is also rather more vital and more applicable to neighbouring fields than could have been expected thirty years ago.

The Proof Theory of Classical and Constructive Inductive Definitions. A Forty Year Saga, 1968 – 2008 Solomon Feferman∗ Department of Mathematics Stanford University Stanford, CA 94305-2125, USA [email protected]

1 Pohlers and The Problem I first met Wolfram Pohlers at a workshop on proof theory organized by Walter Felscher that was held in T¨ubingen in early April, 1973. Among others at that workshop relevant to the work surveyed here were Kurt Sch¨utte, Wolfram’s teacher in Munich, and Wolfram’s fellow student Wilfried Buchholz. This is not meant to slight in the least the many other fine logicians who participated there.1 In T¨ubingen I gave a couple of survey lectures on results and problems in proof theory that had been occupying much of my attention during the previous decade. The following was the central problem that I emphasized there: The need for an ordinally informative, conceptually clear, proof-theoretic reduction of classical theories of iterated arithmetical inductive definitions to corresponding constructive systems. ∗ This

is a somewhat revised text of a lecture that I gave for a general audience at the PohlersFest, M¨unster, 18 July 2008 in honor of Wolfram Pohlers, on the occasion of his retirement from the Institute for Mathematical Logic at the University of M¨unster. Wolfram was an invited participant at a conference in my honor at Stanford in 1998, and it was a pleasure, in reciprocation, to help celebrate his great contributions as a researcher, teacher and expositor. In my lecture I took special note of the fact that the culmination of Wolfram’s expository work with his long awaited Proof Theory text was then in the final stages of production; it has since appeared as Pohlers (2009). In that connection, one should mention the many fine expositions of proof theory that he had previously published, including Pohlers (1987, 1989, 1992, and 1998). 1 That meeting was organized by Walter Felscher under the sponsorship of the Volkswagen Stiftung; there were no published proceedings. It is Pohlers’ recollection that besides him and Felscher, of course, the audience included Wilfried Buchholz, Justus Diller, Ulrich Felgner, Wolfgang Maas, Gert M¨uller, Helmut Pfeiffer, Kurt Sch¨utte and Helmut Schwichtenberg. By the way, Felscher passed away in the year 2000.

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As will be explained below, meeting that need would be significant for the then ongoing efforts at establishing the constructive foundation for and proof-theoretic ordinal analysis of certain impredicative subsystems of classical analysis. I also spoke in T¨ubingen about possible methods to tackle the central problem, including both cut-elimination applied to (prima-facie) uncountably infinite derivations and functional interpretation on the one hand, and the use of naturally developed systems of ordinal notation on the other. I recall that my wife and I had driven to T¨ubingen that morning from Oberwolfach after an unusually short night’s sleep, and that I was going on pure adrenalin, so that my lectures were particularly intense. Perhaps this, in addition to the intrinsic interest of the problems that I raised, contributed to Wolfram’s excited interest in them. Within a year or so he made the first breakthrough in this area (Pohlers 1975), which was to become the core of his Habilitationsschrift with Professor Sch¨utte (Pohlers 1977). The 1975 breathrough was the start of a five year sustained effort in developing a variety of approaches to the above problem by Wolfram Pohlers, Wilfried Buchholz and my student Wilfried Sieg. The results of that work were jointly reported in the Lecture Notes in Mathematics volume 897, Iterated Inductive Definitions and Subsystems of Analysis. Recent proof-theoretical studies (Buchholz et al. 1981). In the next section I will give a brief review of what led to posing the above problem in view of several results by Harvey Friedman, William Tait and me at the 1968 Buffalo conference on intuitionism and proof-theory, with some background from a 1963 seminar on the foundations of analysis led by Georg Kreisel at Stanford in which formal theories of “generalized” inductive definitions (i.e., with arithmetical closure conditions) were first formulated. The goals of proof-theoretic reduction and of proof-theoretic ordinal analysis in one form or another of the relativized Hilbert program (not only for theories of inductive definitions) are here taken at face value, though I have examined both critically; see Feferman (1988, 1993, 2000). In addition to meeting those aims in the problem formulated above are the demands that the solutions be informative and conceptually clear in short, perspicuous. Granted that these are subjective criteria, nevertheless in practice we are able to make reasonably objective judgments of comparison. For example, we greatly valued Sch¨utte’s extension of Gentzen’s cut-elimination theorem for the predicate calculus to “semi-formal” systems with infinitary rules of inference, because it exhibited a natural and canonical role for ordinals as lengths of derivations and bounds of cut-rank (cf. Sch¨utte 1977) in the case of arithmetic and its extensions to ramified analysis. To begin with, the Cantor ordinal ε0 emerged naturally as the upper bound of the lengths of cut-free derivations in the semi-formal system of arithmetic with ω-rule, obtained by eliminating cuts from the (translations into that system of) proofs in Peano Arithmetic PA; by comparison the role of ε0 in Gentzen’s consistency proof of PA still had an ad hoc

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appearance.2 And the determination by Sch¨utte and me in the mid 1960s of Γ0 as the upper bound for the ordinal of predicativity simply fell out of his ordinal analysis of the systems of ramified analysis translated into infinitary rules of inference when one added the condition of autonomy. Incidentally, because of the connection with predicativity, these kinds of proof-theoretical methods due to Sch¨utte — of ordinal analysis via cut-elimination theorems for semi-formal systems with countably infinitary rules of inference — have come to be referred to as predicative. The proof-theoretical work on systems of single and (finitely or transfinitely) iterated arithmetical inductive definitions were the first challenges to obtaining perspicuous ordinal analyses and constructive reductions of impredicative theories. The general problem was both to obtain exact bounds on the provably recursive ordinals and to reduce inductive definitions described ”from above” as the least sets satisfying certain arithmetical closure conditions to those constructively generated “from below”. In the event, the work on these systems took us only a certain way into the impredicative realm, but the method of local predicativity for semi-formal systems with uncountably infinitary rules of inference that Pohlers developed to deal with them turned out to be of wider application. What I want to emphasize in the following is, first of all, that ordinal analysis and constructive reduction are separable goals and that in various cases, each can be done without the other, and, secondly, that the aim to carry these out in ever more perspicuous ways has led to recurrent methodological innovations. The most recent of these is the application of a version of the method of functional interpretation to theories of inductive definitions by Avigad and Towsner (2008), following a long period in which cutelimination for various semi-formal systems of uncountably infinitary derivations had been the dominant method, and which itself evolved methodologically with perspicuity as the driving force. It is not possible in a survey of this length — and at the level of detail dictated by that — to explain or state results in full; for example, I don’t state conservation results that usually accompany theorems on prooftheoretical reduction. Nor is it possible to do justice to all the contributions along the way, let alone all the valuable work on related matters. For example, except for a brief mention in sec. 7 below, I don’t go into the extensive proof-theoretical work on iterated fixed point theories. I hope the interested reader will find this survey useful both as an informative overview and as a point of departure to pursue in more detail not only the topics discussed but also those that are only indicated in passing. Finally, this survey offers an opportunity to remind one of open questions and to raise some interesting new ones.

2 That

role became less mysterious as a result of the work of Buchholz (1997, 2001) explaining Gentzen-style and Takeuti-style reduction steps in infinitary terms.

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2 From 1968 to 1981, with some prehistory In my preface, Feferman (1981), to Buchholz et al. (1981), I traced the developments that led up to that work; in this section I’ll give a brief summary of that material. The consideration of formal systems of “generalized” inductive definitions originated with Georg Kreisel (1963) in a seminar that he led on the foundations of analysis held at Stanford in the summer of 1963.3 Kreisel’s aim there was to assess the constructivity of Spector’s consistency proof of full second-order analysis (Spector 1962) by means of a functional interpretation in the class of so-called bar recursive functionals. The only candidate for a constructive foundation of those functionals would be the hereditarily continuous functionals given by computable representing functions in the sense of (Kleene 1959) or (Kreisel 1959). So Kreisel asked whether the intuitionistic theory of inductive definitions given by monotonic arithmetical closure conditions, denoted ID1 (mon)i below, serves to generate the class of (indices of) representing functions of the bar recursive functionals. Roughly speaking, ID1 (mon), whether classical or intuitionistic, has a predicate PA for each arithmetic A(P, x) (with a placeholder predicate symbol P ) which has been proved to be monotonic in P , together with axioms expressing that PA is the least predicate definable in the system that satisfies the closure condition ∀x(A(P, x) → P (x)). In the event, Kreisel showed that the representing functions for bar recursive functionals of types ≤ 2 can be generated in an ID1 (mon)i but not in general those of type ≥ 3. Because of this negative result, Kreisel did not personally pursue the study of theories of arithmetical inductive definitions any further, but he did suggest consideration of theories of finitely and transfinitely iterated such definitions as well as special cases involving restrictions on the form of the closure conditions A(P, x). For example, those A in which the predicate symbol P has only positive occurrences are readily established to be monotonic in P . And of special interest among such A are those that correspond to the accessible (i.e., well-founded part) of an arithmetical relation. And, finally, paradigmatic for those are the classes of recursive ordinal number classes Oα introduced in Church and Kleene (1936) and continued in Kleene (1938). The corresponding formal systems for α times iterated inductive definitions (α an ordinal) are denoted (in order of decreasing generality) IDα (mon), IDα (pos), IDα (acc) and IDα (O) in both classical and intuitionistic logic, where the restriction to the latter is signalled with a superscript ’i’.4 For limit 3 The

notes for that seminar are assembled in the unpublished volume Seminar on the Foundations of Analysis, Stanford University 1963. Reports, of which only a few mimeographed copies were made; one copy is available in the Mathematical Sciences Library of Stanford University. 4 The positivity requirement has to be modified in the case of intuitionistic systems.

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ordinals λ we shall also be dealing with ID