Set Theory and Syntactic Description 9783110876307, 9789027927040

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SET THEORY AND SYNTACTIC DESCRIPTION

JANUA LINGUARUM STUDIA MEMORIAE N I C O L A I V A N WIJK D E D I C A T A edenda curat

C O R N E L I S H. VAN S C H O O N E V E L D Indiana University

SERIES

MINOR

N R . XXXIV Second Printing

1974

MOUTON THE H A G U E • PARIS

SET THEORY AND SYNTACTIC DESCRIPTION by

WILLIAM S. COOPER University of California

1974

MOUTON THE H A G U E • PARIS

© Copyright 1964 by Mouton A Co., Publishers, The Hague, The Netherlands. No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers.

First printing

1964

Printed in The Netherlands

FOREWORD

These pages will, 1 hope, provide a brief but digestible exposition of the basic mathematical concepts underlying structural descriptions of natural languages. They are written for the reader who may not possess an extensive mathematical background, but who is nonetheless interested in the logical and mathematical foundations of the theory of grammatical description. Since the time of Bloomfield and Sapir, one of the trends of descriptive linguistics has been toward greater rigor, logic, and precision wherever syntax is considered. There has been emphasis on concise notation, clearly stated definitions and postulates, and there has been dissatisfaction with linguistic entities whose logical status is left overly vague. Recently some elegant notational systems for linguistic description have been proposed, whose use gives insight into the theory of grammars, and lends brevity and clarity to their expression. Some of these notations are definitely mathematical in spirit, even if they do not always follow the usual mathematical forms. As an evidence of the new mathematical turn of thought, the phrase "mathematical linguistics" has come into vogue. Yet the linguistic literature seems to contain no clear statement of what sort of mathematics lies at the basis of syntactic description. What existing branch or branches of mathematics are most relevant to syntax? Or must some entirely new branch of mathematics be invented? And if mathematics can be applied to linguistic data, what benefits are to be expected? The chapters to follow survey an area of mathematics which provides mathematical foundations for syntax. The area of mathematics is set theory. No new branch of mathematics seems to be called for. A few standard relations and operations from elementary set theory, together with a few auxiliary operations which can

6

FOREWORD

be interpreted as part of set theory, provide an ample basis for syntactic discussion. Within set theory, a language may be represented by a set of what will be called "sequences". Various other grammatical entities, such as parts of speech, may also be treated as sets of sequences. The notion of a "grammar" of a language has a simple and natural interpretation within set theory it is merely a mathematical definition of the set of sequences which represents the language. Few of the ideas to be presented are original in any fundamental sense. For some decades now, logicians and metamathematicians have been using set theory to describe and discuss their formal languages.1 Logicians usually regard their formal languages as sets of sequences, and the "grammars" with which they describe their languages often have the form of mathematical definitions. Hence our proposal is essentially that these basic ideas be carried over from logic and metamathematics into descriptive linguistics. A few writers have already introduced some of the relevant mathematical notions into the linguistic literature. To mention only two, Bar-Hillel and Chomsky have both introduced set theoretic concepts into their linguistic considerations. The work of BarHillel and his associates is testimony to the fact that mathematical proof can be used to settle complex linguistic issues.2 An important difference between Bar-Hillel's usage of set theory and the usage to be developed here is that our examples will all be concerned with how to formulate descriptions of particular languages or fragments of languages, while Bar-Hillel has been more concerned with comparing various classes, or kinds, of description. Thus, Bar-Hillel's considerations are a step more abstract than ours. Chomsky has used some set-theoretic notation in describing his system of des1

See, for example, Tarski, "The Concept of Truth in Formalized Languages", 1931, reprinted in Logic, Semantics, Metamathematics (London, Oxford University Press, 1956). • See, for example, Y. Bar-Hillel and E. Shamir, "Finite-State Languages: Formal Representations and Adequacy Problems", The Bulletin of the Research Council of Israel, Vol. 8F No. 3 (1960). Bar-Hillel is an authority on set theory, having co-authored an excellent book on the subject: see Y. Bar-Hillel and A. Fraenkel, Foundations of Set Theory (Amsterdam. 1958).

FOREWORD

7

scriptive linguistics, and his concept of a "language" is similar to ours.3 Chomsky's descriptive systems rest upon the concept of the "transformation", and a transformational rule in a Chomskian grammar is meant to be viewed as a sort of rule of derivation. In our system, the transformation is viewed in a slightly different light, and it is presented as only one possible descriptive device among many. 1 am indebted to W. Craig, D. Scott, and S. Lamb, all of the University of California at Berkeley, for helpful conversations; to P. Baxendale and P. Lewis of I.B.M. for many helpful discussions and specific suggestions; to R. B. Lees for some criticism conveyed to me by Miss Baxendale; and to Y. Bar-Hillel for his helpful comments. April 1, 1963 WM. S. COOPER

3

For a paper representative of Chomsky's outlook on grammars, see N. Chomsky, "On Certain Formal Properties of Grammars", Information and Control, 2 (1959), 137-167.

TABLE

OF

CONTENTS

FOREWORD

I . THE ROLE OF SET THEORY

I I . A N EXPOSITION OF SET THEORY

1. 2. 3. 4. 5. 6. 7.

Primitive Constants Non-primitive Constants and Variables. Primitive Notions of Set Theory Class Operations Relations and Functions Numbers Sequences

5

11

17

17 .19 20 21 23 25 25

I I I . SPECIAL OPERATIONS

28

I V . LANGUAGES AND GRAMMARS

33

V . ILLUSTRATIVE GRAMMARS

1. 2. 3. 4. 5. 6. 7.

Immediate Constituents Optional Constituents Repetitive Constituents Discontinuous Constituents Transformations Process Grammars Morphemes

V I . THE SET THEORY THESIS

36

36 39 40 41 42 44 46 50

I

T H E ROLE O F SET T H E O R Y

What are the advantages of building the theory of syntax on mathematical foundations? Let us admit at the start that there are many situations in which mathematical thoroughness is an unnecessary burden. Suppose, for example, that one were to try to teach a student a foreign language by presenting it to him in strict mathematical form, with the utmost detail and rigor. Unless the student is more at home with mathematics than most mathematicians are, this presentation would be bound to distract him from his primary task of learning the speech-habits of the language. But at the other extreme, there are situations where mathematical detail is not only an advantage, it is a necessity. For computer applications such as the machine translation of languages and certain problems in information retrieval, it is necessary that the linguist supply the computer programmers with large quantities of grammatical information, stated unambiguously in minute detail. Certain theoretical issues in linguistics also cannot be approached without a mathematical development of the concepts involved. The situation in linguistics is not much different from physics or some other science: if only a simple intuitive approach is to be used, mathematics is dispensable; but where rigor is needed, so is mathematics. One obvious advantage of mathematical notation is its conciseness, its brevity. But much more important than its brevity is its clarity. Good mathematical notation systems are developed symbol by symbol, by definitions formulated with exacting care. Undefined relations and operations are kept to a tiny minimum, and all other devices are defined in terms of these. Set theory was one of the first systems to receive a truly logical development (in Russell and Whitehead's famous Principia Mathematica), and the logical structure of set theory has been as widely and deeply studied as that

12

THE ROLE OF SET THEORY

of any area of mathematics. To the extent that linguistics is interpreted within set theory, its notation will be as clear and logically satisfying as any system yet devised in mathematics. In view of the great logical rigor obtainable with set theory, there is a substantial hope that set theory might considerably clarify discussion about syntax among linguists who take the time to acquaint themselves with it, and so stimulate clearer technical inter-communication. So far, no notation for syntactic description has yet become generally accepted beyond a few conventions to indicate morphemes, phonemes and the like. By contrast, a fairly standard set-theoretic notation has received acceptance throughout mathematics. The mathematical notation stands available as a common ground for linguists. Moreover, inter-communication among linguists is perhaps not the only kind of communication to consider. The skills of structural linguists are beginning to be sought in the new fields of computer application: Mechanical Translation, Information Retrieval, Voice Recognition, and there are others on the horizon. The development of these applications will require much interdisciplinary effort. Specifically, linguists may have to be able to communicate with computer specialists and electronics technicians. Here some common ground in the form of a standard mathematical notation, while it may not solve all the problems, is better than no common ground at all. As another case in point, linguists have more in common with logicians than is generally realized. The use by linguists of the set-theoretic notation familiar to logicians might hopefully prompt some interdisciplinary communication between these two fields. So for interdisciplinary as well as intradisciplinary communication among linguists, set theory would appear to provide at least as suitable a notation for syntax as any other yet proposed. Some mathematical linguists, notably some in Israel and the USSR, use set theory as their research vehicle as a matter of course, and are already reaping the benefits. However, it cannot yet be said that the use of set theory is truly widespread even among theoretically minded linguists, let alone those with a more empirical orientation. However, the reward to be expected from mathematical notation

THE ROLE OF SET THEORY

13

is not just clarity of communication, but also the availability of deductive power. Mathematical notation is more than a welldefined abbreviation system. With a mathematical notation come axioms, rules of inference, and a stock of theorems. With the aid of these and a little practice, one quickly becomes adept at recognizing when two expressions are equivalent, at reducing a complicated expression to a simpler but equivalent form, at rephrasing an expression in a fashion which circumvents the use of unwanted technical devices, and so forth. The skills of manipulation are somewhat like those taught in high school algebra. In fact, if the special operations to be recommended in chapter 111 are adopted, much of this manipulation will follow some of the very rules of high school algebra. The ability to manipulate set-theoretic expressions can be helpful in a number of situations. Since a grammar is merely a settheoretic definition, the techniques of manipulation can be used to reformulate it in a variety of ways, all of them mathematically equivalent. One possible reason for reformulating a grammar is to simplify it. As a linguist analyzes his data, he may wish to set down his observations using whatever set-theoretic devices first come to mind. Later, his analysis finished, he can consolidate his grammar and reduce it to a simple and elegant form by mathematical manipulation. Another motive for reformulating a grammar might be to compare it against a different grammar of the same language. To pinpoint areas of disagreement between two grammars, one grammar may be manipulated until it is brought as close as possible to the form of the other. Then any disagreements should become analyzable; and if there are none, a proof of equivalence has been effected. There may be practical reasons for reformulating a grammar connected with its intended application - for example a computer programmer might wish to have grammatical information presented to him in some special form which he finds easy to translate into computer operations. Or there may be theoretical motives for reformulation - for example to show that only a special restricted class of descriptive devices is needed to describe a language. And so on. It is this mathematical manipulatability which justifies

14

THE ROLE OF SET THEORY

one in calling the set theory notation a mathematical system rather than just a system of abbreviation. After urging the advantages of set theory, it may be wise to mention what set theory can not be expected to do for linguistics. Set theory cannot settle the old question, "What is a grammatical sentence?" The only assistance set theory can give here is to formulate more precisely the competing definitions of "grammatical sentence". The choice of which criterion of grammaticality is best suited to the purpose at hand, is still left up to the linguist's judgement. Neither can set theory dictate the nature of morphemes, phonemes, or other linguistic entities; however, to the extent that the structure of an entity is posited on grounds of technical convenience, set theory can help by supplying alternative mathematical representations for the entity. After the appropriate mathematical representation for a certain type of linguistic entity has been determined, set theory still does not dictate how the representation is to apply to specific cases - for example, whether "cranberry" is more conveniently treated as one morph or two in English. Although set theory may help clarify the issues involved, set theory of itself specifies no empirical criteria for phoneme-hood, morpheme-hood, word-hood or sentence-hood. More generally, set theory provides no "discovery procedure" for writing grammars. Its function is not to arrive automatically at an analysis for the linguist, but to help him arrive at and present his own analysis - to study it, manipulate it, reformulate it, simplify it, compare it and notice characteristic features of it. It should also be kept in mind that no empirical laws of linguistics can be deduced from the axioms of set theory. Set theory by itself does not state any principles of linguistics except those that are purely logical or mathematical in nature. On the other hand, set theory provides a language in which empirical laws can be stated, once the linguist discovers them. And possibly set theory may be expected to stimulate the discovery of empirical laws by clarifying the concepts involved and allowing the data to be formulated in a manner which brings the laws to the fore. The position of set theory in linguistics is a little like that of mathematics in physics. Perhaps the most fundamental advances of

THE ROLE OF SET THEORY

15

physics occur when it is first discovered how physical concepts may be represented mathematically. The representation of points in time by real numbers, points in space by vectors, and velocity and acceleration by derivative vectors, laid the necessary groundwork for Newton's Laws of Motion. Yet these mathematical representations did not of themselves contain Newton's Laws - they only provided the language in terms of which Newton's Laws could be stated. Notice also that the system of mathematical representation of position, velocity, and so forth, has by itself no predictive power for inferring facts about one part of a system from observations made in another part, nor for predicting the future state of a system when its present state is known. Nevertheless the system of mathematical representation is not without value even by itself, and it is certainly prerequisite to a precise statement of the Laws of Motion, and to their mathematical manipulation. Now set theory provides a system of mathematical representation for descriptive linguistics. It is true that set theory alone can not tell us how to infer facts about one aspect of a natural language when we have observed only some other aspect, nor how to predict the future progress of a presentday language. In this sense, set theory contains no laws of linguistics. But set theory is far from useless nonetheless. It allows one to define the vital concepts of syntax, to describe syntactic phenomena, to compare or reformulate the descriptions, and to manipulate them deductively. As an added benefit, if there are any empirical laws to be discovered, a far-reaching mathematical system such as set theory puts us in a better position to notice them and formulate them. Some systems of linguistic analysis have already been formulated in terms of specialized mathematical notations of a restricted sort. If set theory is regarded merely as an addition to this list, in competition with the rest, the point of using an extremely general formalism such as set theory will have been lost. Set theory is intended to serve a purpose of a different sort from that of these other systems. Previous mathematically-formulated linguistic systems have been designed to provide a restricted mathematical framework suitable only for those specific notions central to the particular system of linguistic analysis. Set theory, by contrast, is intended

16

THE ROLE OF SET THEORY

more as a universal system which does not depend on any particular linguistic notions, but rather provides a foundation broad enough to support the existing systems, and hopefully future systems also. Of course it is natural that a special system of linguistic analysis should be couched in a highly restricted mathematical formalism. And a linguist who finds the system acceptable and is willing to confine himself to it has no need for a less restricted formalism. However, such restricted formalisms are too specialized for the linguist who wishes to combine certain notions from one system with other notions from another, so as to form an eclectic system of his own. Also, for the linguist who is not prepared to confine himself to any combination of existing descriptive devices, but wishes to leave the door open to entirely new descriptive concepts, any prerestricted mathematical formalism will be unacceptable. It has been said that linguistics should not be Procrustean: that a language should not be forced into a preconceived mould; but instead that the descriptive apparatus should be tailored to fit the language being described. If this is so, only a foundational branch of mathematics such as set theory will be found rich enough and flexible enough to provide an adequate basis for syntactic description. To sum up: Set theory is incapable of contributing directly to descriptive linguistics, if by "descriptive linguistics" is meant a store of empirical data, or empirical generalizations about such data. But set theory is capable of contributing indirectly by supplying fundamental mathematical apparatus for a broad (if not all-inclusive) range of descriptive techniques. The usual virtues of precise mathematical definition - i.e., the sharpening of heretofore vague concepts; the clear contrast of subtly inter-related techniques; the avoidance of unfounded disagreements with their resulting factionalism; conciseness; and the possibility of formal manipulation all argue for carrying out linguistic description within a mathematical framework. To find a sufficiently flexible framework, one need look no further than elementary set theory.

II

AN E X P O S I T I O N O F SET T H E O R Y

This chapter is intended as a condensed introduction to a few areas of set theory which relate to descriptive linguistics. The material is simply elementary set theory, whose fundamental operations are well known to every pure mathematician. The presentation is only a sketch. All axioms and proofs will be omitted, but it is hoped that the definitions, sample selection of theorems, and linguistic examples will convey the essential ideas. For the convenience of the reader who may desire further background reading, the notation is fairly standard, and is taken as far as possible from a single reference work.1

1. PRIMITIVE CONSTANTS

In any theory which makes a pretense at rigor, it is necessary that a sharp distinction be made between the undefined terms and those terms which are defined with the aid of the undefined terms. In set theory as it applies to syntactic description, there will be undefined terms of two different sorts. One class of undefined terms will be necessary as part of the technical apparatus itself, while the other class will be definitely linguistic in character. Terms of the first sort will be called primitive technical constants. There will be occasion to discuss them in chapter VI. Undefined terms in the latter class will be called primitive linguistic constants, and they will be said to name primitive elements. The primitive elements may be linguistic entities or abstractions of any kind whatever, and the corresponding set of primitive linguistic constants can be any con1

Namely, P. Suppes' readable textbook, Axiomatic Set Theory (New York, Van Nostrand, 1960).

18

AN EXPOSITION OF SET THEORY

venient set of symbols for the entities. Let us consider three plausible choices for the set of primitive elements: EXAMPLE: One obvious candidate for the set of primitive elements in a discussion of some particular spoken language is the set of phonemes of the language. The usual phoneme symbols would then serve as the primitive linguistic constants.2 If a linguist were to choose the phonemes as his primitive elements, he would presumably intend to forego a treatment of phonology, since it would be awkward at best to define phones in terms of phonemes. On the other hand, he may well intend to discuss the morphemes and morphophonemics, the words, the phrases, and the sentence structures of the language, provided he can manage to define all these in terms of the phonemes. This may not be so difficult as is sometimes thought, when the appropriate mathematical apparatus is at hand. EXAMPLE : But suppose the analysis is to be carried out for a written, rather than a spoken, language. This is the usual situation for computer applications such as Mechanical Translation and Information Retrieval. Here an obvious choice for the primitive elements is the set of all graphemes, or characters, used in the language. With the graphemes as the primitive elements, one would probably wish to let each letter and punctuation mark serve as its own primitive linguistic constant, with special symbols to indicate the blank character, capitalization, etc. For some purposes one wishes to avoid the necessity for a complete analysis. This is the situation when a linguist considers the syntax of a language only at the level of abstraction of morphemes, and leaves aside all consideration of morphophonemics. His primitive elements are then the morphemes whose names serve as the primitive linguistic constants. Other possible choices for the set of primitive (undefined) elements include components, morphophonemes, morphs, lexemes, or words. EXAMPLE:

* Stresses and intonation patterns may also be included, but require special handling to preserve the mathematical fiction of a linear array of entities. For simplicity they will be ignored in this paper.

AN EXPOSITION OF SET THEORY

19

Combinations of primitive elements of different types would be needed in some circumstances - for example one might require both phonemes and graphemes if both the spoken and written aspects of a language were to be described.

2. NON-PRIMITIVE CONSTANTS AND VARIABLES

Besides the primitive linguistic constants, which are to be left undefined, a supply of other constants is needed to name linguistic elements which are defined in the course of the description. We will use upper case English letters, with or without subscripts, for these non-primitive constants, e.g., "A", "N", "N 3 ", "X", "Y x ", etc. The meaning of these symbols remains constant throughout any given description. Also we will freely use various technical constants such as " n " , " + " , "0", braces, and parentheses; the meaning of these technical symbols remains constant not only throughout any given description, but for all descriptions. Occasionally a whole phrase, such as "is a relation" will be regarded as a single technical constant. As variables we use the lower case English letters, with or without subscripts: "a", "n", "x", "yi", etc. Variables are rather sophisticated logical devices, but the reader will not go too far wrong if he compares these variables to the "x's" and "y' s" of high school algebra. However, in high school the variables ranged over numbers, usually the set of all real numbers. Here most variables will range over not numbers but linguistic entities of various sorts. On the few occasions when we will need variables of the high school type, the lower case Greek letters "a" and "P" will be used, and they will be understood to range over the non-negative integers. On all other occasions variables will be lower case English letters ranging over linguistics elements. If any English letters are used as primitive linguistic constants to name, say, certain phonemes, they must be written in some special way which will distinguish them from both the non-primitive constants and the variables. So in this paper, constants denoting phonemes will be italicized; variables will not.

20

AN EXPOSITION OF SET THEORY 3. PRIMITIVE NOTIONS OF SET THEORY

At the basis of set theory is a system of mathematical logic. The commonest system is called "first order predicate calculus". It uses symbols whose intent is crudely approximated by the words "and", "or", "not", "if ... then", "if and only if", "for every ...", and "for some ...". For the sake of readers who may not be acquainted with first order logic, these English approximations will be used in place of logical symbolism in what follows. A concept basic to all of set theory is that of a set (also called class, family, aggregate, or collection). The notion of a set has no definition, but its intuitive sense is that it is simply a collection of elements. The most straight-forward way to specify a set is to list its member elements between braces. For example, the set whose only members are the three English phonemes b, e, and t may be written {b, e, i} or {e, t, b} or in a number of other equivalent ways. A set may have only one member, e.g., {6}. This usage of braces must not be confused with a common linguistic convention, according to which braces indicate that the expression within denotes a morpheme. Primitive elements are regarded as memberless. In addition to these, there is also a special set which contains no members. It is called the null set, and is denoted "0". The zero is an example of a primitive technical constant.3 Another primitive of set theory is the relation of a member to the set in which it belongs. This relation is called the set-membership relation, and is denoted by "e". The expression "x e y" is to be read "x is a member of y", or "y contains x". EXAMPLES: be {b, e, t};

ee{b, e, t}; not e e {b, i}; not e e 0. 3

In some developments of set theory, the null set is nonprimitive, for it can be defined as the only object which has no members. However, here we have other memberless objects, namely the primitive linguistic entities. For this reason, a form of Zermello-Fraenkel set theory has been chosen for which the null set is taken as primitive.

21

AN EXPOSITION OF SET THEORY 4. CLASS OPERATIONS

We have shown how finite sets may be named by listing their members within braces. There is also a more versatile device for specifying sets. Its commonest technical name is abstraction, but the more picturesque name "set-builder" is sometimes heard. A set-builder expression defines a set by specifying the property that an object must have in order to be a member of the set. The symbolism used in a set-builder expression consists of a term followed by a colon followed by a condition involving the variable or variables appearing in the term to the left of the colon, all enclosed in braces. It is the condition to the right of the colon that states the property in question. The formal definition of the set-builder is too involved to state without proper logical notation, 4 but the examples and theorems which follow should convey the idea. In all cases, read "{...:---}" as "the set of all.. ,'s such that ---". EXAMPLES : {x : x = b o r x =

t or

x = e} =

{b, t, e};

{x : x = b} = {b}{x : x e {b, f} and x ^ b} = {t}. THEOREMS: {X : x ^ x } =

0;

{x : x e y} = y. DEFINITION: x n y = { z : z e x and z e y } This defines the intersection operation, n , which (metaphorically speaking) takes two sets and forms the set of all members common to both. EXAMPLES : {b, t, e} N { f , n, b} = {b, {b, t } n «

=

t};

{'};

{b, t } n {e, w} = 0. In the examples, as will be the case in practice, the operation was applied to sets of elements, not to primitive elements themselves. The sample theorems below and for the remainder of the chapter hold for all values of x, y, and z except possibly for primitive elements. * It is given in Suppes. op. cit.

22

AN EXPOSITION OF SET THEORY

THEOREMS: X N X =

X;

xny-ynx; x n 0 = 0. DEFINITION: x + y = { z : z e x o r z e y } This defines the union operation, + , which puts two sets together to form the set of all things contained in either. It is sometimes also called "set-theoretic addition". The symbol " u " is commonly used instead of " + " to distinguish the operation from ordinary arithmetic addition. However, no confusion can arise for us if we remember that the plus sign can only indicate numeric addition if it appears between numbers or numeric variables (Greek letters). In all other cases, plus sign means set-theoretic union. EXAMPLES: {b, /,} + {e, n) = {b, n, t,

e};

{b, t) + {/, n} = {b, t, n}; {b, t} + {t} = {b, i}; {b, t}+0 = {b, t}. THEOREMS: X + X =

X;

x + 0 = x; x + y = y + x. x ~ y = { z : z e x and not z € y }. This defines the set difference operation which takes two sets and forms a set consisting of the first less any members it had in common with the second. DEFINITION:

EXAMPLES: {b, t, e} ~ {t, n} = {b,

e};

{b, t) ~ {b, t, e) = 0. THEOREMS: X ~ x =

x 0 x x

~ ~ ~ ~

0;

0 = x; x = 0; (y + z) = (x ~ y ) n (x ~ z); (y n z) = (x ~ y) + (x ~ z).

DEFINITION : x £ y if and only if x n y = x, This defines the inclusion relation £ , which is to be read "is included in" or "is a subset of".

AN EXPOSITION OF SET THEORY EXAMPLES: {b, t} £

23

{b, t, e}\

{ t } £ {b, t, e}\ {b, t, e) £ {b, t, e}; 0 £ {b, t,e}; not {b, d) £ {b, t, e). THEOREMS: 0 £

X;

x £ x; x £ x + X o y £ x ~ y £ if x £ y

y;

y; x; and y £ z then x £ z.

5. RELATIONS AND FUNCTIONS DEFINITION: =

{x, {x, y } }

This is one of the ways of defining the ordered pair whose left element is x and whose right element is y. It is admittedly somewhat ad hoc, even counter-intuitive. The reader should think of it merely as a set-theoretic structure involving x and y, in which the order of writing x and y is significant; that is, it is not true in general that = , >}. THEOREM: if r is a relation, f = r. DEFINITION: r " X = {z : for some y, y e x and for example, contains nineteen symbols; yet it denotes a sequence which it is customary to write simply as "bet". We may use this latter natural way of naming sequences and still keep the mathematical definition of sequence if we regard "bet" to be merely a shorthand way of writing "{, , }"• Suppose that each of V , "s"\ "s" ",..., "s" stands for some primitive linguistic constant. Then we write DEFINITION

AN EXPOSITION OF SET THEORY 11

1

27

S ..^"- )" for "{, 0 Here the plus sign in the superscript denotes arithmetic addition since it operates on a numeric variable "a", while the other plus sign denotes set-theoretic addition. The first line of the definition declares that anything raised to the power zero is 1, i.e. {0}. The second line is best conceived as an endless series of definitions resulting from the substitution of the integers 0, 1, 2 , . . . for a. That is, x1 = (x° • x) + x°, x2 = (x1 • x) + x1 and so forth. Where A is any set of sequences, A" contains 0, together with all sequences constructable from no more than o occurences of members of A. EXAMPLES: {