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2 INTRODUCTION Crar, 1 and related topics of analysis* Cantor, who is usually considered the founder of set theory as a mathematical discipline, was led by his work into a consideration of infinite sets or classes of arbitrary character. In 1874 he published his famous proof that the set of real numbers cannot be pué into onc-one correspondence with tho set of natural numbers (the non- negative integers). In 1878 he introduced the fundamental notion of two sets being equipollent or having the same power (Machtigheit) % they can be put into one-one correspondence with each other. Clearly two finite sets have the same power just when they have the same number of members, Thus the notion of power leads in the case of infinite sets to a generalization of the notion of a natural number to that of an infinite cardinal number. - Developraent of the general theory of transinite numbers was one of the great accomplisaments of Cantor's mathematical researches. Technical consideration of the many basie concepts of set theory in- troduced by Cantor will be given in due course. From the standpoint of the foundations of mathematics the philosophically revolutionary aspect cf Cantor's work was his bold insistence on the actual infinite, that is, on the existence of infinite sets as mathematical objects on a par with numbers and finite sets. Historically the concept cf infinity bas played a role in the literature of thc foundations of mathematics as important as that of the concept of number. There is scarcely a serious philosopher of mathematies sincc Aristotle who has not been much exercised about this difficult concopt. Any book on set theory is naturally expected to provide an exact analysis of the concepts of number and infinity. But other topies, some controversial and important in foundations research, are also a traditional part of the subject and are consequentiy treated in the chapters that follow. Typical are algebra of sets, general theory of relations, ordering relations in par- ticular functions, finite sets, cardinal numbers, infinite seta, ordinal arithmetic, transfinito induction, definition by transfinite recursion, axiom of choice, Zorn's Lemma. At this point the reader is not expected to know what theso phrases mean, but such a list may still give a clue to the more detailed contents of this book. In this book set theory is developed asiomatically rather than intuitively. Several considerations have guided the choice of an axiomatic approach. One is the author's opinion that the axiomatic development of set theory is among the most impressive accomplishments of modern mathematics. Concepts which were vague and unpleasantly inexact for decades and sometimes even centuries can be given a precise meaning. Adequate axioms for set theory provido one clear, constructive answer to the question: Exactly what assumptions, beyond thosc of elementary logie, are required *For & detailed historicul survey of Cantor's work, ace Jourdain's Introduction to Cantor [1915]. INTRODUCTION 3 for-modern mathematics? The most pressing consideration, já-the. discovery, made around 1900, of various paradoxes in ntultive set theory, which admits the existence of sets of objects “any definite property whatsoever. Some particular restricted tigapproach is needed to avoid these paradoxes, which are discussed “Sand 1.4 below. “2: Logic and Notation. Wo shall use symbols of logic cxtensively ivposes of precision and brevity, particularly in the early chapters. ofs are mainly written in an informal style. The theory developed atéi'às un axiomatie theory of the sort familiar from geometry and “ports of mathematics, and not as a formal logistie system for which “lb tules of syntax and semanties are given. The explicitness of proofs éient to make it a routine maiter for any reader familiar with Hematival logic to provido formalized proofs in some standard system iG However, familiarity with mathematical logic is not required for erstanding any part of the book. this point we introduee the few logical symbols which will be used. ist consider five symbols for the five most common sentential con- rés; The negation of a formula P is written as - P. The conjunction o foímiulas P and (Q is written as P & Q. The disjunction of P and Q v-Q. The implication with P as antecedent and Q as consequent as +Q. The equivalence P if and only if Qas Pe Q. Theuniversal quan- 1: For every v as (Yv), and the existential quantifier For some v as (dv). also use the symbol (E!v) for There is exactly one v such that, This ation may be summarized in the following table. LOGICAL NOTATION It is not the case that P Pand Q PorQ HP then Q Pifandonlyif Q For every v, P For some v, P There is exactly one v such that P sihe sentence: For every x there is a y such that x < y Yinbolized: (VaIne
160) — Fl < The sentence: For every « there is exactly one y such thate + y = 0 is symbolized: (Va) Byte + 4 = 0). A given logical symbol may correspond to several English idioms. “Thus (Vy)P may be read For all v, P as well as For every v, P. Sentences (1) and (2) illustrate the use of parentheses for purposes of punetuation. No formal explanation seems necessary. However, one convention con- cerning the relative dominance of the sentential connectives &, V, — and +» will reduce considerabiy the number of parenthes The convention is that «> and — dominate & and V. Thus, the formula: (rT7S A Similarly, styHOeo(rHOVy 0) may be written: “ sty=00272=0Vyx0 Principles of logie which are needed in the sequel and which may not be familiar to some readers will be intuitively explained when used. One principle used, concerning which there is some disagreement in practice among mathematicians, is that the double bar “=' is taken as the sign of identity. The formula 'z = y' may be read “x is the same as 4, “x is identical with y, or “o is equal to y. The last reading is permi ible here only if ié is understood thal equality means samencss of identity (which is what it does mean in almost al! ordinary mathematical contexts). The exact status é of the relation of identity within set theory is discussed in $2.2. A few remaxks concerning quantifiers may also be helpful. “The scope of a quantifler is the quantifior itself together with the smailest formula É What the smallest formula is, 18: immediately following the quantifier. always indicated by parenthesos. Thus in the formula (4) )te
