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THE INFINITE Thomas Jech THE INFINITE Thomas Jech The Pennsylvania State University century the study of the innite was an exclusive domain of Until the th philosophers and theologians For mathematicians while the concept of innity was crucial in applications of calculus and innite series the innite itself was to paraphrase Gauss just the manner of sp eaking Indeed th century mathe matics even with its frequent use of innite sequences and innitesimals had no real need to study the concept of innity itself functions of real and complex variable and the The increasing use of abstract emerging theory of real numb ers in the work of Cauchy Weierstrass and Dedekind brought ab out the necessity to use innity not only as a manner of sp eaking but rather intro duce into mathematical usage the concept of innite set The earliest mathematician to study sets as ob jects of a mathematical theory Bolzano in the s He gave a convincing philosophical agrument was Bernhard for the existence of innite sets and was also rst to explicitly formulate the concept of onetoone corresp ondence b etween sets It seems that Bolzano was not aware of the p ossibility that there may exist innite sets of dieren t size and that as some historians b elieve he envisioned a mathematical universe where all innite sets are countable While some of his ideas were clearly ahead of his time he did Research for this article was supp orted in part by a grant from the International Research Exchange Board IREX with funds provided by the Andrew W Mellon Foundation the National Endowment for the Humanities and the US Department of State None of these organizations is resp onsible for the views expressed The author is grateful to the Kurt Godel So ciety in Vienna and to the Center for Theoretical Studies in Prague for the opp ortunity to discuss the topics of this to Professor Petr Vopenka for his interpretation of Bolzanos article I am particularly thankful work X Typ eset by A ST M E THOMAS JECH consider paradoxical the fact that the set of all p ositive integers can b e in a oneto one corresp ondence with a smaller subset a fact that had b een already p ointed out by Galileo centuries earlier who realized the signicance of onetoone It was Georg Cantor in the s corresp ondence and who embarked on a systematic study of abstract sets and of cardinal and ordinal numb ers His was also the rst application of set theoretical metho ds namely his nonconstructive pro of of the existence of transcendental numb ers The signicance of Cantors pro of lies in the fact that it employs in the idea that any arbitrary b ounded increasing its pro of of uncountability of R sequence of real numb ers has a least upp er b ound This assumption is related to the concept of real numb ers as Dedekind cuts in Q or as limits of Cauchy sequences of rationals or as innite decimal expansions These three p ossible approaches to the denition of real numb ers are all equivalent in a sense and all require the existence of arbitrary innite sets of integers ery few mathematicians ob ject to the idea of an arbitrary set of integers We V will argue later that the idea of existence of large cardinal numb ers that some mathematicians nd ob jectionable the nevernever land in the words of one detractor is conceptionally no dierent from Cantors idea that is inherent in his pro of A natural question arising from Cantors theory of cardinal numb ers is the fol Do there exist sets of real numb ers that are uncountable yet not equivalent lowing in cardinality to the set of all real numb ers The conjecture that no such sets exist Hypothesis Hilb ert attached such imp ortance has b ecome known as the Continuum to this question that it gured on the top of his famous list of problems in The continuum problem has eventually b een solved in a nontraditional way It was proved that the continuum hyp othesis is neither provable nor refutable thus endent b eing indep To show that a mathematical statement is indep endent one has to sp ecify a THE INFINITE system of p ostulates axioms of the theory which are then shown to b e insucient to prove or to refute the statement For instance the parallels p ostulate is indep en p ostulates The pro of of indep endence is accomplished dent of the rest of Euclids by exhibiting models such as the BolyaiLobacevskij mo del of nonEuclidean ge ometry So clearly the assertion that the continuum hyp othesis is indep endent requires that we sp ecify the axiom system There is a more fundamental reason for axiomatizing set theory With very few insignicant exceptions all mathematical arguments and concepts can b e in principle reduced to concepts and statements formalized in the language of set theory Thus an axiom system for the theory of sets is in fact an axiom system for all of mathematics The axiomatization of set theory based on Cantors denition of sets turns out to b e inconsistent for rather trivial reasons The idea is to consider as sets all collections of ob jects that satisfy an arbitrary prop erty This approach is clearly untenable as exemplied by the set a a ag f whose existence leads to a logical contradiction the widely accepted system of axioms for set theory is the Since the s ZermeloFraenkel axiomatic set theory ZF It consists of several axioms and ax iom schemata that encompass all uses of sets in mathematics This axiomatic theory is almost universally accepted as the logical foundation of current mathe matics Most axioms of ZermeloFraenkel are construction principles that merely formal ize certain manipulations with sets The two notable exceptions are the axiom of innity and the axiom of choice The axiom of innity p ostulates the existence of an innite set or equivalently the existence of the set N of all natural numb ers This axiom is of course the essence of set theory If we replace the axiom of innity by its negation we get a THOMAS JECH theory equivalent to Pe ano arithmetic the axiomatic theory of elementary numb er theory Historically the most interesting axiom of ZF is the axiom of choice Unlike the other axioms it is highly nonconstructive as it p ostulates the existence of choice functions without giving a sp ecic description of such functions It was intro duced by Zermelo in his pro of of wellordering and is in fact equivalent to the principle set can b e well ordered As it seems imp ossible to dene a that states that every well ordering of the set R among others this principle was violently opp osed by some mathematicians of the rst half of this century Some of the ob jections are quite understandable in light of such consequences of the axiom of choice as the HausdorBanachTarski paradoxical decomp osition of the unit ball It should b e noted that the axiom of choice is indep endent of the rest of ZF by the work of G odel and Cohen which puts set theory in situation roughly similar to that of Euclidean v nonEuclidean geometry From the formalists p oint of view it do es not make any dierence whether the axiom of choice is accepted as an axiom or not In view of its usefulness in algebra and analysis the prevailing opinion favors the axiom I would like to p oint out that even such an inno cuous statement as the union of countably many countable sets of real numb ers is countable requires the axiom of choice It is of these ZermeloFraenkel axioms that the continuum hyp othesis was shown Kurt Godel constructed a mo del of set theory to b e indep endent First in the constructible universe in which the continuum hyp othesis holds Godels it is the least p ossible mo del which any other mo del mo del is a canonical mo del that contains all ordinal numb ers has to include In Paul Cohen discovered a metho d of constructing generic extensions of mo dels of set theory including mo dels in which the continuum hyp othesis fails Cohens metho d turned out to b e quite general and resulted in a numb er of indep endence pro ofs It so happ ens that a numb er of conjectures in dierent areas of mathematics cannot b e decided on the THE INFINITE basis of ZermeloFraenkel axioms and have b een proved indep endent by Cohens metho d In addition to the continuum hyp othesis we mention Suslins Problem Borels Conjecture Whiteheads Problem and Kaplanskys Conjecture In view of the apparent incompleteness of ZermeloFraenkel axioms a reasonable suggestion oers itself namely to nd a complete system of axioms in which every p ossible statement of mathematics would b e formally decidable This was the so Program A mortal blow to this program were the two Incomplete called Hilberts ness Theorems of Godel The First Incompleteness Theorem shows in eect that there is no hop e for a complete axiomatization of mathematics It states that every axiomatic theory that includes elementary arithmetic is incomplete if it is consistent In particular b oth Peano arithmetic and ZermeloFraenkel set theory are incomplete and will remain incomplete even if we add further axioms Godels pro of of the Incompleteness Theorem yields an undecidable sentence but unlike the Continuum Hyp othesis in set theory Godels sentence is not particularly meaningful It do es provide a starting p oint in the search for easily understandable statements expressible in the language of arithmetic that are indep endent of Peano arithmetic Such statements have indeed b een found by Harrington and Paris Their examples are combinatorial principles expressible in the language of arith metic and true ie provable in set theory but unprovable in Peano arithmetic It remains to b e seen whether some of the unproved conjectures in elementary numb er theory may p erhaps
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