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Home , Axiom of determinacy, Axiom of reducibility, Constructible universe, Cumulative hierarchy, Inner model theory, The Higher Infinite

How Gödel Transformed Theory Juliet Floyd and Akihiro Kanamori

urt Gödel (1906–1978), with his work on of reals and the like. He stipulated that two sets the constructible L, established have the same power if there is a between the relative of the of them, and, implicitly at first, that one set has a Choice and the . higher power than another if there is an injection KMore broadly, he secured the cumulative of the latter into the first but no bijection. In an hierarchy view of the universe of sets and ensured 1878 publication he showed that R, the plane R × R, the ascendancy of first-order logic as the framework and generally Rn are all of the same power, but for . Gödel thereby transformed set the- there were still only the two infinite powers as set ory and launched it with structured subject mat- out by his 1873 proof. At the end of the publica- ter and specific methods of proof as a distinctive tion Cantor asserted a dichotomy: field of mathematics. What follows is a survey of prior developments in set theory and logic in- Every of real numbers ei- tended to set the stage, an account of how Gödel ther is countable or has the power of the marshaled the ideas and constructions to formu- continuum. late L and establish his results, and a description This was the Continuum Hypothesis (CH) in its of subsequent developments in set theory that res- nascent context, and the continuum problem, to re- onated with his speculations. The survey trots out solve this hypothesis, would become a major mo- in quick succession the groundbreaking work at the tivation for Cantor’s large-scale investigations of beginning of a young subject. infinite numbers and sets. In his Grundlagen of 1883, Cantor developed the Numbers, Types, and Well-Ordering transfinite numbers and the key concept of well- Set theory was born on that day in December 1873 ordering. The progression of transfinite numbers when (1845–1918) established that could be depicted, in his later notation, in terms the continuum is not countable: There is no bijec- of natural extensions of arithmetical operations: tion between the natural numbers N = {0, 1, 2, 3,...} and the real numbers R, since 0, 1, 2,...ω,ω+1,ω+2,...ω+ ω(= ω·2), for any (countable) of reals one can spec- ...ω·3,...ω·ω(= ω2),...ω3,...ωω,... ify nested intervals so that any real in the inter- ≺ section will not be in the sequence. Cantor soon in- A relation is a well-ordering of a set if and only vestigated ways to define between sets if it is a strict linear ordering of the set such that every nonempty has a ≺-least . Well- Juliet Floyd is professor of philosophy at Boston University. orderings carry the sense of sequential counting, Her email address is [email protected]. and the transfinite numbers serve as standards Akihiro Kanamori is professor of mathematics at Boston for gauging well-orderings. Cantor called the set of University. His email address is [email protected]. natural numbers N the first number (I) and

APRIL 2006 NOTICES OF THE AMS 417 the set of numbers whose predecessors are in bi- are now to be the cardinal numbers of the succes- jective correspondence with (I) the second number sive number classes from the Grundlagen and thus class (II). The infinite numbers in the above display to exhaust all the infinite cardinal numbers. Can- ℵ are all in (II). Cantor conceived of (II) as bounded tor pointed out that 2 0 is the of above and showed that (II) itself is not countable. R, but frustrated in his efforts to establish CH he Proceeding upward, Cantor called the set of num- did not even mention the hypothesis, which could ℵ bers whose predecessors are in bijective corre- now have been stated as 2 0 = ℵ1 . Every well-or- spondence with (II) the third number class (III), dered set has an aleph as its cardinal number, but ℵ and so on. Cantor then propounded a basic prin- where is 2 0 in the aleph sequence? ciple in the Grundlagen: CH was thus embedded in the very interstices of the beginnings of set theory. The structures that “It is always possible to bring any well- Cantor built, while now of great intrinsic interest, defined set into the form of a well- emerged largely out of efforts to articulate and es- ordered set.” tablish it. The continuum problem was made the Sets are to be well-ordered and thus to be gauged very first in ’s famous list of 23 prob- by his numbers and number classes. With this lems at the 1900 International Congress of Math- framework Cantor had transformed CH into the ematicians; Hilbert drew out Cantor’s difficulty by positive assertion that (II) and R have the same suggesting the desirability of “actually giving” a power. However, an emerging problem for Cantor well-ordering of R. was that he could not even define a well-ordering (1872–1970), a main architect of R; the continuum, at the heart of mathematics, of the analytic tradition in philosophy, focused in could not be easily brought into the fold of the 1900 on Cantor’s work. Russell was pivoting from transfinite numbers. idealism toward a realism about propositions and two decades after his initial 1873 proof, with it , the thesis that mathematics can Cantor in 1891 came to his celebrated diagonal ar- be founded in logic. Taking a universalist approach gument. In various guises the argument would be- to logic with all-encompassing categories, Russell come fundamental in . Cantor took the class of all classes to have the largest car- himself proceeded in terms of functions, ushering dinal number but saw that Cantor’s 1891 result collections of arbitrary functions into mathemat- leading to higher cardinal numbers presented a ics, but we cast his result as is done nowadays in problem. Analyzing that argument, by the spring terms of the P(x)={y | y ⊆ x} of a set of 1901 he came to the famous Russell’s , x. For any set x, P(x) has a higher power than x. a surprisingly simple counterexample to full com- First, the associating each a ∈ x with prehension, the assertion that for every property {a} is an injection: x →P(x). Suppose now that F A(x) the collection of objects having that prop- is any function: x →P(x). Consider the “diagonal” erty, the class {x | A(x)}, is also an object. Consider set d = {a ∈ x | a/∈ F(a)}. If d itself were a value Russell’s {x | x/∈ x}. If this were an object r, then of F, say d = F(b), then we would have the contra- we would have the contradiction r ∈ r if and only ∈ ∈ diction: b d if and only if b/d. Hence, F cannot if r/∈ r. (1848-1925) was the first to be surjective. systematize quantificational logic in a formalized Cantor had been shifting his notion of set to a language, and he aimed to establish a purely logi- level of abstraction beyond sets of real numbers cal foundation for arithmetic. Russell famously and the like; the diagonal argument can be drawn communicated his paradox to Frege in 1902, who out of the earlier argument, and the new result gen- immediately saw that it revealed a contradiction P eralized the old since (N) and R have the same within his mature logical system. power. The new result showed for the first time that Russell’s own reaction was to build a complex P P there is a set of a higher power than R, e.g. ( (N)). logical structure, one used later to develop math- Cantor’s Beiträge of 1895 and 1897 presented ematics in Whitehead and Russell’s 1910-3 Principia his mature theory of the transfinite. Cantor re- Mathematica. Russell’s ramified theory of types is construed power as cardinal number, now an au- a scheme of logical definitions based on orders tonomous concept beyond une façon de parler and types indexed by the natural numbers. Russell about bijective correspondence. He defined the ad- proceeded “intensionally”; he conceived this dition, multiplication, and exponentiation of car- scheme as a classification of propositions based on dinal numbers primordially in terms of set-theo- the notion of propositional function, a notion not retic operations and functions. As befits the reducible to membership (extensionality). Pro- introduction of new numbers Cantor then intro- ceeding in modern fashion, we may say that the uni- duced a new notation, one using the Hebrew letter verse of the Principia consists of objects stratified aleph, ℵ. ℵ is to be the cardinal number of N and 0 into disjoint types T , where T consists of the in- the successive alephs n 0 dividuals, Tn+1 ⊆{Y | Y ⊆ Tn}, and the types Tn for ℵ ℵ ℵ ℵ i 0, 1, 2,..., α,... n>0 are further ramified into orders On with

418 NOTICES OF THE AMS VOLUME 53, NUMBER 4 i i Tn = i On. An object in On is to be defined either positive use of an arbitrary function operating on in terms of individuals or of objects in some fixed arbitrary of a set having been made explicit, j Om for some j

APRIL 2006 NOTICES OF THE AMS 419 Unlike the development of classical mathematics recursive definition along well-orderings. The proof from marketplace arithmetic and Greek geometry, had an antecedent in the Zermelo 1904 proof, but sets were neither laden with nor bolstered by well- Replacement was necessary even for the very for- worked antecedents. Zermelo axiomatization, un- mulation, let alone the proof, of the theorem. With like Russell’s cumbersome theory of types, provided the ordinals in place von Neumann completed the a simple system for the development of mathe- incorporation of the Cantorian transfinite by defin- matics. Set theory would provide an underpinning ing the cardinals as the initial ordinals, those or- of mathematics, and Zermelo’s axioms would res- dinals not in bijective correspondence with any of onate with mathematical practice. their predecessors. In the 1920s fresh initiatives structured the Replacement has been latterly regarded as some- loose Zermelian framework with new features and how less necessary or crucial than the other axioms, corresponding developments in axiomatics, the the purported effect of the axiom being only on most consequential moves made by John von Neu- large- sets. Initially, mann (1903–1957) in his dissertation, with antic- (1891–1965) and (1887–1963) had ipations by Dimitry Mirimanoff (1861–1945). The independently proposed adjoining Replacement transfinite numbers had been central for Cantor but to ensure that E(a)={a, P(a), P(P(a)),...} would peripheral to Zermelo, and in Zermelo’s system be a set for a, the infinite set given by Zermelo’s ℵ not even 2 0 = ℵ1 could be stated directly. Von , since, as they pointed out, Zer- Neumann reconstrued the transfinite numbers as melo’s axioms cannot establish this. However, even bona fide sets, the ordinals, and established their E(∅) cannot be proved to be a set from Zermelo’s efficacy by formalizing transfinite recursion. axioms, and if his Axiom of Infinity were refor- Ordinals manifest the idea, natural once iterative mulated to accommodate E(∅), there would still be set formation is assimilated, of taking the relation of many finite sets a such that E(a) cannot be proved precedence in a well-ordering simply to be mem- to be a set. Replacement serves to rectify the situ- bership. A set (or class) x is transitive if and only if ation by admitting new infinite sets defined by “re- whenever a ∈ b for b ∈ x, a ∈ x. A set x is a (von placing” members of the one infinite set given by Neumann) ordinal if and only if x is transitive, and the Axiom of Infinity. In any case, the full exercise the membership relation restricted to x = {y | y ∈ x} of Replacement is part and parcel of transfinite re- is a well-ordering of x. The first several ordinals are cursion, which is now used everywhere in modern ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}},..., to be set theory, and it was von Neumann’s formal in- taken as the natural numbers 0,1,2,3, .... The union corporation of this method into set theory, as ne- of these finite ordinals is an ordinal, to be taken as cessitated by his proofs, that brought in Replace- ω; ω ∪{ω} is an ordinal, to be taken as ω +1; and ment. so forth. It has become customary to use the Greek Von Neumann (and others) also investigated the letters α, β, γ,...to denote ordinals; the class of all salutary effects of restricting the universe of sets ordinals is itself well-ordered by membership, and to the well-founded sets. The well-founded sets are ∈ α<βis written for α β; and an ordinal without the sets that belong to some “rank” Vα, these de- an immediate predecessor is a . Von finable through transfinite recursion: Neumann established, as had Mirimanoff before him, ∅ P { | } the key instrumental property of Cantor’s ordinal V0 = ; Vα+1 = (Vα); and Vδ = Vα α<δ numbers for ordinals: Every well-ordered set is for limit ordinals δ. order-isomorphic to exactly one ordinal with mem- V contains every set consisting of natural num- bership. The proof was the first to make full use of ω+1 bers (finite ordinals), and so already at early levels the Axiom of Replacement and thus drew that ax- iom into set theory. there are set counterparts to many objects in math- For a set x and property A(v,w), the property is ematics. That the universe V of all sets is the cu- said to be functional on x if for any a ∈ x, there is ex- mulative hierarchy actly one b such that A(a, b). The Axiom (schema) V = {Vα | α is an ordinal} of Replacement asserts: For any set x and property A(v,w) functional on x, {b |∃a(a ∈ x and A(a, b))} is thus the assertion that every set is well-founded. is a set. This axiom posits sets that result when mem- Von Neumann essentially showed that this asser- bers of a set are “replaced” according to a property; tion is equivalent to a simple assertion about sets, a simple argument shows that Replacement sub- the Axiom of Foundation: Any nonempty set x has sumes Separation. a member y such that x ∩ y is empty. Thus, non- Von Neumann generally ascribed to the ordinals empty well-founded sets have ∈-minimal mem- the role of Cantor’s ordinal numbers, and already bers. If a set x satisfies x ∈ x, then {x} is not well- to incorporate transfinite arithmetic into set the- founded; similarly, if there are x1 ∈ x2 ∈ x1 , then ory he saw the need to establish the Transfinite Re- {x1,x2} is not well-founded. Ordinals and sets con- cursion Theorem, the theorem that validates sisting of ordinals are well-founded, and

420 NOTICES OF THE AMS VOLUME 53, NUMBER 4 well-foundedness can be viewed as a generalization Zermelo’s 1908 axioms when cast in first-order of the notion of being an ordinal that loosens the logic become a countable collection of sentences, connection with transitivity. The Axiom of Foun- and so if they have a model at all, they have a dation eliminates pathologies like x ∈ x and countable model. We thus have the “paradoxical” through the rendition allows existence of countable models for Zermelo’s axioms inductive arguments to establish results about the although they entail the existence of uncountable entire universe. sets. Zermelo found this antithetical and repugnant, In a remarkable 1930 publication Zermelo pro- and proceeded in avowedly second-order terms in vided his final axiomatization of set theory, one that his 1930 work. However, stronger currents were at recast his 1908 axiomatization and incorporated work leading inexorably to the ascendancy of first- both Replacement and Foundation. He herewith order logic. completed his transmutation of the notion of set, his abstract, prescriptive view stabilized by further axioms that structured the universe of sets. Re- Enter Gödel. Gödel virtually completed the math- placement provided the means for transfinite re- ematization of logic by submerging “metamathe- cursion and induction, and Foundation made pos- matical” methods into mathematics. The Com- sible the application of those means to get results pleteness Theorem from his 1930 dissertation about all sets. Zermelo proceeded to offer a strik- established that logical consequence could be cap- ing, synthetic view of a procession of natural mod- tured by formal proof for first-order logic and se- els for his axioms that would have a modern res- cured its key instrumental property of compactness onance and applied Replacement and Foundation for building models. The main advance was of to establish isomorphism and embedding results. course the direct coding, “the arithmetization of Zermelo’s 1930 publication was in part a re- syntax”, which together with a refined version of sponse to Skolem’s advocacy, already in 1922, of Cantor’s diagonal argument led to the celebrated the idea of framing Zermelo’s 1908 axioms in first- 1931 Incompleteness Theorem. This theorem es- order logic. First-order logic is the logic of formal tablished a fundamental distinction between what languages consisting of formulas built up from is true about the natural numbers and what is prov- specified function and relation symbols using log- able and transformed a program advanced by ical connectives and first-order quantifiers ∀ and Hilbert in the 1920s to establish the consistency ∃, quantifiers to be interpreted as ranging over of mathematics by finitary means. Gödel’s work the elements of a domain of discourse. (Second- showed in particular that for a (schematically de- order logic has quantifiers to be interpreted as finable) collection of axioms A, its consistency, that ranging over arbitrary subsets of a domain.) Skolem from A one cannot prove a contradiction, has a for- had proposed formalizing Zermelo’s axioms in the mal counterpart in an arithmetical formula Con(A) ∈ first-order language with and = as binary rela- about natural numbers. Gödel’s “second” theorem tion symbols. Zermelo’s definite properties would asserts that if A is consistent and subsumes the then be those expressible in this first-order lan- elementary arithmetic of the natural numbers, then guage in terms of given sets, and Separation would Con(A) cannot be proved from A. become a schema of axioms, one for each first-order Gödel’s advances in set theory can be seen as formula. Analogous remarks apply to the formal- part of a steady intellectual development from his ization of Replacement in first-order logic. As set fundamental work on incompleteness. His 1931 theory was to develop, the formalization of Zer- paper had a prescient footnote 48a: melo’s 1930 axiomatization in first-order logic would become the standard axiomatization, Zer- As will be shown in Part II of this paper, melo-Fraenkel with Choice (ZFC). The “Fraenkel” the true reason for the incompleteness acknowledges Fraenkel’s early suggestion of in- inherent in all formal systems of math- corporating Replacement. Zermelo-Fraenkel (ZF) is ematics is that the formation of ever ZFC without AC. higher types can be continued into the Significantly, before this standardization both transfinite (cf. D. Hilbert, “Über das Un- Skolem and Zermelo raised issues about the limi- endliche”, Math. Ann. 95, p. 184), while tations of set theory as cast in first-order logic. in any at most count- Skolem had established a fundamental result for ably many of them are available. For it first-order logic with the Löwenheim-Skolem The- can be shown that the undecidable orem: If a countable collection of first-order sen- propositions constructed here become tences has a model, then it has a countable model. decidable whenever appropriate higher Having proposed framing set theory in first-order types are added (for example, the type terms, Skolem pointed out as a palliative for tak- ω to the system P [the simple theory of ing set theory as a foundation for mathematics types superposed on the natural num- what has come to be called the Skolem Paradox: bers as individuals satisfying the Peano

APRIL 2006 NOTICES OF THE AMS 421 axioms]). An analogous situation pre- M |= ϕ[a1,a2,...,an] vails for the axiom system of set theory. asserts that the formula ϕ is satisfied in M ac- Gödel’s letters and lectures clarify that the ad- cording to Tarski’s recursive definition when vi is dition of an infinite type ω to Russell’s theory of interpreted as ai . A subset y of the domain of M types would provide a “definition for ‘truth’ ” for is first-order definable over M if and only if there the theory and hence establish hitherto unprovable is a formula ψ(v0,v1,v2,...,vn) and a1,a2,...,an propositions like those provided by his Incom- in the domain of M such that pleteness Theorem. Inherent in Russell’s theory y = {z | M |= ψ[z,a ,...,a ]}. was the indexing of types by the natural numbers, 1 n and Gödel’s citation in the footnote of Hilbert’s Set theory was launched on an independent 1926 paper in connection with the possibility of ad- course as a distinctive field of mathematics by joining transfinite types would bridge the past and Gödel’s formulation of the class L of constructible the future. Hilbert there had attempted a proof of sets through which he established the relative con- CH using transfinite indexing in his formalism, sistency of AC and CH. He thus attended to the fun- and Gödel would achieve what success is possible damental issues raised at the beginning of set the- in this direction. Gödel never published the an- ory by Cantor and Zermelo. In his first 1938 nounced Part II, which was to have been on truth, announcement Gödel described L as a hierarchy but his engagement with truth and its distinction “which can be obtained by Russell’s ramified hier- from provability could be viewed as his entrée into archy of types, if extended to include transfinite full blown set theory. In a 1933 lecture Gödel ex- orders.” Indeed, with L Gödel had refined the cu- pounded on axiomatic set theory as a natural gen- mulative hierarchy of sets to a cumulative hierar- eralization of the simple theory of types “if certain chy of definable sets which is analogous to the or- superfluous restrictions” are removed: One could ders of Russell’s ramified theory. Gödel’s further innovation was to continue the indexing of the hi- cumulate the types starting with individuals D0 erarchy through all the ordinals. Von Neumann’s and taking Dn+1 = Dn ∪P(Dn), and one could ex- tend the process into the transfinite, mindful that canonical well-orderings would be the spine for a for any type-theoretic system S a new proposition thin hierarchy of sets, and this would be the key (e.g., Con(S)) becomes provable if to S is adjoined to both the AC and CH results. In a 1939 note the “the next higher type and the axioms con- Gödel informally presented L essentially as is done today: For any set x let def(x) denote the collection cerning it”. Thus Gödel came to the cumulative hi- of subsets of x first-order definable over x ac- erarchy as a transfinite extension of the theory of cording to the previous definition. Then define: types that incorporates higher and higher levels of truth. L0 = ∅; Lα+1 = def(Lα),Lδ = {Lα | α<δ} Alfred Tarski (1902–1983) completed the math- for limit ordinals δ; ematization of logic in the early 1930s by provid- ing a systematic “definition of truth”, exercising and the constructible universe philosophers ever since. Tarski simply schema- L = {L | α is an ordinal}. tized truth by taking it to be a correspondence be- α tween formulas of a and set- Gödel pointed out that L “can be defined and its theoretic assertions about an interpretation of the theory developed in the formal systems of set the- language and by providing a recursive definition ory themselves.” This follows by transfinite re- of the satisfaction relation, that which obtains when cursion from the formalizability of def(x) in set the- a formula holds in an interpretation. The eventual ory, the “definability of definability”, which was effect of Tarski’s mathematical formulation of se- later reaffirmed by Tarski’s systematic definition mantics would be not only to make mathematics of the satisfaction relation in set-theoretic terms. out of the informal notion of satisfiability, but also In modern parlance, an is a transitive to enrich ongoing mathematics with a systematic class containing all the ordinals such that, with method for forming mathematical analogues of membership and quantification restricted to it, the several intuitive semantic notions. For coming pur- class satisfies each axiom of ZF. Gödel in effect ar- poses, the following specifies notation and con- gued in ZF to show that L is an inner model and cepts: moreover that L satisfies AC and CH. He thus es- For a first-order language, suppose that M is an tablished the relative consistency Con(ZF) implies interpretation of the language (i.e., a specification Con(ZFC + CH). of a domain as well as interpretations of the func- In his 1940 monograph, based on 1938 lectures, tion and relation symbols), ϕ(v1,v2,...,vn) is a for- Gödel formulated L via a transfinite recursion that mula of the language with the (unquantified) vari- generated L set by set. His incompleteness proof ables as displayed, and a1,a2,...,an are the had featured “Gödel numbering”, the encoding of domain of M. Then formulas by natural numbers, and his L recursion

422 NOTICES OF THE AMS VOLUME 53, NUMBER 4 was a veritable Gödel numbering with ordinals, note. In a comment bringing out the intermixing one that relies on their extent as given beforehand of types and orders, Gödel pointed out that “there to generate a universe of sets. This approach may are sets of lower type that can only be defined with have obfuscated the satisfaction aspects of the the help of quantifiers for sets of higher type.” For construction, but on the other hand it did make example, constructible members of Vω+1 in the more evident other aspects: Since there is a direct, cumulative hierarchy will first appear quite high in definable well-ordering of L, choice functions the constructible Lα hierarchy; resonant with abound in L, and AC holds there. Also, L was seen Gödel’s earlier remarks about truth, members of to have the important property of Vω+1, in particular sets of natural numbers, will en- through the simple operations involved in Gödel’s code truth propositions about higher Lα ’s. Gödel recursion, one consequence of which is that for any had given priority to the ordinals and recursively inner model M, the construction of L in the sense formulated a hierarchy of orders based on defin- of M again leads to the same class L. Decades later ability, and the hierarchy of types was spread out many inner models based on first-order definabil- across the orders. The jumble of the Principia Math- ity would be investigated for which absoluteness ematica had been transfigured into the con- considerations would be pivotal, and Gödel had for- structible universe L. mulated the canonical inner model, rather analo- Gödel’s argument for CH holding in L rests, as gous to the algebraic numbers for fields of char- he himself wrote in a brief 1939 summary, on “a acteristic zero. generalization of Skolem’s method for construct- In a 1939 lecture about L Gödel described what ing enumerable models”, now embodied in the amounts to the Russell orders for the simple situ- well-known Skolem Hull argument and Condensa- ation when the members of a countable collection tion Lemma for L. It is the first significant appli- of real numbers are taken as the individuals and cation of the Löwenheim-Skolem Theorem since new real numbers are successively defined via Skolem’s own to get his paradox. Ironically, though quantification over previously defined real num- Skolem sought through his paradox to discredit set bers, and he emphasized that the process can be theory based on first-order logic as a foundation continued into the transfinite. He then observed for mathematics, Gödel turned paradox into that this procedure can be applied to sets of real method, one promoting first-order logic. Gödel numbers and the like, as individuals, and moreover, showed that in L every subset of Lα belongs to that one can “intermix” the procedure for the real some Lβ for some β of the same power as α (so numbers with the procedure for sets of real num- that in L every real belongs to some Lβ for a count- bers “by using in the definition of a able β, and CH holds). In the 1939 lecture he as- quantifiers that refer to sets of real numbers, and serted that “this fundamental theorem constitutes similarly in still more complicated ways.” Gödel the corrected core of the so-called Russellian axiom called a constructible set “the most general [object] of reducibility.” Thus, Gödel established another that can at all be obtained in this way, where the connection between L and Russell’s ramified the- quantifiers may refer not only to sets of real num- ory of types. But while Russell had to postulate his bers, but also to sets of sets of real numbers and Axiom of Reducibility for his finite orders, Gödel so on, ad transfinitum, and where the indices of it- was able to derive an analogous form for his trans- eration …can also be arbitrary transfinite ordinal finite hierarchy, one that asserts that the types are numbers.” Gödel considered that although this de- delimited in the hierarchy of orders. finition of constructible set might seem at first to Gödel brought into set theory a method of con- be “unbearably complicated”, “the greatest gener- struction and argument and thereby affirmed sev- ality yields, as it so often does, at the same time eral features of its axiomatic presentation. First, the greatest simplicity.” Gödel was picturing Rus- Gödel showed how first-order definability can be sell’s ramified theory of types by first disassociating used in a transfinite recursive construction to es- the types from the orders, with the orders here tablish striking new mathematical results. This sig- given through definability and the types repre- nificantly contributed to a lasting ascendancy for sented by real numbers, sets of real numbers, and first-order logic which beyond its sufficiency as a so forth. Gödel’s intermixing then amounted to a logical framework for mathematics was seen to recapturing of the complexity of Russell’s ramifi- have considerable operational efficacy. Gödel’s con- cation, the extension of the hierarchy into the struction moreover buttressed the incorporation of transfinite allowing for a new simplicity. Replacement and Foundation into set theory. Re- Gödel went on to describe the universe of set the- placement was immanent in the arbitrary extent of ory, “the objects of which set theory speaks”, as the ordinals for the indexing of L and in its formal falling into “a transfinite sequence of Russellian definition via transfinite recursion. As for Foun- [simple] types”, the cumulative hierarchy of sets. dation, underlying the construction was the well- He then formulated the constructible sets as an foundedness of sets. Gödel in a footnote to his 1939 analogous hierarchy, the hierarchy of his 1939 note wrote: “In order to give A [the axiom V = L,

APRIL 2006 NOTICES OF THE AMS 423 that the universe is L] an intuitive meaning, one has described a new inner model, the class of ordinal to understand by ‘sets’ all objects obtained by definable sets. building up the simplified hierarchy of types on an In his 1947 article on the continuum problem of individuals (including types of arbi- Gödel pointed out the desirability of establishing trary transfinite orders).” Some have been baffled the independence of CH, i.e., in addition to his rel- about how the cumulative hierarchy picture came ative consistency result, that also Con(ZF) implies to dominate in set-theoretic practice; although Con(ZFC + the negation of CH). However, Gödel there was Mirimanoff, von Neumann, and espe- stressed that this would not solve the problem. cially Zermelo, the picture came in with Gödel’s The axioms of set theory do not “form a system method, the reasons being both thematic and his- closed in itself”, and so the “very concept of set on torical: Gödel’s work with L with its incisive analy- which they are based” suggests their extension by sis of first-order definability was readily recog- new axioms, axioms that may decide issues like CH. nized as a signal advance, while Zermelo’s 1930 New axioms could even be entertained on the ex- paper with its second-order vagaries remained trinsic basis of the “fruitfulness of their conse- somewhat obscure. As the construction of L was quences”. Gödel concluded by advancing the re- gradually digested, the sense that it promoted of markable opinion that CH “will turn out to be a cumulative hierarchy reverberated to become the wrong” since it has as paradoxical consequences basic picture of the universe of sets. the existence of thin, in various senses he de- scribed, sets of reals of the power of the contin- New Axioms uum. How Gödel transformed set theory can be broadly Later touted as his “program”, Gödel’s advo- cast as follows: On the larger stage, from the time cacy of the search for new axioms mainly had to of Cantor, sets began making their way into topol- do with axioms. These postulate ogy, algebra, and analysis so that by the time of structure in the higher reaches of the cumulative Gödel, they were fairly entrenched in the structure hierarchy, often by positing cardinals whose prop- and language of mathematics. But how were sets erties entail their inaccessibility from below in viewed among set theorists, those investigating strong senses. Speculations about large cardinal sets as such? Before Gödel, the main concerns were possibilities had occurred as far back as the time what sets are and how sets and their axioms can of Zermelo’s first axiomatization of set theory. serve as a reductive basis for mathematics. Even Gödel advocated their investigation, and they can today, those preoccupied with ontology, questions be viewed as a further manifestation of his foot- of mathematical existence, focus mostly upon the note 48a idea of capturing more truth, this time by set theory of the early period. After Gödel, the positing strong closure points for the cumulative main concerns became what sets do and how set hierarchy. In the early 1960s large cardinals were theory is to advance as an autonomous field of vitalized by the infusion of model-theoretic meth- mathematics. The cumulative hierarchy picture ods, which established their central involvement in was in place as subject matter, and the meta- embeddings of models of set theory. The subject mathematical methods of first-order logic mediated was then to become a mainstream of set theory the subject. There was a decided shift toward epis- after the dramatic introduction of a new way of get- temological questions, e.g., what can be proved ting extensions of models of set theory. about sets and on what basis. (1934–) in 1963 established the in- As a pivotal figure, what was Gödel’s own stance? dependence of AC from ZF and the independence What he said would align him more with his pre- of CH from ZFC. That is, Cohen established that decessors, but what he did would lead to the de- Con(ZF) implies Con(ZF + the negation of AC) and velopment of methods and models. In a 1944 ar- that Con(ZF) implies Con(ZFC + the negation of ticle on Russell’s mathematical logic, in a 1947 CH). These results delimited ZF and ZFC in terms article on Cantor’s continuum problem (and in a of the two fundamental issues raised at the be- 1964 revision), and in subsequent lectures and cor- ginning of set theory. But beyond that, Cohen’s respondence, Gödel articulated his philosophy of proofs were soon to flow into method, becoming “conceptual realism” about mathematics. He es- the inaugural examples of , a remarkably poused a staunchly objective “concept of set” ac- general and flexible method for extending models cording to which the axioms of set theory are true of set theory by adding “generic” sets. Forcing has and are descriptive of an objective reality schema- strong intuitive underpinnings and reinforces the tized by the cumulative hierarchy. Be that as it notion of set as given by the first-order ZF axioms may, his actual mathematical work laid the ground- with conspicuous uses of Replacement and Foun- work for the development of a range of models and dation. With L analogous to the field of algebraic axioms for set theory. Already in the early 1940s numbers, forcing is analogous to making tran- Gödel worked out for himself a possible model for scendental field extensions. If Gödel’s construction the negation of AC, and in a 1946 address he of L had launched set theory as a distinctive field

424 NOTICES OF THE AMS VOLUME 53, NUMBER 4 of mathematics, then Cohen’s method of forcing recently provided a variety of propositions of finite began its transformation into a modern, sophisti- combinatorics that are equi-consistent with the ex- cated one. Set theorists rushed in and were soon istence of large cardinals; this incisive work serves establishing a cornucopia of relative consistency re- to affirm the “necessary use” of large cardinal ax- sults, truths in a wider sense, some illuminating ioms even in finite mathematics. In set theory it- problems of classical mathematics. In this sea self, Hugh Woodin has developed a scheme based change the extent and breadth of the expansion of on a new logic in an environment of large cardinals set theory dwarfed what came before, both in terms that argues against CH itself, and with an additional ℵ of the numbers of people involved and the results axiom, that 2 0 = ℵ2 . These results serve as re- established. markable vindications for Gödel’s original hopes Already in the 1960s and into the 1970s large for large cardinals. cardinal postulations were charted out and elabo- rated, investigated because of the “fruitfulness of References their consequences” since they provided quick KURT GÖDEL, [1986], Collected Works, Volume I: Publications proofs of various strong propositions and because 1929–1936, (Solomon Feferman, editor-in-chief), Ox- they provided the consistency strength to establish ford University Press, New York. , [1990], Collected Works, Volume II: Publications new relative consistency results. A subtle connec- ——— 1938–1974, (Solomon Feferman, editor-in-chief), Ox- tion quickly emerged between large cardinals and ford University Press, New York. combinatorial propositions low in the cumulative ——— , [1995] Collected Works, Volume III: Unpublished Es- hierarchy: Forcing showed just how relative the says and Lectures, (Solomon Feferman, editor-in-chief), Cantorian notion of cardinality is, since bijections Oxford University Press, New York. could be adjoined easily, often with little distur- [2002] , Set Theory, third millennium edition, bance to the universe. In particular, large cardinals, revised and expanded, Springer, Berlin. highly inaccessible from below, were found to sat- [2003] AKIHIRO KANAMORI, . Large Car- isfy substantial propositions even after they were dinals in Set Theory from their Beginnings, second edition, Springer-Verlag, Berlin. “collapsed” by forcing to ℵ1 or ℵ2, i.e., bijections [1982] GREGORY H. MOORE, Zermelo’s . Its were adjoined to make the cardinal the first or Origins, Development, and Influence. Springer-Verlag, second uncountable cardinal. Conversely, such New York. propositions were found to entail large cardinal hy- potheses in the clarity of an L-like inner model, sometimes the very same initial large cardinal hy- pothesis. In a subtle synthesis, hypotheses of length concerning the extent of the transfinite were cor- related with hypotheses of width concerning the fullness of power sets low in the cumulative hier- archy, sometimes the arguments providing equi- consistencies. Thus, large cardinals found not only extrinsic but intrinsic justifications. Although their emergence was historically contingent, large car- dinals were seen to form a linear hierarchy, and there was the growing conviction that this hierar- chy provides the hierarchy of exhaustive principles against which all possible consistency strengths can be gauged, a kind of hierarchical completion of ZFC. In the 1970s and 1980s possibilities for new complementarity were explored with the develop- ment of inner for large cardinals, the investigation of minimal L-like inner models hav- ing large cardinals, models that exhibited the kind of fine structure that Gödel had first explored for L. Also, hypotheses about sets of reals were explored because of their fruitful consequences in , the definability theory of the continuum. Then in a grand synthesis, certain large cardinals were found to provide just the con- sistency strength to establish the consistency of ADL(R) , the holding in the minimal inner model L(R) containing all the reals. In a different direction, Harvey Friedman has

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