How Gödel Transformed Set 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 universe L, established have the same power if there is a bijection between the relative consistency of the Axiom of them, and, implicitly at first, that one set has a Choice and the Continuum Hypothesis. 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 set theory. 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 infinite set 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 Georg Cantor (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) sequence 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 bijections between sets if it is a strict linear ordering of the set such that every nonempty subset has a ≺-least element. 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 class (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 cardinal number 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 David Hilbert’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 Bertrand Russell (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 Almost two decades after his initial 1873 proof, with it logicism, 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 mathematical logic. 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 power set P(x)={y | y ⊆ x} of a set of 1901 he came to the famous Russell’s Paradox, x. For any set x, P(x) has a higher power than x. a surprisingly simple counterexample to full com- First, the function 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. Gottlob Frege (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 subsets of a set having been made explicit, j Om for some j<iand m ≤ n, the definitions al- there was open controversy after the appearance j lowing for quantification only over Om.