fea-woodin.qxp 6/29/01 1:11 PM Page 681

The Continuum Hypothesis, Part II W. Hugh Woodin

Introduction So the resolution of the theory of the structure In the first part of this article, I identified the cor- H(ω2), ∈ could well be a far more difficult rect axioms for the structure P(N), N, +, ·, ∈ , challenge than was the resolution of the theory of which is the standard structure for Second Order the structure H(ω1), ∈. Number Theory. The axioms, collectively “Projec- One example of the potential subtle aspects of tive Determinacy”, solve many of the otherwise un- the structure H(ω2), ∈ is given in the following solvable, classical problems of this structure. theorem from 1991, the conclusion of which is in Actually working from the axioms of set theory, essence a property of the structure H(ω2), ∈. ZFC, I identified a natural progression of structures Theorem (Woodin). Suppose that the axiom Mar-  ∈  ∈ increasing in complexity: H(ω), , H(ω1), , tin’s Maximum holds. Then there exists a surjection and H(ω2), ∈, where for each cardinal κ, H(κ) ρ : R → ω2 such that {(x, y) | ρ(x) <ρ(y)} is a pro- denotes the set of all sets whose transitive closure jective set. has cardinality less than κ. The first of these struc- tures is logically equivalent to N, +, ·, the stan- As we saw in Part I, assuming the forcing axiom, dard structure for number theory; the second is Martin’s Maximum, CH holds projectively in that logically equivalent to the standard structure for if X ⊆ R is an uncountable projective set, then Second Order Number Theory; and the third struc- |X| = |R|. This is because Projective Determinacy ture is where the answer to the Continuum Hy- must hold. However, the preceding theorem shows pothesis, CH, lies. The main topic of Part I was the that assuming Martin’s Maximum, CH fails pro- structure H(ω1), ∈. jectively in that there exists a surjection Are there analogs of these axioms, say, some ρ : R → ω2 generalization of Projective Determinacy, for the such that {(x, y) | ρ(x) <ρ(y)} is a projective set. structure H(ω2), ∈? Any reasonable generaliza- tion should settle the Continuum Hypothesis. Such a function ρ is naturally viewed as a “pro- An immediate consequence of Cohen’s method jective counterexample” to CH, for it is a coun- of forcing is that large cardinal axioms are not terexample to the following reformulation of CH: R → R terribly useful in providing such a generalization. Suppose that π : α is a surjection of onto Indeed it was realized fairly soon after the dis- the ordinal α; then α<ω2. covery of forcing that essentially no large cardinal There is a curious asymmetry which follows hypothesis can settle the Continuum Hypothesis. from (the proofs of) these results. Assume there This was noted independently by Cohen and by exist infinitely many Woodin cardinals. Then: Levy-Solovay. Claim (1) There can be no projective “proof” of CH (there can be no projec- W. Hugh Woodin is professor of mathematics at the Uni- tive well-ordering of R of length ω1). versity of California, Berkeley. His e-mail address is [email protected]. Part I of this article appeared in the June/July 2001, Claim (2) There can be a projective issue, pp. 567-576. “proof” of ¬CH (there can be, in The research reported was supported in part by NSF the sense just defined, a projective grant number 9322442. counterexample to CH).

AUGUST 2001 NOTICES OF THE AMS 681 fea-woodin.qxp 6/29/01 1:11 PM Page 682

Therefore, if there exist infinitely many Woodin car- Can the theory of the structure dinals and if the Continuum Hypothesis is to be H(ω2), ∈ be finitely axiomatized decided on the basis of “simple” evidence (i.e., (over ZFC) in a (reasonable) logic which projective evidence), then the Continuum Hy- extends first order logic? pothesis must be false. This is the point of the first The logics arising naturally in this analysis sat- claim. But is this an argument against CH? If so, the isfy two important conditions, Generic Soundness playful adversary might suggest that a similar line and Generic Invariance. As a consequence, any ax-  ∈ of argument indicates that ZFC is inconsistent, for ioms we find will yield theories for H(ω2), , while we can have a finite proof that ZFC is in- whose “completeness” is immune to attack by ap- consistent, we, by Gödel’s Second Incompleteness plications of Cohen’s method of forcing, just as is Theorem, cannot have a finite proof that ZFC is con- the case for number theory. sistent (unless ZFC is inconsistent). There is a key How shall we define the relevant strong logics? difference here, though, which is the point of the There is a natural strategy motivated by the Gödel second claim. If there is more than one Woodin car- Completeness Theorem. If φ is a sentence in the L ∈  dinal, then a projective “proof” that CH is false can language (ˆ=, ˆ) for set theory, then “ZFC φ” in- always be created by passing to a Cohen extension. dicates that there is a formal proof of φ from ZFC. More precisely, if M,E is a model of ZFC to- This is an arithmetic statement. gether with the statement “There exist 2 Woodin The Gödel Completeness Theorem shows that cardinals”, that is, if if φ is a sentence, then ZFC  φ if and only if M,E  φ for every structure M,E such that M,E  ZFC+“There exist 2 Woodin cardinals,” M,E  ZFC. ∗ ∗  then there is a Cohen extension, M ,E , of Therefore a strong logic 0 can naturally be M,E such that M∗,E∗  ZFC + φ, where φ is defined by first specifying a collection of test struc- the sentence which asserts that there exists a sur- tures—these are structures of the form M = M,E, where E ⊂ M × M—and then defining “ZFC  φ” jection ρ : R → ω2 such that {(x, y) | ρ(x) <ρ(y)} 0 is a projective set. This theorem, which is the the- if for every test structure M, if M  ZFC, then orem behind the second claim above, shows that M  φ. what might be called the Effective Continuum Hy- Of course, we shall only be interested in the case pothesis is as intractable as the Continuum Hy- that there actually exists a test structure M such pothesis itself. that M  ZFC. In other words, we require that ZFC These claims are weak evidence that CH is false, be consistent in our logic. so perhaps large cardinal axioms are not quite so The smaller the collection of test structures, useless for resolving CH after all. the stronger the logic, i.e., the larger the set of sen- Of course there is no a priori reason that CH tences φ which are proved by ZFC. Note that if should be decided solely on the basis of projective there were only one test structure, then for each evidence. Nevertheless, in the 1970s Martin con- sentence φ either ZFC 0 φ or ZFC 0 ¬φ. So in jectured that the existence of projective evidence the logic 0 defined by this collection of test against CH will eventually be seen to follow from structures, no propositions are independent of reasonable axioms. the axioms ZFC. By the Gödel Completeness Theorem, first order Axioms for H(ω2) logic is the weakest (nontrivial) logic. Encouraged by the success in Part I in finding the To formulate the notion of Generic Soundness, correct axioms for H(ω1), and refusing to be dis- I first define the cumulative hierarchy of sets: couraged by the observation that large cardinal ax- this is a class of sets indexed by the ordinals. The ioms cannot settle CH, we turn our attention to set with index α is denoted Vα, and the definition H(ω2). Here we have a problem if we regard large is by induction on α as follows: V0 = ∅ ; cardinal axioms as our sole source of inspiration: Vα+1 = P(Vα) ; and if β is a limit ordinal, then Even if there is an analog of Projective Determinacy Vβ = ∪{Vα | α<β}. It is easily verified that the for H(ω2), how can we find it or even recognize it sets Vα are increasing, and it is a consequence of if we do find it? the axioms that every set is a member of Vα for My point is simply that the axiom(s) we seek can- large enough α. not possibly be implied by any (consistent) large It follows from the definitions that Vω = H(ω) cardinal hypothesis remotely related to those cur- and that Vω+1 ⊆ H(ω1). However, Vω+1 = H(ω1). rently accepted as large cardinal hypotheses. Nevertheless, Vω+1 and H(ω1) are logically equiv- Strong Logics alent in that each can be analyzed within the other. The solution is to take an abstract approach. This The relationship between Vω+2 and H(ω2) is far we shall do by considering strengthenings of first more subtle. If the Continuum Hypothesis holds, order logic and analyzing the following question, then these structures are logically equivalent, but which I shall make precise. the assertion that these structures are logically

682 NOTICES OF THE AMS VOLUME 48, NUMBER 7 fea-woodin.qxp 6/29/01 1:11 PM Page 683

equivalent does not imply the Continuum Hy- Definition. For a given strong logic 0, the theory pothesis. of the structure H(ω2), ∈ is “finitely axiomatized Suppose that M is a transitive set such that over ZFC” if there exists a sentence Ψ such that for M,∈  ZFC. The cumulative hierarchy in the some α, Vα  ZFC + Ψ, and for each sentence φ, sense of M is simply the sequence M ∩ Vα indexed ZFC + Ψ 0 “H(ω2), ∈  φ ” by M ∩ Ord. It is customary to denote M ∩ Vα by Mα. If M is countable, then one can always reduce if and only if H(ω2), ∈  φ . to considering Cohen extensions, M∗, which are Universally Baire Sets transitive and for which the canonical embedding There is a transfinite hierarchy which extends the of M into M∗ (given by Cohen’s construction) is hierarchy of the projective sets; this is the hierar- the identity. Thus, in this situation, M ⊆ M∗ and chy of the universally Baire sets. Using these sets, the ordinals of the Cohen extension coincide with I shall define a specific strong logic, Ω-logic. those of the initial model. The precise formulation of Generic Soundness Definition (Feng-Magidor-Woodin). A set A ⊆ Rn involves notation from the Boolean Valued Model is universally Baire if for every continuous function interpretation, due to Scott and Solovay, of Cohen’s F : Ω → Rn, method of forcing. In the first part of this article I noted that Cohen extensions are parameterized where Ω is a compact Hausdorff space, F−1[A] (the by complete Boolean algebras (in the sense of the preimage of A by F) has the property of Baire in Ω; initial structure). Given a complete Boolean alge- i.e., there exists an open set O ⊆ Ω such that the sym- bra B, one can analyze within our universe of sets metric difference F−1[A]  O is meager. the Cohen extension of our universe that B could It is easily verified that every Borel set A ⊆ Rn be used to define in some virtual larger universe is universally Baire. More generally, the univer- where our universe, V, becomes, say, a countable sally Baire sets form a σ-algebra closed under transitive set. V B denotes this potential extension, B preimages by Borel functions and for each ordinal α, Vα denotes the α-th level B B f : Rn → Rm. of V . For each sentence φ, the assertion “Vα  φ” is formally an assertion about the ordinal α and A little more subtle, and perhaps surprising, is the Boolean algebra B; this calculation is the that the universally Baire sets are Lebesgue mea- essence of Cohen’s method. surable. Every analytic set is universally Baire. The fol- Definition. Suppose that 0 is a strong logic. The lowing theorem is proved using Jensen’s Covering logic 0 satisfies Generic Soundness if for each sen- Lemma. tence φ such that ZFC 0 φ, the following holds. Suppose that B is a complete Boolean algebra, α Theorem (Feng-Magidor-Woodin). Suppose that B B is an ordinal, and Vα  ZFC. Then Vα  φ. every projective set is universally Baire. Then every analytic subset of [0, 1] is determined. Our context for considering strong logics will require at the very least that there exists a proper The improvements of this theorem are quite class of Woodin cardinals, and so the requirement subtle; the assumption that every projective set is of Generic Soundness is nontrivial. More precisely, universally Baire does not imply Projective Deter- assuming there exists a proper class of Woodin car- minacy. dinals, for any complete Boolean algebra B there The following theorem of Neeman improves an exist unboundedly many ordinals α such that earlier version of [Feng-Magidor-Woodin] which B Vα  ZFC. required the stronger hypothesis: There exist two The motivation for requiring Generic Sound- Woodin cardinals. ness is simply that if ZFC  φ, then the negation 0 Theorem (Neeman). Suppose that there is a Woodin of φ should not be (provably) realizable by pass- cardinal. Then every universally Baire subset of ing to a Cohen extension. Of course, if  is any 0 [0, 1] is determined. strong logic which satisfies the condition of Generic Soundness, then it cannot be the case that If there exists a proper class of Woodin cardi- either ZFC 0 CH or ZFC 0 ¬CH; i.e., CH remains nals, then the universally Baire sets are closed unsolvable. This might suggest that an approach under continuous images (and so projections). to resolving the theory of H(ω2) based on strong Therefore: logics is futile. But an important possibility arises Theorem. Suppose that there are arbitrarily large through strong logics. This is the possibility that Woodin cardinals. Then every projective set is uni- augmenting ZFC with a single axiom yields a sys- versally Baire. tem of axioms powerful enough to resolve, through inference in the strong logic, all questions Sometimes the Euclidean space R is not the most about H(ω2). illuminating space with which to deal. Let K ⊆ [0, 1]

AUGUST 2001 NOTICES OF THE AMS 683 fea-woodin.qxp 6/29/01 1:11 PM Page 684

be the Cantor set, though any uncountable closed, ing for universally Baire sets, A0 and A1 , nowhere dense subset of [0, 1] would suffice for A0 subsets of K. In the context of deter- is reducible to B if there exists a continuous func- minacy assumptions, Martin proved that this tion partial order is well founded [Moschovakis 1980]. f : K → K In the absence of any determinacy assumptions, Martin’s theorem can be formulated as follows. such that A = f −1(B). The set A is strongly re- ducible to B if the function f can be chosen such Theorem (Martin). Suppose that Ak : k ∈ N is a that for all x, y ∈ K, |f (x) − f (y)|≤(1/2)|x − y|. sequence of subsets of K such that for all k ∈ N, K\ There is a remarkably useful lemma of Wadge both Ak+1 and Ak+1 are strongly reducible to which I state for the projective subsets of the Can- Ak . Then there exists a continuous function K → K −1 tor set and with the reducibilities just defined. g : such that g (A1) does not have the There is a version of this lemma for subsets of property of Baire. [0, 1], but the definitions of reducible and strongly As a corollary we obtain the well-foundedness reducible must be changed. of Woodin cardinal. sets. This is Ω-logic; the “proofs” in Ω-logic are witnessed by universally Baire sets which can be Suppose that A0 and A1 are universally Baire sub- assumed to be subsets of the Cantor set K. The sets of K. Then either A0 is reducible to A1 or A1 ordinal rank of the witness in the hierarchy of is strongly reducible to K\A0. such sets I have just defined provides a quite Suppose that f : K → K is such that for all reasonable notion of the length of a proof in Ω- x, y ∈ K , |f (x) − f (y)|≤(1/2)|x − y| . Then for logic. some x0 ∈ K, f (x0)=x0. This implies that no set The definition of Ω-logic involves the notion of A ⊆ K can be strongly reducible to its comple- an A-closed transitive set where A is universally ment K\A. Therefore, given two universally Baire Baire. subsets of K, A0 and A1, and assuming there is a A-closed Sets Woodin cardinal, exactly one of the following must Suppose that M is a transitive set with the prop- hold. This is easily verified by applying the previ- erty that M,∈  ZFC. ous theorem to the relevant pairs of sets, sorting Suppose that (Ω,F,τ) ∈ M and that through the various possibilities, and eliminating 1. M,∈  “Ω is a compact Hausdorff space”. those that lead to the situation that a set is strongly 2. τ is the topology on Ω; i.e., τ is the set of reducible to its complement. O ∈ M such that M ∈  “O ⊆ Ω and O is 1. Both A0 and K\A0 are strongly reducible to open”. A1, and A1 is not reducible to A0 (or to 3. M,∈  “F ∈ C(Ω, R)”. K\A0). For example, if M is countable and 2. Both A and K\A are strongly reducible to 1 1 M,∈  “Ω is the unit interval [0, 1]”, A0, and A0 is not reducible to A1 (or to K\A1). then Ω =[0, 1] ∩ M . It is easily verified that 3. A0 and A1 are reducible to each other, or [0, 1] ∩ M is dense in [0, 1], and so in this case Ω K\A0 and A1 are reducible to each other. is a countable dense subspace of [0, 1]. Thus one can define an equivalence relation Notice that the element F of M is necessarily a on the universally Baire subsets of the Cantor set function, F : Ω → R. by A0 ∼w A1 if (3) holds, and one can totally Trivially, τ is a base for a topology on Ω yield- order the induced equivalence classes by defin- ing a topological space which of course need not

684 NOTICES OF THE AMS VOLUME 48, NUMBER 7 fea-woodin.qxp 6/29/01 1:11 PM Page 685

be compact, as the preceding example illustrates. for some sentence Ψ, then T Ω φ if and only if Nevertheless, this topological space is necessarily ZFC Ω (Ψ → φ). completely regular. The function F is easily seen Suppose that φ is a sentence (of L(ˆ=, ∈ˆ)) and to be continuous on this space. that ZFC Ω φ. Suppose that A ⊆ K is a universally Let Ω˜ be the Stone-Cˇech compactification of Baire set which witnesses this. Viewing A as a this space, and for each set O ∈ τ let O˜ be the open “proof”, one can naturally define the “length” of subset of Ω˜ defined by O. This is the complement this proof to be the ordinal of A in the hierarchy of the closure, computed in Ω˜ , of Ω\O. of the universally Baire subsets of the Cantor set, The function F has a unique continuous exten- K, given by the relation universally Baire set A∗ ⊆ K such that Later in this article I shall make use of the for all transitive sets M such that M,∈  ZFC, M notion of the length of a proof in Ω-logic when I is A-closed if and only if M is A∗-closed. Thus, for define abstractly the hierarchy of large cardinal our purposes, the distinction between universally axioms. Baire subsets of R versus universally Baire subsets Ω-logic is unaffected by passing to a Cohen of K, the Cantor set, is not relevant. extension. This is the property of Generic Invari- ance. The formal statement of this theorem Ω-logic involves some notation, which I discuss. It is Having defined A-closure, I can now define Ω- customary in set theory to write for a given logic. This logic can be defined without the large sentence φ, “V  φ” to indicate that φ is true, i.e., cardinal assumptions used here, but the definition true in V, the universe of sets. Similarly, if B is a becomes a bit more technical. complete Boolean algebra, “V B  φ” indicates that Definition. Suppose that there exists a proper φ is true in the Cohen extensions of V that B class of Woodin cardinals and that φ is a sen- could be used to define (again in some virtual tence. Then universe where our universe becomes a countable transitive set, as briefly discussed when the  B ZFC Ω φ notation “Vα  φ” was introduced just before the if there exists a universally Baire set A ⊆ R such definition of Generic Soundness). that M,∈  φ for every countable transitive A- Theorem (Generic Invariance). Suppose that closed set M such that M,∈  ZFC. there exists a proper class of Woodin cardinals and that φ is a sentence. Then for each complete There are only countably many sentences in Boolean algebra B , ZFC  φ if and only if the language L(ˆ=, ∈ˆ), and, further, the universally Ω V B “ZFC  φ”. Baire sets are closed under countable unions and Ω preimages by Borel functions. Therefore there Similar arguments establish that if there exists must exist a single universally Baire set AΩ ⊆ R a proper class of Woodin cardinals, then Ω-logic such that for all sentences φ of L(ˆ=, ∈ˆ), ZFC Ω φ satisfies Generic Soundness. if and only if M,∈  φ for every countable tran- The following theorem is a corollary of results sitive set M such that M is AΩ-closed and mentioned in the first part of this article. M,∈  ZFC. Thus Ω-logic is the strong logic Theorem. Suppose that there exists a proper class defined by taking as the collection of test struc- of Woodin cardinals. Then for each sentence φ, tures the countable transitive sets M such that M,∈ is a model of ZFC and M is AΩ-closed. ZFC Ω “H(ω1), ∈  φ ” One can easily generalize the definition of Ω- if and only if H(ω1), ∈  φ . logic to define when T Ω φ where T is an arbi- trary theory containing ZFC. If T is simply ZFC + Ψ A straightforward corollary is that

AUGUST 2001 NOTICES OF THE AMS 685 fea-woodin.qxp 6/29/01 1:11 PM Page 686

ZFC Ω Projective Determinacy, of the projective sets, assuming Projective Deter- minacy, does not require the Axiom of Choice. which vividly illustrates that Ω-logic is stronger These considerations support the claim that than first order logic. the structure H(ω ), ∈ is indeed the next struc- The question of whether there can exist analogs 2 ture to consider after H(ω ), ∈, being the sim- of determinacy for the structure H(ω ), ∈ can 1 2 plest structure where the influence of the Axiom now be given a precise formulation. of Choice is manifest. Can there exist a sentence Ψ such that The Axiom (∗) for all sentences φ either I now come to a central definition, which is that of the axiom (∗). This axiom is a candidate for ZFC + Ψ Ω “H(ω2), ∈  φ ’’ or the generalization of Projective Determinacy to   ∈  ¬ ZFC + Ψ Ω “ H(ω2), φ’’ the structure H(ω2), ∈. The definition of the axiom (∗) involves some more notation from the and such that ZFC + Ψ is Ω-consistent? syntax of formal logic. It is frequently important to monitor the complexity of a formal sentence. Such sentences Ψ will be candidates for the This is accomplished through the Levy hierarchy. generalization of Projective Determinacy to H(ω ). 2 The collection of Σ formulas of our language Notice that I am not requiring that the sentence Ψ 0 L(ˆ=, ∈ˆ) is defined as the smallest set of formulas be a proposition about H(ω ); the sentence can 2 which contains all quantifier-free formulas and refer to arbitrary sets. which is closed under the application of bounded Why seek such sentences? quantifiers. Here is why. By adopting axioms which “settle” Thus, if ψ is a Σ formula, then so are the for- the theory of H(ω ), ∈ in Ω-logic, one recovers 0 2 mulas (∀x ((x ∈ˆx ) → ψ)) and (∃x ((x ∈ˆx ) ∧ ψ)) . for the theory of this structure the empirical com- i i j i i j We shall be interested in formulas which are of pleteness currently enjoyed by number theory. the form (∀x (∃x ψ)), where ψ is a Σ formula. This is because of the generic invariance of Ω- i j 0 These are the Π formulas. Somewhat simpler are logic. 2 the Π formulas and the Σ formulas; these are the More speculatively, such axioms might allow 1 1 formulas of the form (∀x ψ) or of the form (∃x ψ) for the development of a truly rich theory for the i i respectively, where ψ is again a Σ formula. structure H(ω ), ∈, free to a large extent from 0 2 Informally, a Π sentence requires two (nested) the ubiquitous occurrence of unsolvable problems. 2 “unbounded searches” to verify that the sentence Compare, for example, the theory of the projective is true, whereas for a Π sentence only one un- sets as developed under the assumption of Pro- 1 bounded search is required. Verifying that a Σ sen- jective Determinacy with the theory developed of 1 tence is true is even easier. the problems about the projective sets which are For example, consider the structure H(ω), ∈, not solvable simply from ZFC. which I have already noted is in essence the stan- The ideal I , which I now define, plays an es- NS dard structure for number theory. sential and fundamental role in the usual formu- Many of the famous conjectures of modern lation of Martin’s Maximum. mathematics are expressible as Π1 sentences in this Definition. INS is the σ-ideal of all sets A ⊆ ω1 structure. This includes both Goldbach’s Conjec- such that ω1\A contains a closed unbounded ture and the Riemann Hypothesis. However, the set. A set S ⊆ ω1 is stationary if for each closed, Twin Prime Conjecture is expressible by a Π2 sen- unbounded set C ⊆ ω1, S ∩ C = ∅. A set S ⊆ ω1 tence, as is, for example, the assertion that P =NP , is co-stationary if the complement of S is and neither is obviously expressible by a Π1 sen- stationary. tence. This becomes interesting if, say, either of these latter problems were proved to be unsolv- The countable additivity and the nonmaximal- able from, for example, the natural axioms for ity of the ideal I are consequences of the Axiom NS H(ω), ∈. Unlike the unsolvability of a Π sen- of Choice. 1 tence, from which one can infer its “truth”, for Π In my view, the continuum problem is a direct 2 sentences the unsolvability does not immediately consequence of assuming the Axiom of Choice. yield a resolution. This is simply because by assuming the Axiom of If M is a transitive set and P and Q are subsets Choice, the reals can be well ordered and so of M, then one may consider M,P,Q,∈ as a |R| = ℵ for some ordinal α. Which α? This is the α structure for the language L(ˆ=, P,ˆ Q,ˆ ∈ˆ), obtained continuum problem. by adding two new symbols, Pˆ and Qˆ , to L(ˆ=, ∈ˆ). Arguably, the stationary, co-stationary, subsets One defines the Σ formulas and the Π formulas of ω constitute the simplest true manifestation 0 2 1 of this expanded language in the same way as of the Axiom of Choice. A metamathematical analy- above. sis shows that assuming Projective Determinacy, The structure I actually wish to consider is there is really no manifestation within H(ω1) of the Axiom of Choice. More precisely, the analysis H(ω2), INS,X,∈,

686 NOTICES OF THE AMS VOLUME 48, NUMBER 7 fea-woodin.qxp 6/29/01 1:11 PM Page 687

where X ⊆ R is universally Baire. If φ is a sen- cardinal. The axiom which asserts the existence tence in the language L(ˆ=, P,ˆ Q,ˆ ∈ˆ) for this struc- of an uncountable inaccessible cardinal is the ture, then there is a natural interpretation of the weakest traditional large cardinal axiom. assertion that Theorem. Suppose that there exists a proper class ZFC + “H(ω2), INS,X,∈  φ” of Woodin cardinals and that there is an inaccessi- is Ω-consistent. The only minor problem is how ble cardinal which is a limit of Woodin cardinals. to deal with X. But X is universally Baire. Thus I Then ZFC + axiom (∗) is Ω-consistent. define There is an elaborate machinery of iterated forc-  I ∈  ZFC + “ H(ω2), NS,X, φ” ing: this is the technique of iterating Cohen’s to be Ω-consistent if for every universally Baire set method of building extensions [Shelah 1998]. It is A there exists a countable transitive set M such through application of this machinery that, for ex- that ample, the consistency of 1. M is A-closed and M is X-closed; ZFC + “Martin’s Maximum’’ 2. M,∈  ZFC; is established (assuming the consistency of ZFC to- M M gether with a specific large cardinal axiom, much 3. H(ω2) , (INS) ,X∩ M,∈  φ, where stronger than, for example, the axiom that there M H(ω2) = {a ∈ M |M,∈  “a ∈ H(ω2) ”}, is a Woodin cardinal). Iterated forcing can be used to show the consis- and tency of statements of the form “H(ω2), ∈  φ ” M (INS) = {a ∈ M |M,∈  “a ∈INS ”}. for a rich variety of Π2 sentences φ. The previous theorem, on the Ω-consistency of These are the relevant sets as computed in M. the axiom (∗), is proved using the method of forc- With these definitions in hand, I come to the de- ing but not using any machinery of iterated forcing. finition of the axiom (∗). The version I give is an- Further, the theorem is not proved as a corollary of chored in the projective sets; stronger versions of some deep analysis of which Π sentences can hold the axiom are naturally obtained by allowing more 2 in H(ω ). universally Baire sets in the definition. 2 The axiom (∗) settles in Ω-logic the full theory  ∈ Axiom (∗): There is a proper class of of the structure H(ω2), . The stronger version Woodin cardinals, and for each projec- of the following theorem, obtained by replacing H(ω ), ∈ with H(ω ),X,∈ where X is a pro- tive set X ⊆ R, for each Π2 sentence φ, 2 2 if the theory jective set, is also true. Theorem. Suppose that there exists a proper ZFC+ “H(ω ), I ,X,∈  φ” 2 NS class of Woodin cardinals. Then for each sentence is Ω-consistent, then φ, either

ZFC + axiom (∗) Ω “H(ω2), ∈  φ ” or H(ω2), INS,X,∈  φ. What kinds of assertions are there which can be ZFC + axiom (∗) Ω “H(ω2), ∈  ¬φ’’. formulated in the form H(ω2), ∈  φ for some Suppose that φ is a Π2 sentence and that Π2 sentence φ? There are many examples. One ex- X ⊆ R is a projective set such that ample is Martin’s Axiom (ω1). Another, which identifies a consequence of the axiom (∗), is the ZFC + “H(ω2), INS,X,∈  φ” assertion that if A ⊂ R and B ⊂ R are each nowhere is not Ω-consistent. Then the analysis behind the countable and of cardinality ℵ , then A and B are 1 proof of the Ω-consistency of the axiom (∗) yields order isomorphic. A set X ⊂ R is nowhere count- a projective witness for the corresponding Ω-proof. able if X ∩ O is uncountable for each (nonempty) Thus the axiom (∗) is in essence an axiom which open set O ⊆ R. Thus, assuming the axiom (∗), can be localized to H(ω ). More precisely, there is there is exactly one possible order type for nowhere 2 a (recursive) set of axioms, i.e., a recursive theory T, countable subsets of R which have cardinality ℵ1. such that, assuming the existence of a proper I refer the reader to [Shelah 1998] for details, ref- ∗ erences, and other examples. class of Woodin cardinals, the axiom ( ) holds if The axiom (∗) is really a maximality principle and only if H(ω ), ∈  T. somewhat analogous to asserting algebraic clo- 2 sure for a field. Finally, assuming there is a proper class of A cardinal κ is an inaccessible cardinal if it Woodin cardinals, the axiom (∗) is equivalent to a is a limit cardinal with the additional property strong form of a bounded version of Martin’s Max- that any cofinal subset of κ necessarily has imum, so again seemingly disparate threads are cardinality κ. For example, ω is an inaccessible woven into a single tapestry.

AUGUST 2001 NOTICES OF THE AMS 687 fea-woodin.qxp 6/29/01 1:11 PM Page 688

ℵ The Axiom (∗) and 2 0 reasonably take this as the definition of 0 . This (Ω) There is a Π2 sentence ψAC , which if true in the suggests the definition of 0 . ℵ structure H(ω ), ∈ implies that 2 0 = ℵ . 2 2 Definition. Suppose that there exists a proper The statement “H(ω ), ∈  ψ ” is Ω-consis- 2 AC class of Woodin cardinals. Then tent, and so as a corollary the axiom (∗) implies ℵ 0 ℵ (Ω) 2 = 2 . 0 = {kφ | ZFC Ω φ}. Definition ψAC : Suppose S and T are each Suppose that M is a transitive set and A ⊆ M. stationary, co-stationary, subsets of ω1 . Then Then the set A is definable in the structure M,∈ ⊆ there exist: a closed unbounded set C ω1 ; a if there is a formula ψ(x1) of L(ˆ=, ∈ˆ) such that well-ordering L, < of cardinality ω ; and a 1 A = {a |M,∈  ψ[a]}. bijection π : ω1 → L such that for all α ∈ C, ∗ The following theorem is one version of Tarski’s α ∈ S ↔ α ∈ T , theorem on the undefinability of truth. where α∗ is the countable ordinal given by the order type of {π(β) | β<α} as a suborder of Theorem (Tarski). Suppose that M is a transitive set ⊆ { | ∈  } L, <. with H(ω) M. Then the set kφ M, φ is not definable in the structure M,∈. By standard methods ψ can be shown to be AC { |  } expressible in the required form (as a Π sen- For each sentence Ψ, the set kφ ZFC + Ψ φ 2  ∈ tence). is definable in the structure H(ω), . Thus by Tarski’s theorem, for each sentence Ψ the set ℵ Lemma. Suppose that ψAC holds. Then 2 0 = ℵ2 . {φ | ZFC + Ψ  “H(ω), ∈  φ”} There is a subtle aspect to this lemma. Suppose is not equal to the set {φ |H(ω), ∈  φ}. This is that CH holds and that x : α<ω  is an enu- α 1 a special case of Gödel’s First Incompleteness The- meration of R. orem. Thus ψ must fail. However, it is possible that AC Analogous considerations apply in our situation, there is no counterexample to ψAC which is de- finable from the given enumeration x : α<ω . and so our basic problem of determining when the α 1  ∈ I note that Martin’s Maximum can be shown to theory of the structure H(ω2), can be finitely axiomatized, over ZFC, in Ω-logic naturally leads imply ψAC . This gives a completely different view (Ω) of why the axiom Martin’s Maximum implies that to the problem: How complicated is 0 ? ℵ This set looks potentially extremely compli- 2 0 = ℵ2 . cated, for it is in essence Ω-logic. Note that since And What about CH? {φ | ZFC Ω “H(ω1), ∈  φ ”} The basic question is the following. Is there an ana- log of the axiom (∗) in the context of CH? Con- is equal to the set {φ |H(ω1), ∈  φ} by Tarski’s (Ω) tinuing the analogy with the theory of fields, one theorem, 0 is not definable in the structure seeks to complete the similarity: H(ω1), ∈. The calculation of the complexity of 0(Ω) in- ?+CH real closed + ordered ∼ . volves adapting the Inner Model Program to ana- axiom (∗) algebraically closed lyze models of Determinacy Axioms rather than More generally: Under what circumstances can the models of Large Cardinal Axioms. theory of the structure H(ω2), ∈ be finitely ax- This analysis, which is a bit involved and tech- iomatized, over ZFC, in Ω-logic? nical, yields the following result where c+ denotes Formally the sentences of our language, L(ˆ=, ∈ˆ), the least cardinal greater than c. Suppose that are (certain) finite sequences of elements of the un- there exists a proper class of Woodin cardinals. derlying alphabet, which in this case can be taken Then 0(Ω) is definable in the structure H(c+), ∈. to be N. There is a natural (recursive) bijection of Now if the Continuum Hypothesis holds, then + N with the set of all finite sequences from N. This c = ω1 and so H(c )=H(ω2). Therefore, if the associates to each sentence φ of L(ˆ=, ∈ˆ) a positive Continuum Hypothesis holds, then 0(Ω) is definable integer kφ, which is the Gödel number of φ. in the structure H(ω2), ∈. To address the questions above, I require a de- Appealing to Tarski’s theorem, we obtain as a finition generalizing the definition of 0 where 0 corollary our main theorem. is the set Theorem. Suppose that there exists a proper class {kφ | φ is a Σ1 sentence and H(ω), ∈  φ}. of Woodin cardinals, Vκ  ZFC + Ψ, and for each L ∈ˆ Assuming ZFC is consistent, then the set sentence φ of (ˆ=, ) either ZFC + Ψ  “H(ω ), ∈  φ ” or {kφ | φ is a sentence and ZFC  φ} Ω 2 ZFC + Ψ  “H(ω ), ∈  ¬φ”. is recursively equivalent to 0 (a simple, though Ω 2 somewhat subtle, claim). In fact, one could Then CH is false.

688 NOTICES OF THE AMS VOLUME 48, NUMBER 7 fea-woodin.qxp 6/29/01 1:11 PM Page 689

There are more precise calculations of the com- Suppose that κ is an ordinal and that φ is a Σ2 plexity of 0(Ω) than I have given. For the indicated formula. Then “V  φ[κ]” indicates that φ is true application on CH, one is actually interested in the of κ in V, the universe of sets. Similarly, if B is a complexity of sets X ⊆ N which are Ω-recursive. complete Boolean algebra, then “V B  φ[κ]” indi- Ultimately, it is not really CH which is the critical cates that φ is true of κ in the Cohen extensions issue, but effective versions of CH. of V that B could be used to define. An inaccessible cardinal κ is strongly inacces- The Ω Conjecture sible if for each cardinal λ<κ, 2λ <κ. Perhaps Ω-logic is not the strongest reasonable ∃ logic. Definition. ( x1φ) is a large cardinal axiom if φ(x ) is a Σ -formula; and, as a theorem of ZFC, ∗ 1 2 Definition (Ω -logic). Suppose that there exists a if κ is a cardinal such that V  φ[κ], then κ is un- proper class of Woodin cardinals and that φ is a countable, strongly inaccessible, and for all com- sentence. Then plete Boolean algebras B of cardinality less than B  ZFC Ω∗ φ κ, V φ[κ].

if for all ordinals α and for all complete Boolean Definition. Suppose that (∃x1φ) is a large cardi- B B algebras B, if Vα  ZFC, then Vα  φ. nal axiom. Then V is φ-closed if for every set X there exist a transitive set M and κ ∈ M ∩ Ord Generic Soundness is immediate for Ω∗ -logic, ∗ such that and evidently Ω -logic is the strongest possible M,∈  ZFC, logic satisfying this requirement. The property of generic invariance also holds X ∈ Mκ and such that M,∈  φ[κ]. for Ω∗ -logic. The connection between Ω-logic and first order Theorem (Generic Invariance). Suppose that logic is now easily identified. there exists a proper class of Woodin cardinals and that φ is a sentence. Then for each complete Lemma. Suppose that there exists a proper class of Woodin cardinals and that Ψ is a Π sentence. Then Boolean algebra B, ZFC  ∗ φ if and only if 2 Ω  B ZFC Ω Ψ if and only if there is a large cardinal V  “ZFC  ∗ φ”. Ω axiom (∃x1φ) such that Having defined Ω∗-logic, a natural question arises. ∗  Is Ω -logic the same as Ω-logic (at least for Π2- ZFC Ω “V is φ-closed” sentences)? The restriction to Π2 sentences is a and such that ZFC + “V is φ-closed”  Ψ. necessary one. An immediate corollary of this lemma is that the Ω Conjecture: Suppose that there ex- Ω Conjecture is equivalent to the following con- ists a proper class of Woodin cardinals. jecture, which actually holds for all (conventional) Then for each Π2 sentence φ, large cardinal axioms currently within reach of ZFC Ω∗ φ if and only if ZFC Ω φ. the Inner Model Program. If the Ω Conjecture is true, then I find the ar- Conjecture: Suppose that there exists gument against CH, based on strong logics, to be a proper class of Woodin cardinals. Sup- a more persuasive one. One reason is that the Ω pose that (∃x1φ) is a large cardinal Conjecture implies that if theory of the structure axiom such that V is φ-closed. H(ω2), ∈ is finitely axiomatized, over ZFC, in Ω∗ -logic, then CH is false. Then ZFC Ω “V is φ-closed.” Connections with the Logic of Large Cardinal Axioms The equivalence of this conjecture with the Ω Con- Ω-logic is intimately connected with an abstract no- jecture is essentially a triviality. tion of what a large cardinal axiom is. If the Ω Con- Nevertheless, reformulating the Ω Conjecture jecture is true, then the validities of ZFC in Ω- in this fashion does suggest a route toward prov- logic—these are the sentences φ such that ing the Ω Conjecture. Moreover, the reformula- ZFC Ω φ—calibrate the large cardinal hierarchy. tion, in conjunction with the preceding lemma, To illustrate this claim I make the following ab- shows quite explicitly that if the Ω Conjecture is stract definition of a large cardinal axiom, essen- true, then Ω-logic is simply the natural logic tially identifying large cardinal axioms with one associated to the set of large cardinal axioms fundamental feature of such axioms. This is the (∃x1φ) for which V is φ-closed. feature of “generic stability”. It is precisely this as- The Ω Conjecture and the Hierarchy of Large pect of large cardinal axioms which underlies the Cardinals fact that such axioms cannot settle the Continuum To the uninitiated the plethora of large cardinal ax- Hypothesis. A formula φ is a Σ2 formula if it is of ioms seems largely a chaotic collection founded on the form (∃xi(∀xj ψ)) where ψ is a Σ0 formula. a wide variety of unrelated intuitions. An enduring

AUGUST 2001 NOTICES OF THE AMS 689 fea-woodin.qxp 6/29/01 1:11 PM Page 690

mystery of large cardinals is that empirically they remarkable event, it seems relatively few believe really do seem to form a well-ordered hierarchy. that such a resolution will ever happen. The search for an explanation leads to the fol- Of course, for the dedicated skeptic there is al- lowing question. ways the “widget possibility”. This is the future where it is discovered that instead of sets we Is it possible to formally arrange the should be studying widgets. Further, it is realized ∃ large cardinal axioms ( x1φ) into a that the axioms for widgets are obvious and, more- well-ordered hierarchy incorporating over, that these axioms resolve the Continuum the known comparisons of specific Hypothesis (and everything else). For the eternal axioms? skeptic, these widgets are the integers (and the Con- If the Ω Conjecture is true, then the answer is affir- tinuum Hypothesis is resolved as being meaning- mative, at least for those axioms suitably realized less). Widgets aside, the incremental approach within the universe of sets. More precisely, suppose sketched in this article comes with a price. What there exists a proper class of Woodin cardinals. The about the general continuum problem; i.e. what large cardinal axioms (∃x1φ) such that about H(ω3), H(ω4), H(ω(ω1+2010)), etc.? ZFC Ω “V is φ-closed” The view that progress towards resolving the Continuum Hypothesis must come with progress are naturally arranged in a well-ordered hierarchy on resolving all instances of the Generalized Con- by comparing the minimum possible lengths of the tinuum Hypothesis seems too strong. The under-  Ω-proofs, ZFC Ω“V is φ-closed”. standing of H(ω) did not come in concert with an If the Ω Conjecture holds in V, then this hierarchy understanding of H(ω1), and the understanding includes all large cardinal axioms (∃x φ) such that 1 of H(ω1) failed to resolve even the basic myster- the universe V is φ-closed. This, arguably, accounts ies of H(ω2). The universe of sets is a large place. for the remarkable success of the view that all We have just barely begun to understand it. large cardinal axioms are comparable. Of course, it is not the large cardinals themselves (κ such References that φ[κ] holds) which are directly compared, but [Feng, Magidor, and Woodin 1992] Q. FENG, M. MAGIDOR, the auxiliary notion that the universe is φ-closed. and W. H. WOODIN, Universally Baire sets of reals, Set Nevertheless, restricted to those large cardinal Theory of the Continuum (H. Judah, W. Just, and H. Woodin, eds.), Math. Sci. Res. Inst. Publ., vol. 26, axioms (∃x1φ) currently within reach of the Inner Springer-Verlag, Heidelberg, 1992, pp. 203–242. Model Program, this order coincides with the usual [Kanamori 1994] A. KANAMORI, The Higher Infinite: Large order which is (informally) defined in terms of Cardinals in Set Theory from Their Beginnings, Per- consistency strength. spect. Math. Logic, Springer-Verlag, Berlin, 1994. Finally, this hierarchy explains, albeit a poste- [Levy and Solovay 1967] A. LEVY and R. SOLOVAY, Measur- riori, the intertwining of large cardinal axioms and able cardinals and the continuum hypothesis, determinacy axioms. Israel J. Math. 5 (1967), 234–248. Resolving the Ω Conjecture is essential if we are [Martin 1976] D. MARTIN, Hilbert’s first problem: The Con- tinuum Hypothesis, Mathematical Developments to advance our understanding of large cardinal Arising from Hilbert’s Problems (F. Browder, ed.), axioms. If the Ω Conjecture is true, we obtain, at Proc. Sympos Pure Math., vol. 28, Amer. Math. Soc., last, a mathematically precise definition of this Providence, RI, 1976, pp. 81–92. hierarchy. But, as one might expect, with this [Moschovakis 1980] Y. MOSCHOVAKIS, Descriptive Set progress come problems (of comparing specific Theory, Stud. Logic Found. Math., vol. 100, North- large cardinal axioms) which seem genuinely out Holland, 1980. of reach of current methods. If the Ω Conjecture [Neeman 1995] I. NEEMAN, Optimal proofs of determi- is refuted from some large cardinal axiom (which nacy, Bull. Symbolic Logic 1 (3) (1995), 327–339. [Shelah 1998] S. SHELAH, Proper and Improper Forcing, 2nd likely must transcend every determinacy axiom), ed., Perspect. Math. Logic, Springer-Verlag, Berlin, then the explicit hierarchy of large cardinal axioms 1998. as calibrated by the validities of Ω-logic is simply [Wadge 1972] W. WADGE, Degrees of complexity of an initial segment of something beyond. subsets of the Baire space, Notices Amer. Math. Soc. 19 (1972), A-714. Concluding Remarks [Woodin 1999] W. HUGH WOODIN, The Axiom of Determi- So, is the Continuum Hypothesis solvable? Per- nacy, Forcing Axioms, and the Nonstationary Ideal, haps I am not completely confident the “solution” Ser. Logic Appl., vol. 1, de Gruyter, 1999. I have sketched is the solution, but it is for me con- vincing evidence that there is a solution. Thus, I now believe the Continuum Hypothesis is solvable, which is a fundamental change in my view of set theory. While most would agree that a clear reso- lution of the Continuum Hypothesis would be a

690 NOTICES OF THE AMS VOLUME 48, NUMBER 7