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A Woodin Cardinal? John R WHAT IS... ? a Woodin Cardinal? John R. Steel All mathematical statements can be expressed Large cardinal hypotheses attempt to capture in the language of set theory (LST), whose vari- the idea that there are more sets than one can ables are understood as ranging over sets, and possibly imagine, by means of the following in- whose only non-logical symbol ∈ stands for the formal reflection principle: suitable properties of membership relation. The vast majority of math- V reflect to some Vα. For one example, V is in- ematical proofs require no more than the axioms finite, and the ordinary Axiom of Infinity asserts of Zermelo-Fraenkel set theory with Choice, or that some Vα shares this property. For another, ZFC. In this way, set theory provides a foundation V is a model of second-order ZFC (also called for all of mathematics. Nevertheless, a surprising Kelley-Morse set theory), so our informal princi- number of basic questions about sets in general ple leads to the assertion that some Vκ satisfies are not decided by the axioms of ZFC; moreover second-order ZFC. This is equivalent to κ being an many of the more abstract questions of analysis, inaccessible cardinal. algebra, and topology are left similarly undecid- The stronger large cardinal hypotheses assert ed. Perhaps the most famous of the undecided the existence of elementary embeddings j : V → M questions is Cantor’s Continuum Problem: what that are nontrivial, i.e., not the identity. Here M is a is the cardinality of the set of all real numbers? transitive class. Elementarity means that whenever LST Another more concrete such question is whether ϕ(v1,...,vn) is a formula of with the displayed all sets of real numbers that are projective are free variables, and a1,...,an are sets, then Lebesgue measurable. (A set is projective if and V ⊨ ϕ[a1,...,an] ⇔ M ⊨ ϕ[j(a1),...,j(an)]. only if it can be built up from a countable inter- (For N a transitive set or class, and b ,...,b ∈ N, section of open sets by taking continuous images 1 n we say N ⊨ ϕ[b ,...,b ] if and only if ϕ(v ,...,v ) and complements finitely many times.) 1 n 1 n is true when its quantifiers are understood as The most fruitful way to extend ZFC so as to ranging over N, its variable vi is understood as remove some of this incompleteness is to strength- naming bi , and the ∈ symbol of LST is understood en its axiom asserting that there are infinite sets. as standing for set-membership.) For such a j, the Large cardinal hypotheses do this. critical point of j, or crit(j), is the least ordinal For α an ordinal number, we define Vα by κ such that j(κ) 6= κ. Since j is order-preserving, transfinite induction: V0 =∅, Vα+1 ={x | x ⊆ Vα}, this implies κ < j(κ). The reflection here occurs at and Vλ = Sα<λ Vα for λ a limit ordinal. Thus Vα κ: if ϕ(κ) holds in V , and M resembles V enough consists of those sets that can be built up in < α that M ⊨ ϕ[κ], then M ⊨∃α<v1ϕ(α)[j(κ)], so stages by taking sets of objects previously formed. by the elementarity of j, there is an α < κ such Each Vα is transitive (i.e., if x ∈ Vα, then x ⊆ Vα), that ϕ(α) holds in V . The more M resembles V , the and α ≤ β ⇒ Vα ⊆ Vβ. It follows from the Axiom more reflection we have. By a result of K. Kunen, ZFC of Foundation of that every set is in some Vα. M = V is impossible, but weaker conditions on M We write V for the union of all the Vα, the universe lead to plausible and useful principles. of all sets. V is not itself a set, but rather what is For example, κ is measurable if and only if sometimes called a proper class. there is any transitive M and nontrivial embed- ding j : V → M with crit(j) = κ. If there is such a John R. Steel is professor of mathematics at the Uni- j,M with Vβ ⊆ M, we say κ is β-strong, and we say versity of California at Berkeley. His email address is κ is strong if and only if κ is β-strong for all β. [email protected]. If there is such a j,M such that every λ-sequence 1146 Notices of the AMS Volume 54, Number 9 of elements of M belongs to M, then we say κ B is projective if and only if membership in B can is λ-supercompact, and we say κ is supercompact be defined within LSA from some real parameter. if and only if κ is λ-supercompact for all λ. We Most questions about projective sets of reals can have listed these properties in order of increasing be phrased in LSA; for example, the Lebesgue strength; indeed, if κ is strong, then Vκ ⊨ “there is measurability and determinacy of projective sets a measurable cardinal”, and if κ is supercompact, can be expressed using sentences in LSA. Then, by then Vκ ⊨ “there is a strong cardinal”. work of Martin, Woodin, and the author, we have S. Ulam first isolated a property equivalent to that the following are equivalent: measurability in 1930: κ is measurable if and only (a) PD, if there is a κ-additive, 2-valued measure defined (b) for each n, every consequence in LSA of on all subsets of κ that gives singletons measure 0. the theory ZFC plus “there are n Woodin (To get a measure from an elementary embedding, cardinals” is true. ⇔ set µ(A) = 1 κ ∈ j(A), for A ⊆ κ. Conversely, Thus PD is precisely the “instrumentalist’s trace” of one gets elementary embeddings from measures Woodin cardinals in the language of second-order using the ultrapower construction.) Embeddings arithmetic. corresponding to stronger large cardinal proper- Underlying the proof that Woodin cardinals ties can be captured in a similar way by systems imply PD is a structure known as the iteration of measures. tree. Roughly speaking, an iteration tree on M is Let κ < δ be cardinals, and A ⊆ Vδ; then κ is a tree of models with root M that is generated A-strong in δ if and only if for all β < δ there is by a certain process. This process involves using an elementary j : V → M such that crit(j) = κ and a system of measures coding an embedding in j(A) ∩ Vβ = A ∩ Vβ. (The case A = Vδ implies κ is one model to generate an embedding with domain β-strong for all β < δ.) We say δ is A-Woodin if and in some other model, and thus equips the tree only if there is a κ < δ that is A-strong in δ, and with commuting elementary embeddings from the we say δ is Woodin in case it is A-Woodin for all models earlier on a given branch to those later on A ⊆ Vδ. The hypothesis that there are Woodin car- that branch. A simple example is an alternating dinals is strictly between the existence of strong chain on M, an iteration tree having two distinct and supercompact cardinals in strength. branches, the “even” branch consisting of models Woodin cardinals were discovered by W. H. Mn for n even (with M0 = M), and the “odd” branch Woodin in 1984. New techniques due to M. Fore- consisting of M0 together with the Mn for n odd. If man, M. Magidor, and S. Shelah had just shown B is a subset of the Bairespace, then an alternating that the Lebesgue measurability of projective sets chain representation of B is a continuous function of real numbers follows from large cardinal hy- A on NN such that for each x ∈ NN, A(x) is an potheses much weaker than had been previously alternating chain on some Vδ, and x ∈ B if and suspected. Woodin showed that in fact the exis- only if the direct limit along the even branch of tence of infinitely many Woodin cardinals implies A(x) is wellfounded. If B has an alternating chain all projective sets of reals are Lebesgue measur- representation, then B is determined. The proof of able. About a year later, in 1985, D. A. Martin and PD from Woodin cardinals goes by showing that if the author showed that the existence of infinite- δ is Woodin, and there are infinitely many Woodin ly many Woodin cardinals implies all projective cardinals above δ, then every projective set has an subsets of the Baire space NN are determined (PD), alternating chain representation on Vδ. a stronger regularity property that, by work of In general, an iteration tree may have transfinite many people in the 1960s and 1970s, is the basis length. The construction of iteration trees, and the for a thorough and detailed structure theory for analysis of the properties of arbitrary iteration ZFC projective sets. (By itself, decides very little trees, is at the heart of many basic open problems about the projective sets.) in pure large cardinal theory. The extent to which Of course, it follows that any large cardinal δ is Woodin (that is, the complexity of those sets hypothesis that implies there are infinitely many A ⊆ V such that δ is A-Woodin) is mirrored in the PD δ Woodin cardinals also implies . In fact, building complexity of the iteration trees one can generate on work of Martin, Woodin had already shown on Vδ.
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