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The Hypothesis, Part I W. Hugh Woodin

Introduction of these and related issues, see [Kanamori, Arguably the most famous formally unsolvable 1994]. problem of is Hilbert’s first prob- The independence of a φ from the axioms of theory is the arithmetic statement: lem: ZFC does not prove φ, and ZFC does not prove ¬φ. Cantor’s : Suppose that Of course, if ZFC is inconsistent, then ZFC proves X ⊆ R is an . Then there exists a bi- anything, so independence can be established only jection π : X → R. by assuming at the very least that ZFC is consis- tent. Sometimes, as we shall see, even stronger as- This problem belongs to an ever-increasing list sumptions are necessary. of problems known to be unsolvable from the The first result concerning the Continuum (usual) axioms of . Hypothesis, CH, was obtained by Gödel. However, some of these problems have now (Gödel). Assume ZFC is consistent. Then been solved. But what does this actually mean? so is ZFC + CH. Could the Continuum Hypothesis be similarly solved? These questions are the subject of this ar- The modern era of set theory began with Cohen’s ticle, and the discussion will involve ingredients discovery of the method of and his appli- from many of the current areas of set theoretical cation of this new method to show: investigation. Most notably, both Theorem (Cohen). Assume ZFC is consistent. Then Axioms and Axioms play central roles. so is ZFC + “CH is false”. For the problem of the Continuum Hypothesis, I shall focus on one specific approach which has de- I briefly discuss the methodology for estab- veloped over the last few years. This should not lishing that a proposition is unsolvable, reviewing be misinterpreted as a claim that this is the only some basic notions from mathematical . It is approach or even that it is the best approach. How- customary to work within set theory, though ulti- mately the , fundamentally arithmetic ever, it does illustrate how the various, quite dis- statements, can be proved in theory. tinct, lines of investigation in modern set theory L(ˆ=, ∈ˆ) denotes the for set the- can collectively yield new, potentially fundamen- ory; this is a countable collection of formulas. The tal, insights into questions as basic as that of the formulas of L(ˆ=, ∈ˆ) with no unquantified occur- Continuum Hypothesis. rences of variables are sentences. Both =ˆ and ∈ˆ are The generally accepted axioms for set theory— simply symbols of this formal language with no but I would call these the twentieth-century other a priori significance. choice—are the Zermelo-Fraenkel Axioms together From elementary logic one has the notion of a with the of Choice, ZFC. For a discussion structure for this language. This is a pair M = M,E , where M is a nonempty set and W. Hugh Woodin is professor of mathematics at the E ⊆ M × M is a binary relation on the set M. If φ University of California, Berkeley. His e-mail address is is a sentence in the language L(ˆ=, ∈ˆ), then the [email protected]. structure M is a model of φ, written “M|= φ” if

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the sentence is true when interpreted as an as- for it is equal to the set {[j]∼∞ | (j,i) ∈ E0}, which sertion within the structure M,E , where the sym- evidently has at most i. bol ∈ˆ is interpreted by the binary relation E and Building New Models of ZFC =ˆ is interpreted by equality on M. Of course, one Is there a model of ZFC? A consequence of Gödel’s could consider structures of the form M,E,∼ , Second Incompleteness Theorem is that one can- where ∼ is an equivalence relation on M intended not hope to prove the existence of a model of ZFC as the of =ˆ, but then one could pass working just from the axioms of ZFC. Nevertheless, to the quotient structure M/ ∼,E/∼ . So nothing one can still study the problem of building new is really gained by this attempt at generality. (hopefully interesting) models of ZFC from given A theory is a set of sentences, and for a given models of ZFC. theory T, I write “M,E |= T” to indicate that Gödel solved the substructure problem in 1938, M,E |= φ for each sentence φ ∈ T. showing that if M,E |= ZFC, then there exists ∗ ⊆ ZFC denotes a specific (infinite) theory. A model M M such that  ∗ ∗ ∗ of ZFC is simply a structure M,E such that M ,E∩ (M × M ) |= ZFC + CH.  | M,E = ZFC. Over 25 years later Cohen, arguably the Galois This can be defined quite naturally without re- of set theory, solved the extension problem. The weakest version of Cohen’s extension theorem is course to formal logic. For example, one of the ax- actually formally equivalent to the statement of ioms of ZFC is the Axiom of , which Cohen’s theorem given at the beginning of this ar- is formally expressible as ticle. This weak version simply asserts that if  | ∀x1∀x2(x1=ˆx2 ↔∀x3(x3∈ˆx1 ↔ x3∈ˆx2)) M,E = ZFC, then there exists a structure ∗∗ ∗∗ and which informally is just the assertion that two M ,E |= ZFC + “CH is false’’, sets are equal if they have the same elements. such that M ⊆ M∗∗ and such that Thus E = E∗∗ ∩ (M × M). Cohen’s method has turned out to be quite pow- M,E |= “’’ erful: it and its generalizations are the basic tools if and only if for all a ∈ M and for all b ∈ M, if for establishing independence. Moreover, essen- tially no other effective method for building ex- {c ∈ M | (c,a) ∈ E} = {c ∈ M | (c,b) ∈ E}, tensions of models of ZFC is known. An important point is that neither Cohen’s then a = b. method of extension nor Gödel’s method of re- Therefore R,< |= “Axiom of Extensionality’’ ; striction affects the arithmetic statements true in but if we define E ⊂ N × N by the structures, so the intuition of a true model of E = {(n, m) | For some prime p, number theory remains unchallenged. It seems that most mathematicians do believe n+1| n+2  } p m and p m , that arithmetic statements are either true or false. then N,E |= “Axiom of Extensionality’’ . No generalization of Cohen’s method has yet been Continuing by examining the remaining axioms, discovered to challenge this view. But this is not one can develop naturally the notion of a model to say that such a generalization will never be of ZFC. found. The empirical completeness of arithmetic cou- One can fairly easily define a model which sat- pled with an obvious failure of completeness for isfies all of the axioms of ZFC except for the cru- set theory has led some to speculate that the phe- cial Axiom of . For example, let E denote 0 nomenon of independence is fundamental, in par- the binary relation on N just specified, and define ticular that the continuum problem is inherently an equivalence relation ∼ by i ∼ j if 0 0 vague with no solution. It is in this view a ques- {k | (k, i) ∈ E } = {k | (k, j) ∈ E } . Define a binary 0 0 tion that is fundamentally devoid of meaning, anal- ∈ ∈ relation E1 by (i,k) E1 if (j,k) E0 for some ogous to asking, “What is the color of π?” ∼ ∼ ∼ j 0 i, and define 1 from E1 as 0 was defined Wherein lies the ? I shall begin by de- from E0. Continue by induction to define increas- scribing some classical questions of Second Order ∼ ∈ N  ∈ N ing n: n and En : n . Let Number Theory—this is the theory of the ∼∞= ∪{∼n| n ∈ N}, together with all sets of integers—which are also not solvable from ZFC. Here I maintain there is a and let E∞ = ∪{En | n ∈ N}. The quotient struc- solution: there are axioms for Second Order Num- ture N/ ∼∞,E∞/ ∼∞ satisfies all of the axioms ber Theory which provide a theory as canonical as of ZFC except the . The Axiom of that of number theory. These relatively new axioms Infinity fails to hold, since for each i ∈ N the set provide insights to Second Order Number Theory

of equivalence classes {[j]∼∞ | (j,i) ∈ E∞} is finite, which transcend those provided by even ZFC.

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Can these axioms be extended to more compli- cardinals, as are ω and ω1. The assertion that a cated sets in order to solve the continuum prob- set X has cardinality ℵ1 is the assertion that there lem? This question will be the focus of the second exists a of X with ω1. Similarly, X has car- ℵ part of this article. dinality 2 0 , or c, if there is a bijection of X with Some Preliminaries P(N), powerset of N, which is the set of all For purposes of this article it is convenient to work of N. within set theory. This can initially be conceptu- The ordinals height in the of ally confusing, for we shall work within set theory sets. Suppose that M is a and that attempting to analyze set theory. M,∈ |= ZFC. Then it follows that the set So let us assume that the universe of sets ex- {a ∈ M |M,∈ |=“a is an ordinal’’} ists and that the axioms represent true assertions about this universe. We shall initially assume just is precisely the set of all ordinals, α ∈ M. Moreover, the axioms of ZFC. Eventually we shall augment this is an initial segment of the ordinals. Thus the these axioms by some modest large cardinal ax- height of M is precisely the ordinal M ∩ Ord, where ioms. The discussion to take place simply refers Ord denotes the of all ordinals. to objects in this universe. Definition. Suppose κ is an infinite cardinal. H(κ) Definition. A set X is transitive if each of X denotes the set of all sets X whose transitive clo- is also a of X. The transitive closure of a set sure has cardinality less than κ. X is the set ∩{Y | Y is transitive and X ⊆ Y }. Every set belongs to H(κ) for sufficiently large Suppose that M,E is a model of ZFC. Then the cardinals κ. This in the context of the other axioms model M,E is transitive if M is a transitive set is equivalent to the . and The answer to the continuum problem lies in understanding H(ω ), where ω is the smallest E = {(a, b) | a ∈ M,b ∈ M, and a ∈ b}, 2 2 cardinal greater than ω1 . This suggests an so that ∈ˆ is interpreted by actual set member- incremental approach. One attempts to under- ship. Transitive models of ZFC are particularly stand in turn the structures H(ω), H(ω1), and nice, but they are even harder to come by than ar- then H(ω2). A little more precisely, one seeks to bitrary models. The existence of a model of ZFC find the relevant axioms for these structures. Since does not imply the existence of a transitive model the Continuum Hypothesis concerns the structure of ZFC. of H(ω2), any reasonably complete collection of The theorems of Cohen and Gödel on perturb- axioms for H(ω2) will resolve the Continuum ing models of ZFC are best understood when the Hypothesis. initial structure M,E is transitive and M is The first of these structures, H(ω), is a countable. In the case that M,E is transitive, the familiar one in disguise: N, +, · . In fact, it can ∗ ∗ structure M ,E produced by Gödel’s con- be shown that the structures H(ω), ∈ and struction is again transitive. If M,E is transitive N/ ∼∞,E∞/ ∼∞ are isomorphic, where the latter and M is countable, then the structures is defined in the discussion immediately preced- ∗∗ ∗∗ M ,E produced by Cohen’s method can, ing the discussion of building new models of ZFC. without loss of generality, be assumed to also be Thus number theory is simply set theory in the transitive. presence of the negation of the Axiom of Infinity. The ordinals are those sets X which are transi- The next structure, H(ω1), is also a familiar tive and totally ordered by the membership one. It is essentially just the structure relation. Thus a transitive set X is an ordinal if and P(N), N, +, ·, ∈ , which is the standard structure only if for all a ∈ X and for all b ∈ X, if a = b, then for Second Order Number Theory. either a ∈ b or b ∈ a. A consequence of the axioms Of course, neither N, +, · nor P(N), N, +, ·, ∈ is that if (L, <) is a totally ordered set which is well is a structure for the language L(ˆ=, ∈ˆ), but each is ordered (every (nonempty) subset of L has a <-least naturally a structure for a specific formal language element), then there is an ordinal X such that which is easily defined. the total orders (L, <) and (X,∈) are isomorphic. There are natural questions about H(ω1) which Collectively the ordinals are well ordered by the are not solvable from ZFC. However, there are ax- membership relation, and this ordering is exactly ioms for H(ω1) which resolve these questions, the ordering arising from the comparison of the providing a theory as canonical as that of number order types of well-orders. theory, and which are clearly true. But the truth of The first three ordinals are ∅, {∅}, {∅, {∅}}. these axioms became evident only after a great deal The finite ordinals are the nonnegative integers; ω of work. For me, a remarkable aspect of this is that denotes the least infinite ordinal, and ω1 denotes it demonstrates that the discovery of mathemati- the least uncountable ordinal. Finally, an ordinal cal truth is not a purely formal endeavor. κ is a cardinal if there is no bijection of κ with α The second part of this article will focus on the for any ordinal α<κ. The finite ordinals are attempt to find a generalization of these axioms

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for H(ω2). Here is where the answer to the Why consider the projective sets? The answer continuum problem lies, for the Continuum Hy- is simply that the structure of H(ω1) can be pothesis is expressible as a proposition about reinterpreted as the structure of the projective H(ω2). The surprising answer is that there are sets. More formally, the projective sets correspond generalizations but that any generalization which to sets A ⊆ H(ω1) for which there is a formula yields a theory which is strongly canonical in a cer- φ(x1,x2) of L(ˆ=, ∈ˆ) and an element a ∈ H(ω1) tain specific sense must imply that CH is false. such that In the course of this discussion I will have to de- A = {b ∈ H(ω ) |H(ω ), ∈ |= φ[a, b]}. fend the claim that H(ω2), ∈ , rather than 1 1 P(R), R, +, ·, ∈ , is the correct immediate gener- Such sets A are definable, from parameters, in the  ∈ alization of the structure H(ω1), (or, equiva- structure H(ω1), ∈ . This is a common method of lently, of P(N), N, +, ·, ∈ ). The structure logic: study a structure by studying the sets and P(R), R, +, ·, ∈ , which is naturally given by the relations which can be defined in the structure. powerset of R, has more traditionally been The Axiom of Choice implies the existence of viewed as the next stop on the journey into the many bizarre sets. A well-known example is the transfinite. The method used to analyze the Banach-Tarski Paradox: there exists a finite parti- possibilities for strongly canonical theories for tion of the unit ball of R3 into pieces from which  ∈ H(ω2), actually shows that there can be no two copies of the unit ball can be manufactured P R R · ∈ strongly canonical theory for ( ), , +, , . If using only rigid motions. Such a partition is a  ∈ CH holds, then these two structures, H(ω2), paradoxical partition. P R R · ∈ and ( ), , +, , , are in essence the same (each To guide our discussion, consider the following can be interpreted in the other), just as are the question. structures H(ω1), ∈ and P(N), N, +, ·, ∈ . If CH fails, then these two structures can be very dif- Question. Can there be a paradoxical partition of ferent, with the former structure possibly being the unit ball of R3 into pieces, each of which is a fundamentally simpler than the latter structure. projective set? Every analytic subset of Rn is Lebesgue mea- The First Step, H(ω1) surable; this was first proved by Luzin in 1917. As Consider the following basic operations for sub- a corollary there can be no paradoxical partition sets of Rn; these generate the projective sets from of the unit ball of R3 into pieces, each of which is the closed sets. in the σ-algebra generated by analytic sets. This (Projection) Suppose X ⊆ Rn+1. The projec- is because any paradoxical partition must include tion of X to Rn is the of X under the pieces which are not Lebesgue measurable. projection , Of course, our pilot question on projective π : Rn+1 → Rn , paradoxical partitions really suggests the more fundamental question: defined by π(a1,... ,an,an+1)= (a1,... ,an) . (Complements) Suppose X ⊆ Rn . The com- Question. Are the projective sets Lebesgue plement of X is the set measurable? ∗ By the 1920s the difficulty of this question was X = {(a1,... ,an) | (a1,... ,an) ∈/ X}. apparent: [Luzin, 1925] one does not know and n Definition (Luzin). A set X ⊆ R is a projective set one will never know [of the projective if for some k it can be generated from a sets whether or not they are each n+k closed subset of R in finitely many steps, ap- Lebesgue measurable.] plying the basic operations of taking projections and complements. Curiously, Gödel’s basic method of showing the (relative) of CH with ZFC yielded a I caution that because we are working in the surprising bonus to which Gödel himself attached Euclidean spaces, it generally requires three a significance comparable to that of his results on applications of our basic operations to reach anything CH. interesting. The projection of a closed subset of Rn+2 yields a subset of Rn+1 which is easily seen to Theorem (Gödel). Assume ZFC is consistent. be expressible as a countable of closed sets. Then so is ZFC + “There is a nonmeasurable Complementing and projecting again take one projective set”. beyond the Borel sets and into the analytic sets. More formally, a set X ⊆ Rn is an analytic set if there In fact, an immediate corollary of the proof of exists a closed set C ⊆ Rn+2 such that X is the pro- this theorem is the (relative) consistency with the jection of Rn+1\Y, where Y is the projection of C. axioms of set theory of the statement:

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There exists a paradoxical partition of Let SEQ be the set of all finite binary sequences. the unit ball of R3 into pieces, each of A strategy τ is a which is the projection of the comple- →{ } ment of an analytic set. τ : SEQ 0, 1 .

So Luzin’s theorem on the Lebesgue measurabil- A run  i : i ∈ N of the game is generated ity of the analytic sets was in fact the strongest according to τ by Player I if 1 = τ(∅) and for all theorem one could hope to prove working just k ∈ N, from the axioms ZFC. 2k+1 = τ( 1, ··· , 2k). The consistency with ZFC that every projective set is Lebesgue measurable, while true, cannot Similarly the run is generated according to τ by be proved assuming just the consistency of ZFC. Player II if for all k ∈ N, 2k = τ( 1, ··· , 2k−1). Nevertheless, an immediate corollary of Solovay’s A strategy τ is a winning strategy for Player I if results on the measure problem for the projective every run of the game generated according to τ by sets is that if ZFC is consistent, then so is ZFC, Player I is winning for Player I. Similarly, τ is a together with the statement winning strategy for Player II if every run of the game generated according to τ by Player II is There is no paradoxical partition of the winning for Player II. unit ball of R3 into pieces, each of which is a projective set. Definition. Suppose that A ⊆ [0, 1]. The game GA is determined—briefly, A is determined—if there Thus already at H(ω ) there are natural struc- 1 is a winning strategy for one of the players. tural questions which are formally unsolvable. The resolution of these questions, if indeed they can The Axiom of Choice yields a set A such that be resolved, requires the discovery of new axioms. GA is not determined. This is a simple diagonal- ℵ Both Gödel’s method of restriction and Cohen’s ization . There are only 2 0 many method of extension can alter H(ω1) in the sense possible strategies, and for each strategy τ the of the models, even in the case that the initial and assertion that τ is a winning strategy for one of ℵ final models are transitive. Informally, restricting the players in the game GA effectively makes 2 0 generally deletes sets of integers, and new sets of many predictions about membership in A. integers can appear in a Cohen extension. Thus it However, the undetermined set given by the is not at all obvious that these questions about the Axiom of Choice is not in general a projective set. projective sets are any more tractable than the This (essentially, Mycielski–Steinhaus, 1962) sug- Continuum Hypothesis. gests the following axiom: [Gödel, 1947] There might exist axioms Projective Determinacy: Suppose that so abundant in their verifiable conse- A is a projective subset of [0, 1]. Then quences, shedding so much light upon the game G is determined. a whole discipline, and furnishing such A powerful methods for solving given It has to be acknowledged at this point in our problems (and even solving them, as far discussion that the axiom Projective Determinacy as possible, in a constructivistic way) is not only not obviously true, it is not even that quite irrespective of their intrinsic obviously consistent. It is, however, a fruitful necessity they would have to be as- axiom, but (a logician’s joke) so is the axiom 0=1. sumed at least in the same sense as In this article, the first of a two-part , any established physical theory. my main objective is to present some of the I now discuss one candidate for such an axiom. compelling evidence that the axiom Projective Determinacy is the “right axiom” for the projective Determinacy sets. The evidence I present is really a small frac- Fix A ⊆ [0, 1]. I define the game G , which is an A tion of what is now available. The subject of the infinite game with two players. projective sets has expanded in ways not foreseen Player I and Player II alternate choosing or even imagined by its founders. i ∈{0, 1}, In 1964 Mycielski and Swierczkowski proved that if the axiom Projective Determinacy holds, so that Player I chooses for i odd, and Player II i then every projective set is Lebesgue measurable. chooses for i even. Player I begins by choosing i Consequently, this axiom implies that there is ; Player II then picks and so forth. 1 2 R3 Player I wins if no paradoxical partition of the unit ball of into projective pieces. ∞ −i The axiom also implies that every uncountable 2 ∈ A; ℵ i projective set has cardinality 2 0 . In fact much i=1 more is true. Davis proved, again shortly after the otherwise Player II wins. axiom was introduced, that the axiom implies

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that every uncountable projective set contains an 1. The Axiom of Choice fails projectively in that uncountable closed subset. there is no projective well-ordering of the reals. This establishes that there is no formal coun- 2. The Axiom of Choice holds projectively in that terexample to CH to be found among the projec- if F : R → V is a general projective set, then tive sets (assuming Projective Determinacy). there is a choice function for F which is also Surprisingly, the actual relationship between a general projective set. CH and the projective sets is quite subtle, even Projective Determinacy and Large Cardinals assuming Projective Determinacy. This will be the One might attempt to analyze Projective Deter- starting point for the second part of this article. minacy incrementally. One can also naturally analyze versions of the In 1953 Gale-Stewart proved that every open Axiom of Choice for the projective sets. subset of [0, 1] is determined, and they boldly Suppose that A ⊆ R2. For each x ∈ R let asked whether every Borel set is determined. Two Ax = {y ∈ R | (x, y) ∈ A} decades later, in a technical tour de force, Martin proved the answer is affirmative. A remarkable be the section of A at x. aspect of Martin’s proof is that Friedman { |  ∅} Let B be the projection of A, B = x Ax = . [Friedman, 1971] had previously shown that the → R A function f : B uniformizes the set A if for determinacy of all Borel sets could not be proved ∈ ∈ each x B, f (x) Ax ; see the figure below. The in with the Axiom of Choice. This is the axiom system ZC, i.e., ZFC without the Axiom(s) of Replacement. Most mathematicians, realizing it or not, work in this restricted axiom system. Roughly, Martin’s method was to associate to a Borel set A, by induction on the Borel rank of A, an open set A∗ ⊂ ZN where Z is discrete. A∗ is constructed so that from the determinacy of the game associated to A∗ (here the players play elements of Z), one infers the determinacy of A. The Gale-Stewart Theorem applies to show that A∗ is determined, and so one obtains the determi- nacy of A. As A ranges over the possible Borel sets, the associated sets Z range over

α {P (R) | α<ω1}, function f is projective if its graph is a projective increasing in cardinality with the Borel rank of A. subset of R2 . Here Pα(R) denotes the α-th iterated powerset of R, In 1930 Luzin asked if every projective subset which is defined by induction as follows, where of the plane can be uniformized by a projective for each set X, P(X)={Y | Y ⊆ X}. P(X) is the function. Nearly half a century later Moschovakis powerset of X. P0(R)=R, Pα+1(R)=P(Pα(R)), and proved that the answer is affirmative if Projective Pα R ∪{Pβ R | } Determinacy holds. ( )= ( ) β<α Thus, assuming Projective Determinacy, one if α>0 and α is a limit ordinal. has a complete analysis of the Axiom of Choice at In Zermelo set theory one cannot prove that the projective level, which can be summarized as Pω(R) even exists, and so as Friedman’s theorem follows. For this summary it is convenient to predicted, Martin’s proof cannot be implemented n generalize the notion of a projective subset of R assuming just the axioms ZC. to the notion of a general projective set. So let’s The determinacy of all analytic sets A ⊆ [0, 1] say that a set A is a general projective set if there cannot be proved in ZFC. This is because the R → exists a surjection π : M, where M is the theory ZFC is simply not strong enough. Thus { } transitive closure of A , such that Martin’s theorem on the determinacy of all Borel sets is the strongest theorem one can hope to {(x, y) | π(x) ∈ π(y)} prove without appealing to new axioms. Here enter large cardinal axioms, which informally are ax- is a projective subset of R2 . The two notions co- ioms which assert the existence of “large” cardi- incide for subsets of Rn; a set A ⊆ Rn is a general nals. Perhaps best known among these is the axiom projective set if and only if it is a projective set. which asserts the existence of a measurable car- Projective sets are by definition subsets of Rn; dinal. An uncountable cardinal κ is a measurable general projective sets can be ordinals, functions, cardinal if there exists a nonprincipal U etc. Here is the promised summary. on the subsets of κ which is κ-complete; i.e., if

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X ⊂ U and X has cardinality less than κ, then precisely, fixing a large cardinal axiom Ψ, for each ∩{A ⊂ κ | A ∈ X}∈U. model M,E of ZFC + Ψ one seeks a substructure Around 1970, five years before he proved M∗,E∩ (M∗ × M∗) which is also a model of all Borel sets are determined, Martin proved that ZFC + Ψ and which satisfies various sentences true if there is a , then all analytic in Gödel’s substructure. For example, if Ψ is the sets are determined (every Borel set is an analytic large cardinal axiom “There is a Woodin set, and so this also established that all Borel sets cardinal”, simply requiring that in the substructure are determined, but not in the theory ZFC). produced, the sentence “There is a projective But Solovay showed that one cannot hope to set which is not Lebesgue measurable” holds yields prove Projective Determinacy from just the a difficult problem. For stronger large cardinal existence of a measurable cardinal. The reason, axioms there are generalizations of this require- as before, is a lack of strength: ZFC + “Projec- ment which yield similarly nontrivial problems. tive Determinacy’’ implies the consistency of For the large cardinal axiom “There is a mea- ZFC + “There is a measurable cardinal’’. There- surable cardinal”, the correct generalization of fore, by Gödel’s Second Incompleteness Theorem Gödel’s construction was discovered by Solovay one cannot infer Projective Determinacy from the and then further analyzed in work of Kunen and existence of a measurable cardinal. Still stronger Silver. With these results the Program axioms are necessary, and the special case of began. 1 The fact that from infinitely many Woodin car- proving the determinacy of all ∼Σ2 subsets of [0, 1] 1 dinals one can prove the projective sets are became a central problem (the ∼Σ2 sets are those projective sets which can be represented as the Lebesgue measurable is strong evidence that from projection of the of an analytic set). the same assumption one should be able to prove Some speculated that no large cardinal axiom Projective Determinacy. In 1985, using techniques known was sufficient in strength to imply this developed in the Inner Model Program, Martin–Steel fragment of Projective Determinacy. In 1978 succeeded in doing this. Surprisingly, the Martin succeeded in proving the determinacy of combinatorial properties of Woodin cardinals 1 responsible for their discovery—for example, all ∼Σ2 sets by using essentially the strongest large cardinal hypothesis known at the time. Finally, in those aspects yielding the measurability of all 1983 I proved the determinacy of all projective sets projective sets—play no role in this determinacy using large cardinal axioms in a natural hierarchy proof. which begins with the large cardinal axiom used Theorem (Martin–Steel). Assume there exist 1 by Martin to establish the determinacy of all ∼Σ2 infinitely many Woodin cardinals. Then every sets. projective set is determined. The natural character of these determinacy proofs suggested that essentially optimal large How are large cardinals used in determinacy cardinal assumptions were being used. But this proofs? The basic strategy is the same as for picture, though quite appealing (at least to Martin Martin’s proof of the determinacy of all Borel sets, and me), was completely wrong. The large cardi- though Martin’s proof of the determinacy of all nal axioms used to prove Projective Determinacy, analytic sets from the existence of a measurable including the axiom used by Martin to prove the cardinal is a more accurate prototype. Given a set ⊆ determinacy of all Σ1 sets, were far stronger than A [0, 1], one again associates to the set A an ∼2 ∗ ⊂ N necessary. The first indications that the picture was open set A Z , where Z is a discrete set care- fully constructed such that from the determinacy incorrect came from a surprising direction: ∗ the evidence was discovered by Foreman– of the game naturally associated to A , one obtains the determinacy of the initial set A. The Gale- Magidor–Shelah in their seminal work on Martin’s ∗ Maximum, a maximal forcing axiom which I shall Stewart Theorem shows that A is determined, and so as before one obtains the determinacy of A. discuss below. This ultimately led to the following For a typical projective set A the associated set Z theorem from 1984. is very large, of cardinality on the order of the Theorem (Shelah–Woodin). Assume there exist large cardinals one is assuming to exist. infinitely many Woodin cardinals. Then every The connection between Projective Determi- projective set is Lebesgue measurable. nacy and large cardinal axioms is fundamental. This claim is supported by the following theorem I shall not give the definition of a Woodin from 1987, which shows that this time the picture cardinal, but instead simply note that this large is correct. cardinal axiom is much weaker than those used in the determinacy proofs just discussed. Theorem (Woodin). The following are equivalent: The Inner Model Program attempts to general- 1. Projective Determinacy. ize Gödel’s substructure construction to models satisfying various large cardinal axioms (and the 2. For each k ∈ N there exists a countable tran- stronger the axiom, the harder the problem). More sitive set M such that

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M,∈ |= ZFC + “There exist k Woodin then the Continuum Hypothesis necessarily fails cardinals” in the associated extension. Another quite interesting feature of this extension is that in this and such that M is countably iterable. extension there exists a countably additive 3 The notion that M is countably iterable is a measure on R , extending , technical one from the Inner Model Program. which measures all projective sets and which is The Program, which is more invariant under rigid motions. So in this exten- ambitious than the Inner Model Program, sion the following statement holds: originates in seminal work of Dodd and Jensen. It There is no paradoxical partition of the is more ambitious simply because it attempts to unit ball of R3 into projective pieces. build the substructures of the Inner Model Program without necessarily assuming that the (relevant) This extension, first defined and analyzed by Solovay, is sometimes referred to as a Solovay large cardinal axioms even hold in the initial extension. structure. If Ω is a compact Hausdorff space, then the The extension of the Core Model Program to regular open algebra of Ω is the complete Boolean the realm of Woodin cardinals is due primarily to algebra given by the lattice of regular open subsets Steel [Steel, 1996], and with this development it of Ω (open sets O ⊆ Ω which are the interior of has become clear that Projective Determinacy is their closure) ordered by set inclusion. Forcing ax- actually implied by a vast number of seemingly ioms are naturally motivated by the following unrelated combinatorial . Projective feature of Cohen’s original extension; this exten- Determinacy is ubiquitous in set theory. sion is defined from b ∈ M such that In fact, this is an instance of a more pervasive M,E |=“b is the regular open algebra phenomenon where propositions are calibrated, on the of consistency, by large cardinal of the product space [0, 1]’’. axioms. There are now numerous examples of ω2 this phenomenon. Many early results of this kind The interesting feature of this extension is that were proved using Jensen’s Covering Lemma. In in it the following generalization of the Baire fact, using the Covering Lemma, one can prove Category Theorem holds: the determinacy of all analytic sets from a perhaps The [0, 1] is not the ℵ bewildering array of propositions. I refer the reader 1 union of meager sets. to [Kanamori, 1994] for further details. A recent example where methods from the Core A compact Hausdorff space Ω is ccc if every Model Program are used to establish Projective collection of pairwise disjoint open subsets of Ω Determinacy involves axioms which attempt to is countable. solve the continuum problem by explicitly making Martin’s Axiom (ω1): Suppose that Ω the Continuum Hypothesis false. is a compact Hausdorff space which is Forcing Axioms ccc. Then Ω is not the union of ℵ1 many Forcing Axioms are in essence axioms asserting meager subsets of Ω. generalizations of the . The connection lies in the technical aspects of A simple motivation for such an axiom is that if CH is to be false, then sets of cardinality ℵ should Cohen’s method of constructing extensions of 1 behave, as much as possible, like countable sets. models of ZFC. One can attempt to strengthen the axiom, al- Suppose that M,E |= ZFC. The Cohen exten- lowing a larger class of compact Haudorff spaces. sions associated to the structure M,E correspond But one cannot allow arbitrary compact Hausdorff to complete Boolean algebras in the sense of M,E , spaces. The maximum possible class was identified ∈ i.e., to elements b M such that by Foreman–Magidor–Shelah. The definition M,E |=“b is a complete ’’. involves the notion of a closed unbounded subset of ω1 : a cofinal set C ⊂ ω1 is closed and un- If b is trivial, for example, if bounded if it is closed in the order of ω1. M,E |=“b is a finite Boolean algebra’’, Suppose that B is a complete Boolean algebra. Every set S⊆ B has a greatest lower bound,  then the associated extension is simply M,E . denoted by S, and a least upper bound which is But if denoted by S. 1 M,E |=“b is the measure algebra Definition (Foreman–Magidor–Shelah). Suppose B of the product space [0, 1]’’, that is a complete Boolean algebra. The Boolean algebra B is stationary set preserving if the ω2 following holds. Suppose that b is a nonzero ele- 1The Boolean algebra of Borel subsets modulo null sets. ment of B and that bα : α<ω1 is a sequence of

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elements of B. Then there exist 0

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correct axioms for this structure, be extended to an understanding of the structure H(ω2), ∈ ? This question will be the main topic of the second, and final, part of this article.

References [Cohen 1966] P. COHEN, Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966. [Foreman, Magidor, Shelah 1988] M. FOREMAN, M. MAGIDOR, and S. SHELAH, Martin’s Maximum, saturated ideals and non-regular ultrafilters I. Ann. of Math. 127 (1988), 1–47. [Friedman 1971] H. FRIEDMAN, Higher set theory and math- ematical practice, Ann. of Math. Logic 2 (1971), 325–357. [Gale and Stewart 1953] D. GALE and F. STEWART, Infinite games with perfect information, Contributions to the Theory of Games (Harold W. Kuhn and Alan W. Tucker, eds.), Ann. of Math. Stud., vol. 28, Princeton University Press, Princeton, NJ, 1953, pp. 245–266. [Gödel 1940] K. GÖDEL, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothe- sis with the Axioms of Set Theory, Ann. of Math. Stud., vol. 3, Princeton University Press, Princeton, NJ, 1940, pp. 33–101. [Gödel 1947] K. GÖDEL, What is Cantor’s continuum prob- lem? Amer. Math. Monthly 54 (1947), 515–545. [Kanamori 1994] A. KANAMORI, : Large Cardinals in Set Theory from Their Beginnings, Perspect. Math. Logic, Springer-Verlag, Berlin, 1994. [Luzin 1925] N. LUZIN, Sur les ensembles projectifs de M. Henri Lebesgue, C. R. Hebdomadaires des Seances Acad. Sci. Paris 180 (1925), 1572–1574. [Martin 1975] D. MARTIN, Borel determinacy, Ann. of Math. 102 (1975), 363–371. [Martin and Steel 1989] D. MARTIN and J. STEEL, A proof of projective determinacy, J. Amer. Math. Soc. 2 (1989), 71–125. [Moschovakis 1980] Y. MOSCHOVAKIS, Descriptive Set Theory, Stud. Logic Found. Math., vol. 100, North- Holland, Amsterdam, 1980. [Mycielski and Steinhaus 1962] J. MYCIELSKI and H. STEINHAUS, A mathematical axiom contradicting the axiom of choice, Bull. Acad. Polonaise Sci., Ser. Sci. Math., Astron. Phys. 10 (1962), 1–3. [Solovay 1970] R. SOLOVAY, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1–56. [Steel 1996] J. STEEL, The Core Model Iterability Problem, Lecture Notes Logic, vol. 8, Springer-Verlag, Heidel- berg, 1996. [Woodin 1999] W. HUGH WOODIN, The Axiom of Determi- nacy, Forcing Axioms, and the Nonstationary Ideal, Ser. Logic Appl., vol. 1, de Gruyter, Berlin, 1999.

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