The Transfinite Universe W. Hugh Woodin - Logic at Harvard
The Transfinite Universe W. Hugh Woodin - Logic at Harvard
The Transfinite Universe W. Hugh Woodin - Logic at Harvard
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Chapter 1<strong>The</strong> <strong>Transfinite</strong> <strong>Universe</strong>W. <strong>Hugh</strong> <strong>Woodin</strong>Professor of M<strong>at</strong>hem<strong>at</strong>icsDepartment of M<strong>at</strong>hem<strong>at</strong>icsUniversity of California, BerkeleyBerkeley, CA USA<strong>The</strong> 20 th century choice for the axioms 1 of Set <strong>The</strong>ory are the Zermelo-Frankelaxioms together with the Axiom of Choice, these are the ZFC axioms. Thisparticular choice has led to a 21 th century problem:<strong>The</strong> ZFC Delemma: Many of the fundamental questions of Set<strong>The</strong>ory are formally unsolvable from the ZFC axioms.Perhaps the most famous example is given by the problem of the ContinuumHypothesis: Suppose X is an infinite set of real numbers, must it be the caseth<strong>at</strong> either X is countable or th<strong>at</strong> the set X has cardinality equal to the cardinalityof the set of all real numbers?One interpret<strong>at</strong>ion of this development is:Skeptic’s Attack: <strong>The</strong> Continuum Hypothesis is neither true norfalse because the entire conception of the universe of sets is a completefiction. Further, all the theorems of Set <strong>The</strong>ory are merelyfinitistic truths, a reflection of the m<strong>at</strong>hem<strong>at</strong>ician and not of anygenuine m<strong>at</strong>hem<strong>at</strong>ical “reality”.Here and in wh<strong>at</strong> follows, the “Skeptic” simply refers to the meta-m<strong>at</strong>hem<strong>at</strong>icalposition which denies any genuine meaning to a conception of uncountable sets.<strong>The</strong> counter-view is th<strong>at</strong> of the “Set <strong>The</strong>orist”:1 This paper is dedic<strong>at</strong>ed to the memory of Paul J. Cohen.1
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 2<strong>The</strong> Set <strong>The</strong>orist’s Response: <strong>The</strong> development of Set <strong>The</strong>ory,after Cohen, has led to the realiz<strong>at</strong>ion th<strong>at</strong> formally unsolvableproblems have degrees of unsolvability which can be calibr<strong>at</strong>ed bylarge cardinal axioms.Elabor<strong>at</strong>ing further, as a consequence of this calibr<strong>at</strong>ion, it has been discoveredth<strong>at</strong> in many cases very different lines of investig<strong>at</strong>ion have led to problemswhose degree of unsolvability is the same. Thus the hierarchy of large cardinalaxioms emerges an intrinsic, fundamental conception within Set <strong>The</strong>ory. Toillustr<strong>at</strong>e this I discuss two examples.An excellent reference for both the historical development and the backgroundm<strong>at</strong>erial for much of wh<strong>at</strong> I will discuss is the book by Kanamori [5].<strong>The</strong> present paper is not intended to be a survey: my intent is to discuss somevery recent results which I think have the potential to be relevant to the conceptof the <strong>Universe</strong> of Sets. <strong>The</strong> danger of course is th<strong>at</strong> this invariably involvesspecul<strong>at</strong>ion and this is compounded whenever such specul<strong>at</strong>ion is based on researchin progress (as in my manuscript [18]).1.1 <strong>The</strong> examples<strong>The</strong> first example, infinitary combin<strong>at</strong>orics. A n<strong>at</strong>ural class of objectsfor study are the subsets of ω 1 which is the least uncountable ordinal. Recallth<strong>at</strong> ω 1 is the set of all countable ordinals and so the collection of all subsets ofω 1 is exactly the collection of all sets of countable ordinals. I shall be concernedwith two varieties of subsets of ω 1 which I define below.Definition 1 (1) A set C ⊆ ω 1 is closed if, for all α < ω 1 , if C ∩ α is cofinalin α then α ∈ C.(2) A set S ⊆ ω 1 is st<strong>at</strong>ionary if S ∩C ≠ ∅ for all closed, cofinal sets C ⊆ ω 1 .⊓⊔<strong>The</strong> sets, S ⊆ ω 1 , which are st<strong>at</strong>ionary and co-st<strong>at</strong>ionary are in many respectsthe simplest manifest<strong>at</strong>ion of the Axiom of Choice. For example onecan show, without appealing to the Axiom of Choice, th<strong>at</strong> if there exists awellordering of the real numbers, then such a set S must exist. <strong>The</strong> converseis not true as the existence of such a set S does not imply the existence of awellordering of the real numbers. Recall th<strong>at</strong> a wellordering of the real numbersis total order of the real numbers rel<strong>at</strong>ive to which every nonempty set of realnumbers has a least element.<strong>The</strong>refore, it is n<strong>at</strong>ural to ask how complic<strong>at</strong>ed the structure of the st<strong>at</strong>ionarysubsets of ω 1 (modulo non-st<strong>at</strong>ionary subsets of ω 1 ) is or even if there can
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 3exist a small gener<strong>at</strong>ing family for these sets. Consider the following St<strong>at</strong>ionaryBasis Hypothesis:(SBH) <strong>The</strong>re exists ω 1 many st<strong>at</strong>ionary sets, 〈S α : α < ω 1 〉, suchth<strong>at</strong> for every st<strong>at</strong>ionary set S ⊆ ω 1 , there exists α < ω 1 such th<strong>at</strong>S α ⊆ S modulo a nonst<strong>at</strong>ionary set.<strong>The</strong> assertion th<strong>at</strong> S α ⊆ S modulo a nonst<strong>at</strong>ionary set is simply the assertionth<strong>at</strong> the set,S α \S = {β < ω 1 | β ∈ S α and β /∈ S} ,is not st<strong>at</strong>ionary. Such a sequence 〈S α : α < ω 1 〉 of st<strong>at</strong>ionary subsets of ω 1would give in a n<strong>at</strong>ural sense, a basis for the st<strong>at</strong>ionary subsets of ω 1 which isof cardinality ℵ 1 .<strong>The</strong>re is a remarkable theorem of Shelah, [14]:<strong>The</strong>orem 2 (Shelah) <strong>The</strong> hypothesis SBH implies th<strong>at</strong> CH is false.⊓⊔This theorem in conjunction with the subsequent consistency results of [17],suggests the following intriguing conjecture: <strong>The</strong> hypothesis SBH implies th<strong>at</strong>2 ℵ0 = ℵ 2 .<strong>The</strong> second example, infinite games. Suppose A ⊆ P(N) where P(N)denotes the set of all sets σ ⊆ N and N = {1, 2, . . . , k, . . .} is the set of alln<strong>at</strong>ural numbers.Associ<strong>at</strong>ed to the set A is an infinite game involving two players, Player Iand Player II. <strong>The</strong> players altern<strong>at</strong>e declaring <strong>at</strong> stage k whether k ∈ σ or k /∈ σas follows:Stage 1: Player I declares 1 ∈ σ or declares 1 /∈ σ;Stage 2: Player II declares 2 ∈ σ or declares 2 /∈ σ;Stage 3: Player I declares 3 ∈ σ or declares 3 /∈ σ; . . .After infinitely many stages a set σ ⊆ N is specified. Player I wins this run ofthe game if σ ∈ A; otherwise Player II wins. (Note: Player I has control ofwhich odd numbers are in σ, and Player II has control of which even numbersare in σ.).A str<strong>at</strong>egy is simply a function which provides moves for the players givenjust the current st<strong>at</strong>e of the game. More formally a str<strong>at</strong>egy is a functionτ : [N]
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 4and declaring “k /∈ σ” if τ(a, k) = 0, wherea = {i < k | “i ∈ σ” was declared <strong>at</strong> stage i”} .<strong>The</strong> str<strong>at</strong>egy τ is a winning str<strong>at</strong>egy for Player I if by following the str<strong>at</strong>egy<strong>at</strong> each stage k where it is Player I’s turn to play (i.e., for all odd k), Player Iwins the game no m<strong>at</strong>ter how Player II plays. Similarly τ is a winning str<strong>at</strong>egyfor Player II if by following the str<strong>at</strong>egy <strong>at</strong> each stage k where it is PlayerII’s turn to play (i.e., for all even k), Player II wins the game no m<strong>at</strong>ter howPlayer I plays. <strong>The</strong> game is determined if there is a winning str<strong>at</strong>egy for one ofthe players. Clearly it is impossible for there to be winning str<strong>at</strong>egies for bothplayers.It is easy to specify sets A ⊆ P(N) for which the corresponding game isdetermined. For example if A = P(N) then any str<strong>at</strong>egy is a winning str<strong>at</strong>egyfor Player I. On the other hand, if A is countable, then one can fairly easilyshow th<strong>at</strong> there exists a str<strong>at</strong>egy which is a winning str<strong>at</strong>egy for Player II.<strong>The</strong> problem of specifying a set A ⊆ P(N) for which the correspondinggame is not determined, turns out to be quite a bit more difficult. <strong>The</strong> Axiomof Determinacy, AD, is the axiom which asserts th<strong>at</strong> for all sets A ⊆ P(N),the game given by A, as described above, is determined. This axiom was firstproposed by Mycielski and Steinhaus, [10], and contradicts the Axiom of Choice.So the problem here is whether the Axiom of Choice is necessary to constructa set A ⊆ P(N) for which the corresponding game is not determined. Clearlyif the Axiom of Choice is necessary, then the existence of such set A is quite asubtle fact.Three problems and three formal theories. I now add a third problemto the list and specify formally a list of three problems. As indic<strong>at</strong>ed, the firstand third problems are within ZFC and the second problem is within just thetheory ZF (this is the theory given by the axioms ZFC without the Axiom ofChoice).Problem 1: (ZFC) Does SBH hold?Problem 2: (ZF) Does AD hold?Problem 3: (ZFC) Do there exist infinitely many <strong>Woodin</strong> cardinals?I shall not give the formal definition of a <strong>Woodin</strong> cardinal here as it is a largecardinal notion whose definition is a bit technical; see [5] for one definition.<strong>The</strong> first problem, Problem 1, is formally unsolvable if assuming the axiomsZFC one can neither prove or refute the SBH. Similarly, Problem 2 is formally
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 5unsolvable if assuming the axioms ZF, one can neither prove or refute AD.Finally, Problem 3 is formally unsolvable if assuming the axioms ZFC one canneither prove or refute the existence of infinitely many <strong>Woodin</strong> cardinals. Ineach case the assertion of formal unsolvability is simply a st<strong>at</strong>ement of Number<strong>The</strong>ory. <strong>The</strong> remarkable fact is th<strong>at</strong> these three assertions of Number <strong>The</strong>oryare equivalent, and this is a theorem of Number <strong>The</strong>ory from the classical(Peano) axioms for Number <strong>The</strong>ory. Thus, two completely different lines ofinvestig<strong>at</strong>ion have resulted in problems whose degree of formal unsolvability isthe same, and this is exactly calibr<strong>at</strong>ed by a large cardinal axiom.Assuming th<strong>at</strong> the axioms ZFC are formally consistent, then, for the threeproblems indic<strong>at</strong>ed here, the only possible formal solutions are as follows: “No”for the first and third problems, and “yes” for the second problem. <strong>The</strong>refore, itis more n<strong>at</strong>ural to rephrase these assertions of formal unsolvability as assertionsth<strong>at</strong> particular theories are formally consistent. I have implicitly defined threetheories, and the assertions of unsolvability discussed above correspond to theassertions th<strong>at</strong> these theories are each formally consistent.<strong>The</strong>ory 1: ZFC + SBH.<strong>The</strong>ory 2: ZF + AD.<strong>The</strong>ory 3: ZFC + “<strong>The</strong>re exist infinitely many <strong>Woodin</strong> cardinals”.<strong>The</strong> following theorem is the theorem which implies th<strong>at</strong> the degree of unsolvabilityof the three problems th<strong>at</strong> I have listed is the same; see [5] for adiscussion of this theorem.<strong>The</strong>orem 3 <strong>The</strong> three theories, <strong>The</strong>ory 1, <strong>The</strong>ory 2, and <strong>The</strong>ory 3, areequiconsistent.⊓⊔1.2 A prediction and a challenge for the SkepticAre the three theories I have defined really formally consistent? <strong>The</strong> claim th<strong>at</strong>they are consistent is a prediction which can be refuted by finite evidence (aformal contradiction). Just knowing the first two theories are equiconsistentdoes not justify this prediction <strong>at</strong> all. I claim:It is through the calibr<strong>at</strong>ion by a large cardinal axiom in conjunctionwith our understanding of the hierarchy of such axioms astrue axioms about the universe of sets th<strong>at</strong> this prediction isjustified.As a consequence of my belief in this claim, I make a prediction:
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 6In the next 10,000 years there will be no discovery of an inconsistencyin these theories.This is a specific and unambiguous prediction about the physical universe. Furtherit is a prediction which does not arise by a reduction to a previously heldtruth (as for example is the case for the prediction th<strong>at</strong> no counterexample toFerm<strong>at</strong>’s Last <strong>The</strong>orem will be discovered).This is a genuinely new prediction which I make based on the developmentof Set <strong>The</strong>ory over the last 50 years and which I make based on my belief th<strong>at</strong>the conception of the transfinite universe of sets is meaningful. Finally, I makethis prediction independently of all specul<strong>at</strong>ion of wh<strong>at</strong> comput<strong>at</strong>ional devicesmight be developed in the next 10, 000 years which increase the effectiveness ofresearch in M<strong>at</strong>hem<strong>at</strong>ics—it is a prediction based on M<strong>at</strong>hem<strong>at</strong>ics and not onconsider<strong>at</strong>ion of the M<strong>at</strong>hem<strong>at</strong>ician.Now the Skeptic might object th<strong>at</strong> this prediction is not interesting or n<strong>at</strong>uralbecause the formal theories are not interesting or n<strong>at</strong>ural. But such objectionsare not allowed in Physics: the ultim<strong>at</strong>e physical theory should explainall (physical) aspects of the physical universe, not just those which we regardas n<strong>at</strong>ural. How can we apply a lessor standard for the ultim<strong>at</strong>e m<strong>at</strong>hem<strong>at</strong>icaltheory? In fact, I make the stronger prediction:<strong>The</strong>re will be no discovery ever of an inconsistency in these theories.One can arguably claim th<strong>at</strong> if this stronger prediction is true, then it is aphysical law.Skeptic’s Retre<strong>at</strong>: OK, I accept the challenge noting th<strong>at</strong> I onlyhave to explain the predictions of formal consistency given by thelarge cardinal axioms. <strong>The</strong> formal theory of Set <strong>The</strong>ory as given bythe axioms ZFC is so “incomplete” th<strong>at</strong>: Any large cardinal axiom,in the n<strong>at</strong>ural formul<strong>at</strong>ion of such axioms, is either consistent withthe axioms of Set <strong>The</strong>ory, or there is an elementary proof th<strong>at</strong> theaxiom cannot hold.We shall see this is a very shrewd counter-<strong>at</strong>tack, even framed within the specificcontext of the current list of large cardinal axioms where it is a much moreplausible position. I shall need to review some elementary concepts from Set<strong>The</strong>ory. This is necessary to specify the basic templ<strong>at</strong>e for large cardinal axioms.1.3 <strong>The</strong> cumul<strong>at</strong>ive hierarchy of sets.As is customary in modern Set <strong>The</strong>ory, V denotes the <strong>Universe</strong> of Sets. <strong>The</strong>purpose of this not<strong>at</strong>ion is to facilit<strong>at</strong>e the (m<strong>at</strong>hem<strong>at</strong>ical) discussion of Set
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 7<strong>The</strong>ory, it does not presuppose any meaning to the concept of the <strong>Universe</strong> ofSets.<strong>The</strong> ordinals calibr<strong>at</strong>e V through the definition of the cumul<strong>at</strong>ive hierarchyof sets, [19]. <strong>The</strong> relevant definition is given below.Definition 4 Define for each ordinal α a set V α by induction on α.(1) V 0 = ∅.(2) V α+1 = P(V α ) = {X | X ⊆ V α }.1. If β is a limit ordinal, then V α = ⋃ {V β | β < α}. ⊓⊔It is a consequence of the ZF axioms th<strong>at</strong> for every set a there must exist anordinal α such th<strong>at</strong> a ∈ V α .A set N is transitive if every element of N is a subset of N. Transitive setsare fragments of V which are analogous to initial segments. For each ordinal αthe set V α is a transitive set.Every ordinal is a transitive set; in fact, the ordinals are precisely thosetransitive sets X with the property th<strong>at</strong> for all a, b ∈ X, if a ≠ b then eithera ∈ b or b ∈ a. Thus if X is an ordinal and if Y ∈ X, then necessarily Y is anordinal. This defines a n<strong>at</strong>ural order on the ordinals. If α and β are ordinalsthen α < β if α ∈ β. Thus every ordinal is simply the set of all ordinals whichare smaller than the given ordinal, rel<strong>at</strong>ive to this order.<strong>The</strong> simplest (proper) class is the class of all ordinals. This class is a transitiveclass and more generally a class M ⊆ V is a transitive class if everyelement of M is a subset of M. <strong>The</strong> basic templ<strong>at</strong>e for large cardinal axioms isas follows.<strong>The</strong>re is a transitive class M and an elementary embeddingj : V → Mwhich is not the identity.With the exception of the definition of a Reinhardt cardinal which I shall cometo below, one can always assume th<strong>at</strong> the classes, M and j, are classes whichare logically definable from parameters by formulas of a fixed bounded levelof complexity (Σ 2 -formulas). Moreover the assertion th<strong>at</strong> j is an elementaryembedding, is the assertion:For all formulas φ(x) and for all sets a, VM |= φ[j(a)];|= φ[a] if and only ifand this is equivalent to the assertion:
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 8For all formulas φ(x), for all ordinals α, and for all sets a ∈ V α , V α |= φ[a]if and only if j(V α ) |= φ[j(a)].<strong>The</strong>refore, this templ<strong>at</strong>e makes no essential use of the notion of a class. It issimply for convenience th<strong>at</strong> I refer to classes (and this is the usual practice inSet <strong>The</strong>ory).Suppose th<strong>at</strong> M is a transitive class and th<strong>at</strong> j : V → M is an elementaryembedding which not the identity. Suppose th<strong>at</strong> j(α) = α for all ordinalsα. <strong>The</strong>n one can show by transfinite induction th<strong>at</strong> for all ordinals α, theembedding j is the identity on V α . <strong>The</strong>refore, since j is not the identity, theremust exist an ordinal α such th<strong>at</strong> j(α) ≠ α and since j is order-preserving onthe ordinals this is equivalent to the requirement th<strong>at</strong> α < j(α). <strong>The</strong> least suchordinal is called the critical point of j and it can be shown th<strong>at</strong> this must be acardinal. <strong>The</strong> critical point of j is the large cardinal specified and the existenceof the transitive class M and the elementary embedding j are the witnesses forthis.A cardinal κ is a measurable cardinal if there exists a transitive class M andan elementary embedding j : V → M such th<strong>at</strong> κ is the critical point of j.It is by requiring M to be closer to V th<strong>at</strong> one can define large cardinalaxioms far beyond the axiom, “<strong>The</strong>re is a measurable cardinal”. In generalthe closer one requires M to be to V , the stronger the large cardinal axiom.In [12], the n<strong>at</strong>ural maximum axiom was proposed (M = V ). <strong>The</strong> associ<strong>at</strong>edlarge cardinal axiom is th<strong>at</strong> of a Reinhardt cardinal.Definition 5 A cardinal κ is a Reinhardt cardinal if there is an elementaryembedding, j : V → V such th<strong>at</strong> κ is the critical point of j.⊓⊔<strong>The</strong> definition of a Reinhardt cardinal makes essential use of classes, butthe following vari<strong>at</strong>ion does not and is more useful for this discussion. <strong>The</strong>definition requires a logical notion. Suppose th<strong>at</strong> α and β are ordinals suchth<strong>at</strong> α < β. <strong>The</strong>n we write V α ≺ V β to mean th<strong>at</strong> for all formulas φ(x) and forall a ∈ V α , V α |= φ[a] if and only if V β |= φ[a].Definition 6 A cardinal κ is a weak Reinhardt cardinal if there exist γ > λ > κsuch th<strong>at</strong>(1) V κ ≺ V λ ≺ V γ ,(2) there exists an elementary embedding, j : V λ+2 → V λ+2 such th<strong>at</strong> κ is thecritical point of j.⊓⊔<strong>The</strong> definition of a weak Reinhardt cardinal only involves sets. <strong>The</strong> rel<strong>at</strong>ionshipbetween Reinhardt cardinals and weak Reinhardt cardinals is unclear;
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 9but, given the original motiv<strong>at</strong>ion for the definition of a Reinhardt cardinal,one would conjecture th<strong>at</strong> <strong>at</strong> least in terms of consistency strength, Reinhardtcardinals are stronger than weak Reinhardt cardinals. Hence my choice of terminology.In any case, the concept of a weak Reinhardt cardinal is better suitedto illustr<strong>at</strong>e the key points I am trying to make.<strong>The</strong> following theorem is an immedi<strong>at</strong>e corollary of the fundamental inconsistencyresults of Kunen [7].<strong>The</strong>orem 7 (Kunen) <strong>The</strong>re are no weak Reinhardt cardinals.⊓⊔<strong>The</strong> proof is elementary so this does not refute the Skeptic’s Retre<strong>at</strong>. ButKunen’s proof makes essential use of the Axiom of Choice. <strong>The</strong> problem isopen without this assumption, and this is not just an issue for weak Reinhardtcardinals, which is just a notion of large cardinal defined in this paper. <strong>The</strong>rereally is no known interesting example of a strengthening of the definition ofa Reinhardt cardinal th<strong>at</strong> yields a large cardinal axiom which can be refutedwithout using the Axiom of Choice. <strong>The</strong> difficulty is th<strong>at</strong> without the Axiomof Choice it is extraordinarily difficult to prove anything about sets.Kunen’s proof leaves open the possibility th<strong>at</strong> the following large cardinalaxiom might be consistent with the Axiom of Choice. This therefore is essentiallythe strongest large cardinal axiom not known to be refuted by the Axiomof Choice; see [5] for more on this as well as for the actual st<strong>at</strong>ement of Kunen’stheorem.Definition 8 A cardinal κ is an ω-huge cardinal if there exists λ > κ and anelementary embedding j : V λ+1 → V λ+1 such th<strong>at</strong> κ is the critical point of j. ⊓⊔One could strengthen this axiom still further by requiring in addition th<strong>at</strong>for some γ > λ, we have V κ ≺ V λ ≺ V γ and so m<strong>at</strong>ch in form the definition of aweak Reinhardt cardinal, the only modific<strong>at</strong>ion being th<strong>at</strong> λ + 2 is replaced byλ + 1. This change would not affect any of the claims below concerning ω-hugecardinals.<strong>The</strong> issue of whether the existence of a weak Reinhardt cardinal is consistentwith the axioms ZF is an important issue for the Set <strong>The</strong>orist because by theresults of [18], the theoryZF + “<strong>The</strong>re is a weak Reinhardt cardinal”proves the formal consistency of the theoryZFC + “<strong>The</strong>re is a proper class of ω-huge cardinals”.This number-theoretic st<strong>at</strong>ement is a theorem of Number <strong>The</strong>ory. But, asindic<strong>at</strong>ed above, the notion of an ω-huge cardinal is essentially the strongestlarge cardinal notion which is not known to be refuted by the Axiom of Choice.
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 10<strong>The</strong>refore the number theoretic assertion th<strong>at</strong> the theoryZF + “<strong>The</strong>re is a weak Reinhardt cardinal”is consistent is a stronger assertion than the number theoretic assertion th<strong>at</strong>the theoryZFC + “<strong>The</strong>re is a proper class of ω-huge cardinals”is consistent. More precisely, the former assertion implies, but is not impliedby, the l<strong>at</strong>ter assertion; unless of course the theoryZFC + “<strong>The</strong>re is a proper class of ω-huge cardinals”is formally inconsistent. This raises an interesting question:How could the Set <strong>The</strong>orist ever be able to argue for the predictionth<strong>at</strong> the existence of weak Reinhardt cardinals is consistent with axiomsof Set <strong>The</strong>ory without the Axiom of Choice?Moreover this one prediction implies all the predictions (of formal consistency)the Set <strong>The</strong>orist can currently make based on the entire large cardinal hierarchyas presently conceived (in the context of a universe of sets which s<strong>at</strong>isfies theAxiom of Choice). My point is th<strong>at</strong> by appealing to the Skeptic’s Retre<strong>at</strong>, onecould reasonably claim th<strong>at</strong> the theoryZF + “<strong>The</strong>re is a weak Reinhardt cardinal”,is formally consistent—and in making this single claim one would subsumeall the claims of consistency th<strong>at</strong> the Set <strong>The</strong>orist can make based on ourcurrent understanding of the <strong>Universe</strong> of Sets (without abandoning the Axiomof Choice).<strong>The</strong> only tools currently available seem powerless to resolve this issue. Reinterpretingthe number theoretic st<strong>at</strong>ement th<strong>at</strong> the theory,ZF + “<strong>The</strong>re is a weak Reinhardt cardinal”,is formally consistent, in a way th<strong>at</strong> allows the Set <strong>The</strong>orist to argue for thetruth of this st<strong>at</strong>ement, seems equally hopeless. Finally, unlike the axiom ADthere is no candid<strong>at</strong>e presently known for a fragment of V for which the existenceof weak Reinhardt cardinals is the correct (or even a possible) axiom.1.4 Probing the <strong>Universe</strong> of Sets: the InnerModel Program<strong>The</strong> Inner Model Program is the detailed study of large cardinal axioms. <strong>The</strong>first construction of an inner model is due to Gödel, [3] and [4]. This construc-
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 11tion founded the Inner Model Program, and the transitive class constructedis denoted by L. This is the minimum possible universe of sets containing allordinals.If X is a transitive set, then Def(X) denotes the set of all A ⊆ X such th<strong>at</strong>A is logically definable in the structure (X, ∈) from parameters. <strong>The</strong> definitionof L is simply given by replacing the oper<strong>at</strong>ion P(X) in the definition of V α+1by the oper<strong>at</strong>ion Def(X). More precisely:Definition 9 (1) Define L α by induction on the ordinal α:(a) L 0 = ∅ and L α+1 = Def(L α );(b) If α is a limit ordinal then L α = ⋃ {L β | β < α}.(2) L is the class of all sets a such th<strong>at</strong> a ∈ L α for some ordinal α. ⊓⊔It is perhaps important to note th<strong>at</strong> while there must exist a proper classof ordinals α such th<strong>at</strong>L α = L ∩ V α ,this is not true for all ordinals α.<strong>The</strong> question of whether V = L is an important one for Set <strong>The</strong>ory. <strong>The</strong>answer has profound implic<strong>at</strong>ions for the conception of the <strong>Universe</strong> of Sets.<strong>The</strong>orem 10 (Scott, [13]) Suppose there is a measurable cardinal.V ≠ L.<strong>The</strong>n⊓⊔<strong>The</strong> Axiom of Constructibility is the axiom which asserts V = L; more preciselythis is the axiom which asserts th<strong>at</strong> for each set a there exists an ordinal αsuch th<strong>at</strong> a ∈ L α . Scott’s <strong>The</strong>orem provided the first indic<strong>at</strong>ion th<strong>at</strong> the Axiomof Consructibility is independent of the ZFC axioms. At the time there was nocompelling reason to believe th<strong>at</strong> the existence of a measurable cardinal wasconsistent with the ZFC axioms, so one could not make the claim th<strong>at</strong> Scott’stheorem established the formal independence of the Axiom of Consructibilityfrom the ZFC axioms. Of course it is an immedi<strong>at</strong>e corollary of Cohen’s resultsth<strong>at</strong> the Axiom of Consructibility is formally independent of the ZFC axioms.<strong>The</strong> modern significance of Scott’s theorem is more profound: Scott’s theoremestablishes th<strong>at</strong> the Axiom of Consructibility is false. This claim (th<strong>at</strong> V ≠ L)is not universally accepted, but in my view no one has come up with a credibleargument against this claim.<strong>The</strong> Inner Model Program seeks generaliz<strong>at</strong>ions of L for the large cardinalaxioms; in brief it seeks generaliz<strong>at</strong>ions of the Axiom of Consructibilitywhich are comp<strong>at</strong>ible with large cardinal axioms (such as the axioms for measurablecardinals and beyond). It has been a very successful program and
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 12its successes have led to the realiz<strong>at</strong>ion th<strong>at</strong> the large cardinal hierarchy is avery “robust” notion. <strong>The</strong> results which have been obtained provide some ofour deepest glimpses into the <strong>Universe</strong> of Sets, and its successes have led to a“meta-prediction”:A Set <strong>The</strong>orist’s Cosmological Principle: <strong>The</strong> large cardinalaxioms for which there is an inner model theory are consistent; thecorresponding predictions of unsolvability are true because the axiomsare true.Despite the the r<strong>at</strong>her formidable merits as indic<strong>at</strong>ed above, there is a fundamentaldifficulty with the prospect of using the Inner Model Program to counterthe Skeptic’s Retre<strong>at</strong>. <strong>The</strong> problem is in the basic methodology of the InnerModel Program, but to explain this I must give a (brief) description of the(technical) templ<strong>at</strong>e for inner models.<strong>The</strong> inner models which are the goal and focus of the Inner Model Programare defined layer by layer working up through the hierarchy of large cardinalaxioms, which in turn is n<strong>at</strong>urally revealed by the construction of these innermodels. Each layer provides the found<strong>at</strong>ion for the next, and L is the first layer.Roughly (and in practice) in constructing the inner model for a specific largecardinal axiom, one obtains an exhaustive analysis of all weaker large cardinalaxioms. <strong>The</strong>re can be surprises here in th<strong>at</strong> seemingly different notions of largecardinals can coincide in the inner model. Finally as one ascends through thehierarchy of large cardinal axioms, the construction generally becomes moreand more difficult.1.5 <strong>The</strong> building blocks for inner models: ExtendersSuppose th<strong>at</strong> M is a transitive class and th<strong>at</strong> j : V → M is an elementaryembedding with critical point κ. As with the basic templ<strong>at</strong>e for large cardinalaxioms I discussed above, here and below one can restrict to the classes whichare definable classes (by Σ 2 -formulas) etc., so th<strong>at</strong> no essential use of classes isinvolved.It is immedi<strong>at</strong>e from the definitions th<strong>at</strong> for all ordinals γ, j(γ) is an ordinaland moreover j(γ) ≥ γ. Suppose th<strong>at</strong> κ < γ < j(κ) and th<strong>at</strong> P(γ) ⊆ M whereP(γ) = {A | A ⊆ γ}. <strong>The</strong> function:E(A) = j(A) ∩ γwith domain P(γ) is the extender E of length γ defined from j. Note th<strong>at</strong>since γ > κ, necessarily E is not the identity function. Extenders are nontrivial
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 13fragments of the elementary embedding j. (<strong>The</strong> concept of an extender isdue to Mitchell.) <strong>The</strong> definition th<strong>at</strong> I have given is really th<strong>at</strong> of a strongextender because of the assumption th<strong>at</strong> P(γ) ⊆ M. This I do for expositoryreasons. In the case th<strong>at</strong> γ < j(κ), which is the present case, one could dropthis requirement without affecting much of the discussion.<strong>The</strong> importance of the concept of an extender is the following. Suppose th<strong>at</strong>E is an extender of length γ derived from an elementary embedding j : V → Mand th<strong>at</strong> N is a transitive class such th<strong>at</strong> N |= ZFC. Suppose th<strong>at</strong> E ∩ N ∈ Nand th<strong>at</strong> γ = κ + 1 where κ is the critical point of j. <strong>The</strong>n there exists <strong>at</strong>ransitive class M E ⊆ N and an elementary embedding,j E : N → M Esuch th<strong>at</strong> E ∩ N is the extender of length γ derived from j E . <strong>The</strong> point hereof course is th<strong>at</strong> the assumption is only th<strong>at</strong> E ∩ N ∈ N as opposed to themuch stronger assumption th<strong>at</strong> E ∈ N. Both M E and j E can be chosen to bedefinable classes of N (by Σ 2 -formulas) using just the parameter, E ∩ N.Without the assumption th<strong>at</strong> γ = κ + 1, which is a very special case, theseclaims still hold provided th<strong>at</strong> one drops the requirement P(γ) ⊆ M in thedefinition of an extender th<strong>at</strong> I have given.<strong>The</strong>se remarks suggest th<strong>at</strong> one should seek, as generaliz<strong>at</strong>ions of L, transitiveclasses N such th<strong>at</strong> N contains enough extenders of the form, E ∩ N, forsome extender E ∈ V to witness th<strong>at</strong> the targeted large cardinal axiom holdsin N. One can then regard such transitive classes as refinements of V whichare constructed to “preserve” certain extenders of V . <strong>The</strong> complic<strong>at</strong>ion is inspecifying just which extenders are to be preserved.For each set A one can n<strong>at</strong>urally define a class L[A] which is L rel<strong>at</strong>ivizedto the set A, as follows:Definition 11 (1) Define L α [A] by induction on the ordinal α:(a) L 0 = ∅ and L α+1 [A] = Def(X)∩P(L α [A]) where X = L α [A]∪{L α [A] ∩ A};(b) If α is a limit ordinal then L α [A] = ⋃ {L β [A] | β < α}.(2) L[A] is the class of all sets a such th<strong>at</strong> a ∈ L α [A] for some ordinal α. ⊓⊔If F is a function, then L[F ] is defined to be L[A] where A = F . Thus, ifthe domain of F is disjoint from L then L[F ] = L.Constructing from a single extender E yields L[E], which is a true generaliz<strong>at</strong>ionof L and solves the inner model problem for the large cardinal axiom:“<strong>The</strong>re is a measurable cardinal”. This claim follows from the results andmethods of [6] and is illustr<strong>at</strong>ed in part by the theorem below.
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 14<strong>The</strong>re is a fe<strong>at</strong>ure of inner models for large cardinals which is implicit inthis example:For a specific large cardinal axiom there is in general no unique innermodel for th<strong>at</strong> axiom, but r<strong>at</strong>her a family of inner models. But all theseinner models are equivalent in a n<strong>at</strong>ural (but technical) sense.One illustr<strong>at</strong>ion of this is given by the following theorem, which is a modernformul<strong>at</strong>ion of the fundamental results of Kunen [6] on the inner model problemfor one measurable cardinal. For each extender E, let κ E denote the least ordinalα such th<strong>at</strong> E(α) ≠ α. This coincides with critical point of the elementaryembedding, j : V → M from which E is derived.<strong>The</strong>orem 12 Suppose th<strong>at</strong> E and F are extenders.(1) If κ E = κ F , then L[E] = L[F ].(2) If κ E < κ F , then L[F ] ⊂ L[E] and there is an elementary embedding,j : L[E] → L[F ].⊓⊔For the generaliz<strong>at</strong>ions of L[E] which one must consider to solve the innermodel problem for large cardinals beyond the level of measurable cardinals,this ambiguity is much more subtle and lies <strong>at</strong> the core of the difficulty in evendefining the inner models.By the theorem above one cannot use a single extender to build an innermodel for essentially any large cardinal axiom beyond the level of a singlemeasurable cardinal. For example suppose th<strong>at</strong> E is an extender. Combiningelements of Gödel’s basic analysis of L, generalized to an analysis of L[E] with<strong>The</strong>orem 12, it follows th<strong>at</strong> the inner model L[E] will fail s<strong>at</strong>isfy the largecardinal axiom “<strong>The</strong>re are two measurable cardinals”. <strong>The</strong>re is an obviousremedy: to reach inner models for stronger large cardinal axioms one shoulduse sequences,Ẽ = 〈E α : α < θ〉,where each E α is the extender derived from some elementary embedding asabove. <strong>The</strong> complic<strong>at</strong>ion is in how to actually define the sequence; in fact, onemust ultim<strong>at</strong>ely allow the sequence to contain partial extenders which cre<strong>at</strong>esstill further complic<strong>at</strong>ions.A partial extender of length γ is the extender of length γ derived from a Σ 0 -elementary embedding j : N → M where N and M are transitive sets which areonly assumed to be closed under the Gödel oper<strong>at</strong>ions, and, instead of requiringP(γ) ⊆ M, one requires th<strong>at</strong>P(γ) ∩ N = P(γ) ∩ M.
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 15<strong>The</strong> requirement th<strong>at</strong> j be a Σ 0 -elementary embedding is the requirement th<strong>at</strong>for a very restricted collection of formulas, φ(x 0 ), and for all a ∈ N N |= φ[a]if and only if M |= φ[j(a)]. <strong>The</strong> relevant formulas are the Σ 0 -formulas.<strong>The</strong> difficulty mentioned above is in determining exactly when such partialextenders are acceptable. In fact, things get so complic<strong>at</strong>ed th<strong>at</strong>, unlike thesitu<strong>at</strong>ion with measurable cardinals, one can only define the inner model by simultaneouslydeveloping the detailed analysis of the inner model in an elabor<strong>at</strong>einduction.<strong>The</strong> current st<strong>at</strong>e of the art is found in the inner models defined by Mitchelland Steel, [9]. <strong>The</strong> definition of these inner models is the culmin<strong>at</strong>ion of a nearly20 year program of developing the theory of inner models. <strong>The</strong> Mitchell-Steelinner models can accommod<strong>at</strong>e large cardinals up to the level of superstrongcardinals, but existence has only been proved—from the relevant large cardinalaxioms—<strong>at</strong> the level of a <strong>Woodin</strong> cardinal which is a limit <strong>Woodin</strong> cardinals. Inthis program of establishing existence of Mitchell-Steel inner models, the bestresults to d<strong>at</strong>e are due to I. Neeman, [11].<strong>The</strong> distinction between developing the theory of the inner models and provingexistence of the inner models is perhaps a confusing one <strong>at</strong> first glance. <strong>The</strong>precise explan<strong>at</strong>ion requires details of the Mitchell-Steel <strong>The</strong>ory which are beyondthe scope of the present discussion. Roughly, the Mitchell-Steel <strong>The</strong>oryreduces the problem of the existence of the generaliz<strong>at</strong>ion of L for the largecardinal axiom under consider<strong>at</strong>ion to a specific combin<strong>at</strong>orial hypothesis, providedth<strong>at</strong> the large cardinal axiom is <strong>at</strong> the level of a superstrong cardinal orbelow. This combin<strong>at</strong>orial (iter<strong>at</strong>ion) hypothesis can be specified without anyreference to the Mitchell-Steel <strong>The</strong>ory and, more generally, without referenceto inner model theory <strong>at</strong> all. <strong>The</strong>re is of course the possibility th<strong>at</strong> this issymptom<strong>at</strong>ic of a far more serious problem and th<strong>at</strong> by answering one of thetest questions of the Inner Model Program neg<strong>at</strong>ively one can prove th<strong>at</strong> theInner Model Program as presently conceived fails for some large cardinal axiombelow the level of a superstrong cardinal.1.6 <strong>The</strong> Inner Model Program, the Core ModelProgram and the Skeptic’s Retre<strong>at</strong>As I have claimed, there is a fundamental problem with appealing to the InnerModel Program to counter the Skeptic’s Retre<strong>at</strong>. <strong>The</strong> precise n<strong>at</strong>ure of theproblem is subtle so I shall begin by describing wh<strong>at</strong> might seem to be a plausibleversion of the problem. I then briefly will try to describe the actual problem.This will involve the Core Model Program which is a variant of the Inner Model
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 16Program.Suppose (for example) th<strong>at</strong> a hypothetical large cardinal axiom “Ω” providesa counterexample to the Skeptic’s Retre<strong>at</strong>, and this is accomplished by the InnerModel Program.To use the Inner Model Program to refute the existence of an “Ωcardinal”one first must be able to successfully construct the innermodels for all smaller large cardinals, and this hierarchy would befully revealed by the construction.Perhaps this could happen, but it can only happen once. This is the problem.Having refuted the existence of an “Ω-cardinal”, how could one then refute theexistence of any smaller large cardinals, for one would have solved the innermodel problem for these smaller large cardinals. This would refute the Set<strong>The</strong>orist’s Cosmological Principle. So the fundamental problem is:<strong>The</strong> Inner Model Program seems inherently unable, by virtue of itsinductive n<strong>at</strong>ure, to provide a framework for an evolving understandingof the boundary between the possible and the impossible (largecardinal axioms).Upon close inspection it is perhaps not entirely convincing th<strong>at</strong> there is aproblem here. Arguably there is the potential for a problem, but the specificdetails of how the Inner Model Program might succeed in countering the Skeptic’sRetre<strong>at</strong> are clearly critical in determining whether or not there really is aproblem.Though idea th<strong>at</strong> the Inner Model Program could ever yield an inconsistencyresult has always seemed unlikely, there is another way th<strong>at</strong> the Inner ModelProgram might succeed in establishing inconsistency results in a manner th<strong>at</strong>refutes the Skeptic’s Retre<strong>at</strong>. <strong>The</strong> Core Model Program can be described asfollows:Suppose th<strong>at</strong> L[Ẽ] is an inner model as constructed by the Inner ModelProgram. In general would one not expect th<strong>at</strong> the inner model L[Ẽ] to containeven all the real numbers; for example if the Continuum Hypothesis is false inV , then necessarily there are real numbers which are not in L[Ẽ]. <strong>The</strong>refore,every extender on the sequence Ẽ when restricted to L[Ẽ] is necessarily a partialextender in V . This suggests th<strong>at</strong> one might <strong>at</strong>tempt to construct the innermodel L[Ẽ] without using extenders <strong>at</strong> all, just partial extenders.While this might seem reasonable, there is of course a problem. If thereare no extenders in V , then there are no measurable cardinals in V ; and soone cannot in general expect to be able to build an inner model in which there
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 17are measurable cardinals. But suppose one assumes th<strong>at</strong> there is no propertransitive class N in which a particular large cardinal axiom holds. <strong>The</strong>n areasonable conjecture is th<strong>at</strong> there is an inner model of the form L[Ẽ] which is“close” to V .One measure of the closeness of an inner model N to V is a weak coveringprinciple. This requires a definition. A cardinal γ is singular if there existsa cofinal set X ⊆ γ such th<strong>at</strong> |X| < γ, and γ + refers to the least cardinalκ such th<strong>at</strong> κ > γ. Suppose th<strong>at</strong> N is a (proper) transitive class and th<strong>at</strong>N |= ZFC. <strong>The</strong>n weak covering holds for V rel<strong>at</strong>ive to the inner model N if forall uncountable singular cardinals γ, if γ = |V γ |, then(γ + ) N = γ + .Allowing th<strong>at</strong> case th<strong>at</strong> Ẽ = ∅, so th<strong>at</strong> L[Ẽ] = L, this becomes a veryinteresting problem. <strong>The</strong> program to solve this family of problems is the CoreModel Program. Both the Inner Model Program and the Core Model Programseek to construct exactly the same form of an inner model: the only differenceis in the assumptions from which one starts. <strong>The</strong> Inner Model Program startswith the assumption th<strong>at</strong> a particular large cardinal axiom holds in V , whereasthe Core Model Program starts with the assumption th<strong>at</strong> a particular largecardinal axiom does not hold in any transitive class N ⊆ V . It is customaryto refer to the transitive classes constructed by the methods of the Core ModelProgram as core models.<strong>The</strong> Core Model Program was inspired by Jensen’s Covering Lemma andbegan with the results of Dodd and Jensen [2]. <strong>The</strong> strongest results to d<strong>at</strong>e areare primarily due to Steel, who extended the Core Model Program to the levelof <strong>Woodin</strong> cardinals in [15]. As with the Inner Model Program, the solutionsprovided by the Core Model Program increase in complexity as the associ<strong>at</strong>edlarge cardinal axiom is strengthened.<strong>The</strong> Core Model Program has been quite successful, and out of it havecome a number of deep combin<strong>at</strong>orial theorems. For example the methodsand constructions of Core Model Program play an essential role in the proof of<strong>The</strong>orem 3. One might suspect th<strong>at</strong> utility of the Core Model Program is limitedfor proving the kinds of theorems th<strong>at</strong> an inconsistency result would require,because of the requisite hypothesis th<strong>at</strong> there be no proper transitive class Nin which a specific large cardinal axiom holds. But despite this requirement,the Core Model Program has yielded some surprising theorems of Set <strong>The</strong>ory.Recall th<strong>at</strong> the Generalized Continuum Hypothesis (GCH) is the assertionth<strong>at</strong> for all infinite cardinals, γ, 2 γ = γ + where 2 γ is the cardinalityof P(γ) = {X | X ⊆ γ}. <strong>The</strong> following theorem is an example of a theoremproved by the methods of the Core Model Program.
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 18<strong>The</strong>orem 13 ([16]) Suppose th<strong>at</strong> there exists a countable set A such th<strong>at</strong> A isa set of ordinals and V = L[A]. <strong>The</strong>n the GCH holds.⊓⊔On general grounds, to prove the theorem it suffices from the hypothesis ofthe theorem to just prove th<strong>at</strong> the Continuum Hypothesis holds. <strong>The</strong> specialcase where A ⊆ ω follows from Gödel’s analysis of L, generalized to the analysisof L[A]. For the general case where one does not assume A ⊆ ω, there is noelementary proof of the Continuum Hypothesis known.By a theorem of Jensen [1], for any sentence φ, if the sentence is consistentwith the axioms ZFC, then the existence of a proper class N within which thesentence holds is consistent with the hypothesis of <strong>The</strong>orem 13. If in additionthe sentence is consistent with the axioms ZFC + GCH, then one can evenrequire th<strong>at</strong> the transitive class N be close to L[A]. For example one can requireth<strong>at</strong> N and L[A] have the same cardinals. While this additional consistencyassumption may seem like a very restrictive assumption, <strong>at</strong> least for the currentgener<strong>at</strong>ion of large cardinal axioms, it is not.<strong>The</strong>refore, it is perhaps not unreasonable th<strong>at</strong> the Core Model Programmight yield th<strong>at</strong> some proposed large cardinal axiom is inconsistent and in doingso, refute the Skeptic’s Retre<strong>at</strong>. But to accomplish this, the Core ModelProgram would seem to have to produce an “ultim<strong>at</strong>e” core model correspondingto the ultim<strong>at</strong>e inner model. But, if this is an inner model of the form L[Ẽ],for some sequence of (partial) extenders, as is the case for essentially all coremodels which have been constructed to d<strong>at</strong>e, then the n<strong>at</strong>ure of the extenderson the sequence Ẽ should reveal the entire large cardinal hierarchy—and weagain are in a situ<strong>at</strong>ion where further progress looks unlikely.Thus, it would seem th<strong>at</strong> the Skeptic’s Retre<strong>at</strong> is in fact a powerful counter<strong>at</strong>tack.But there is something wrong here, some fundamental misconception.<strong>The</strong> answer lies in understanding large cardinal axioms which are much strongerthan those within reach of the Mitchell-Steel hierarchy of inner models. Ironically,one outcome of my proposed resolution to this misconception is th<strong>at</strong> forthese large cardinal axioms, the Set <strong>The</strong>orist’s Cosmological Principle is eitherfalse or useless. I shall discuss these ramific<strong>at</strong>ions after <strong>The</strong>orem 22.1.7 Supercompact cardinals and beyondWe begin with a definition.Definition 14 (Solovay) A cardinal κ is a supercompact cardinal if for eachordinal α there exists a transitive class M and an elementary embeddingj : V → M such th<strong>at</strong>
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 19(1) κ is the critical point of j and j(κ) > α;(2) M contains all functions, f : α → M. ⊓⊔Still stronger are extendible cardinals, huge cardinals, and n-huge cardinalswhere n < ω. <strong>The</strong>se I shall not define here. As I have already indic<strong>at</strong>ed, thestrongest large cardinal axioms not known to be inconsistent with the Axiomof Choice are the family of axioms asserting the existence of ω-huge cardinals.<strong>The</strong>se axioms have seemed so far beyond any conceivable inner model theoryth<strong>at</strong> they simply are not understood.<strong>The</strong> possibilities for an inner model theory <strong>at</strong> the level of supercompactcardinals and beyond has been essentially a complete mystery until recently.<strong>The</strong> reason lies in the n<strong>at</strong>ure of extenders. Suppose th<strong>at</strong> E is an extender oflength γ derived from an elementary embedding j : V → M with critical pointκ and such th<strong>at</strong> P(γ) ⊆ M. If γ ≤ j(κ) then E is a short extender, otherwiseE is a long extender. Up to this point I have only considered short extenders.<strong>The</strong> properties of long extenders can be quite subtle and it is for this reasonth<strong>at</strong> I impose the requirement P(γ) ⊆ M. Even with this requirement manysubtleties remain. For example, <strong>The</strong>orem 12, which I implicitly st<strong>at</strong>ed for shortextenders, is false for long extenders. One can prove the following vari<strong>at</strong>ionprovided the extenders are not too long. For expository purposes let me definean extender E to be a suitable extender if E is the extender of length γ derivedfrom an elementary embedding j : V → M such th<strong>at</strong> P(γ) ⊆ M and suchth<strong>at</strong> γ < j(α) for some α < j(κ), where κ is the critical point of j. Suitableextenders can be long extenders but they cannot be too long.<strong>The</strong>orem 15 Suppose th<strong>at</strong> E and F are suitable extenders. <strong>The</strong>nR ∩ L[E] ⊆ R ∩ L[F ] or R ∩ L[F ] ⊆ R ∩ L[E]. ⊓⊔Without the restriction to suitable extenders, it is not known if this theoremholds.<strong>The</strong> following lemma of Magidor reformul<strong>at</strong>ed in terms of suitable extendersgives a useful reformul<strong>at</strong>ion of supercompactness, [18]. <strong>The</strong> st<strong>at</strong>ement ofthe lemma involves the following not<strong>at</strong>ion which I previously defined for shortextenders. Suppose E is an extender of length γ derived from an elementaryembedding j : V → M. <strong>The</strong>n κ E is the critical point of j. Suppose th<strong>at</strong> α < γ.<strong>The</strong>n by the definition of E, and since α = {β | β < α} ⊆ γ, we find th<strong>at</strong> α isin the domain of E and either E(α) = γ or E(α) < γ. Moreover the followinghold:1. E(α) < γ if and only if j(α) < γ and E(α) = j(α),
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 202. E(α) = γ if and only if j(α) ≥ γ.Thus, κ E is simply the least α such th<strong>at</strong> E(α) ≠ α.Lemma 16 Suppose th<strong>at</strong> δ is a cardinal. <strong>The</strong>n the following are equivalent:(1) δ is supercompact;(2) For each ordinal γ > δ, there exists a suitable extender E of length γ suchth<strong>at</strong> E(κ E ) = δ.⊓⊔My convention in wh<strong>at</strong> follows is th<strong>at</strong> a class E of extenders witnesses th<strong>at</strong>δ is a supercompact cardinal if, for each γ > δ, there exists an extender E ∈ Esuch th<strong>at</strong>1. E has length γ, and2. E(κ E ) = δ and for some α < δ, E(α) = γ.Note th<strong>at</strong> condition (2) implies th<strong>at</strong> E is a suitable extender.<strong>The</strong> Mitchell-Steel inner models are constructed from sequences of shortextenders. But to build inner models <strong>at</strong> the level of supercompact cardinals andbeyond one must have long extenders on the sequence, and this cre<strong>at</strong>es seriousobstacles if the these extenders are “too” long. In fact, Steel has isol<strong>at</strong>ed aspecific obstacle which becomes severe <strong>at</strong> the level of one supercompact cardinalwith a measurable cardinal above.But by some fairly recent theorems from [18] something completely unexpectedand remarkable happens. Suppose th<strong>at</strong> N is a transitive class, for somecardinal δ,N |= “δ is a supercompact cardinal”,and th<strong>at</strong> this is witnessed by class of all E ∩ N such th<strong>at</strong> E ∩ N ∈ N and suchth<strong>at</strong> E is an extender. <strong>The</strong>n the transitive class N is close to V and N inheritsessentially all large cardinals from V .For example, suppose th<strong>at</strong> for each n there is a proper class of n-hugecardinals in V . <strong>The</strong>n in N, for each n, there is a proper class of n-huge cardinals.<strong>The</strong> amazing thing is th<strong>at</strong> this must happen no m<strong>at</strong>ter how N is constructed.This would seem to undermine my earlier claim th<strong>at</strong> inner models should beconstructed as refinements of V which preserve enough extenders from V towitness th<strong>at</strong> the targeted large cardinal axiom holds in the inner model. Itdoes not, and the reason is th<strong>at</strong> by simply requiring th<strong>at</strong> E ∩N ∈ N for enoughsuitable extenders from V to witness th<strong>at</strong> the large cardinal axiom, “<strong>The</strong>re is asupercompact cardinal”, holds in N, one (and this is the surprise) necessarilymust have E ∩N ∈ N for a much larger class of extenders of V . So the principle
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 21th<strong>at</strong> there are enough extenders of N which are of the form E ∩ N for someextender E ∈ V , to witness the targeted large cardinal axiom holds in N, ispreserved. <strong>The</strong> change, in the case th<strong>at</strong> N is constructed from a sequence ofextenders, is th<strong>at</strong> these extenders do not have to be on the sequence from whichN is constructed. In particular, in the case th<strong>at</strong> the sequence of extenders fromwhich N is constructed contains only suitable extenders, large cardinal axiomscan be witnessed to hold in N by the “phantom” extenders (these are extendersof N which are not on the sequence) which cannot be witnessed to hold by anyextender on the sequence.As a consequence of this, one can completely avoid the cited obstacles because:One does not need to have the kinds of long extenders on the sequencewhich give rise to the obstacles.Specifically, one can restrict consider<strong>at</strong>ion to extender sequences of just suitableextenders and this is a paradigm shift in the whole conception of inner models.<strong>The</strong> analysis yields still more. Suppose th<strong>at</strong> there is a positive solution (inZFC) to the inner model problem for just one supercompact cardinal. Note th<strong>at</strong>this seems <strong>at</strong> the edge of feasibility without encountering the serious obstaclesraised by long extenders. More precisely, suppose th<strong>at</strong>, if κ is a supercompactcardinal, then (provably) there is a definable sequenceẼ = 〈E α : α ∈ Ord〉of (partial) extenders such th<strong>at</strong> L[Ẽ] is an L-like inner model in which κ issupercompact, and, th<strong>at</strong> this is witnessed by the extenders on the sequence(which are the restriction of true extenders to the associ<strong>at</strong>ed inner model L[Ẽ]).<strong>The</strong>n as a corollary one would would obtain a proof of the following conjecture:Conjecture (ZF) <strong>The</strong>re are no weak Reinhardt cardinals.⊓⊔It is possible to isol<strong>at</strong>e a specific conjecture which must be true if there isa positive solution to the inner problem for one supercompact cardinal, as describedabove, and which itself suffices for this inconsistency result. To explainthis further I must give one last definition.This is the definition of the class, HOD, which origin<strong>at</strong>es in remarks of Gödel<strong>at</strong> the Princeton Bicentennial Conference in December, 1946. <strong>The</strong> first detailedreference appears to be [8] (see the review of [8] by G. Kreisel).Definition 17 (ZF) (1) For each ordinal α, let HOD α be the set of all setsa such th<strong>at</strong> there exists a set A ⊆ α such th<strong>at</strong>(a) A is definable in V α from ordinal parameters;
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 22(b) a ∈ L α [A].(2) HOD is the class of all sets a such th<strong>at</strong> a ∈ HOD α for some α. ⊓⊔<strong>The</strong> definition of HOD α combines fe<strong>at</strong>ures of the definition of L α and fe<strong>at</strong>uresof the definition of V α . I caution th<strong>at</strong>, just as is the case for L α , in generalwe haveHOD α ≠ HOD ∩ V α ,though for a proper class of ordinals α it is true th<strong>at</strong> HOD α = HOD ∩ V α .<strong>The</strong> class HOD is quite interesting for a number of reasons one of which isillustr<strong>at</strong>ed by the following observ<strong>at</strong>ion of Gödel which as indic<strong>at</strong>ed is st<strong>at</strong>edwithin just the theory ZF, in other words without assuming the Axiom ofChoice.<strong>The</strong>orem 18 (ZF) HOD |= ZFC.⊓⊔This theorem gives a completely different approach to showing th<strong>at</strong> if ZF isformally consistent then so is ZFC.One difficulty with HOD is th<strong>at</strong> the definition of HOD is not absolute;for example, in general HOD is not even the same as defined within HOD.As a consequence almost any set theoretic question one might n<strong>at</strong>urally askabout HOD is formally unsolvable. Two immedi<strong>at</strong>e such questions are whetherV = HOD and, more simply, whether HOD contains all the real numbers. Bothof these questions are formally unsolvable but are of evident importance becausethey specifically address the complexity of the Axiom of Choice. If V = HODthen there is no mystery as to why the Axiom of Choice holds, but of courseone is left with the problem of explaining why V = HOD. If V = L then it iseasy to verify th<strong>at</strong> V = HOD. Further the inner models of Mitchell-Steel (inthe situ<strong>at</strong>ions where existence can be proved) can always be constructed to becontained in HOD even though the axiom V = HOD can fail in a Mitchell-Steelinner model.I now present a key conjecture. This conjecture involves the notion th<strong>at</strong> anuncountable cardinal γ is a regular cardinal. This is the property th<strong>at</strong> for allX ⊆ γ, if |X| < γ, then X is bounded in γ. Altern<strong>at</strong>ively, referring to notionsalready defined, an uncountable cardinal γ is a regular cardinal if it is not asingular cardinal.<strong>The</strong> HOD Conjecture: (ZFC) Suppose th<strong>at</strong> κ is a supercompact cardinal.<strong>The</strong>n there exists a regular cardinal γ > κ which is not a measurable cardinalin HOD.⊓⊔
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 23If there is a supercompact cardinal in V then the HOD Conjecture impliesth<strong>at</strong> HOD is “close” to V . Assuming th<strong>at</strong> a slightly stronger large cardinalhypothesis holds in V (and th<strong>at</strong> the HOD Conjecture also holds in V ), thenone actually obtains th<strong>at</strong> HOD is quite close to V . As evidence for this l<strong>at</strong>terclaim, I note the following theorem. Part (3) of this theorem is a very strongclosure condition for HOD and part (2) follows directly from part (3). <strong>The</strong>analogous closure condition holds for any transitive (proper) class N |= ZFCsuch th<strong>at</strong> for some cardinal δ,N |= “δ is a supercompact cardinal”,and such th<strong>at</strong> this is witnessed by class of all restrictions E ∩ N, where E is anextender and E ∩ N ∈ N. Referring back to the discussion on the distinctionbetween the Core Model Program and the Inner Model Program, this closurecondition establishes th<strong>at</strong> in the context of the existence of just one supercompactcardinal, there is no difference between the primary objectives of thesetwo programs.<strong>The</strong>orem 19 (ZFC) Suppose th<strong>at</strong> there is an extendible cardinal and th<strong>at</strong> theHOD Conjecture holds. <strong>The</strong>n the following hold:(1) <strong>The</strong>re exists an ordinal α such th<strong>at</strong> for all cardinals γ > α, if γ is asinular cardinal then γ + = (γ + ) HOD ;(2) suppose for each n there is a proper class of n-huge cardinals. <strong>The</strong>n foreach n,HOD |= “<strong>The</strong>re is a proper class of n-huge cardinals.”;(3) there exists an ordinal α such th<strong>at</strong> for all γ > α, ifj : HOD ∩ V γ+1 → HOD ∩ V j(γ)+1is an elementary embedding with critical point above α, then j ∈ HOD. ⊓⊔If the HOD Conjecture is provable in ZFC then there is a striking corollaryin ZF:<strong>The</strong>orem 20 (ZF) Suppose th<strong>at</strong> λ is a limit of supercompact cardinals and th<strong>at</strong>there is an extendible cardinal below λ. <strong>The</strong>n there is no elementary embedding,j : V λ+2 → V λ+2 ,which is not the identity.⊓⊔
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 24This corollary would prove the conjecture above (th<strong>at</strong> there are no weak Reinhardtcardinals) and would give an inconsistency result which more closelym<strong>at</strong>ches the version of <strong>The</strong>orem 7 th<strong>at</strong> Kunen actually proved (assuming theAxiom of Choice).<strong>The</strong>orem 21 (Kunen, [7]) Suppose th<strong>at</strong> λ is an ordinal. <strong>The</strong>n there is noelementary embedding j : V λ+2 → V λ+2 which is not the identity.⊓⊔<strong>The</strong> connection between the HOD Conjecture and the inner model problemfor one supercompact cardinal is illustr<strong>at</strong>ed by the next theorem. Arguably,st<strong>at</strong>ement (3) of this theorem would follow from any reasonable solution to theinner model problem for one supercompact cardinal.<strong>The</strong>orem 22 (ZFC) Suppose th<strong>at</strong> there is an extendible cardinal. <strong>The</strong>n thefollowing are equivalent;(1) <strong>The</strong> HOD Conjecture holds;(2) there is a cardinal δ such th<strong>at</strong>HOD |= “δ is a supercompact cardinal”and this is witnessed by the class of all E∩HOD such th<strong>at</strong> E is an extenderand E ∩ HOD ∈ HOD.(3) there exists a class N ⊆ HOD and there exists a cardinal δ such th<strong>at</strong>N |= ZFC + “δ is a supercompact cardinal”and this is witnessed by the class of all E ∩ N such th<strong>at</strong> E is an extenderand E ∩ N ∈ N.⊓⊔<strong>The</strong>se developments come with a price. For the large cardinal axiomsstronger than the axiom which asserts the existence of one supercompact cardinal,the Set <strong>The</strong>orist’s Cosmological Principle must either be abandoned orrevised. <strong>The</strong> reason is th<strong>at</strong> the solution to the inner model problem for thespecific axiom “<strong>The</strong>re exists one supercompact cardinal” necessarily will solvethe inner model problem (as currently defined) for essentially all of the knownlarge axioms up to and including the axiom “<strong>The</strong>re is an ω-huge cardinal”.We therefore face a very simple dichotomy of possibilities: the inner modelproblem for the axiom “<strong>The</strong>re exists one supercompact cardinal” is solvableor it is not. <strong>The</strong> other possibility—this is the possibility th<strong>at</strong> this solvabilityquestion is itself unsolvable—is not an option based on any reasonable notionof m<strong>at</strong>hem<strong>at</strong>ical truth. <strong>The</strong> simple reason is th<strong>at</strong> if the solvability question is
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 25itself formally unsolvable then the inner model problem for the axiom “<strong>The</strong>reexists one supercompact cardinal” is not solvable. <strong>The</strong> situ<strong>at</strong>ion here is exactlylike the situ<strong>at</strong>ion for number (but not all) of the prominent open questions ofmodern m<strong>at</strong>hem<strong>at</strong>ics. For example, if the “Riemann Hypothesis” is formallyunsolvable (and there is absolutely no evidence for this), then the RiemannHypothesis is true.Wh<strong>at</strong>ever the outcome to this dichotomy of possibilities, one outcome seemscertain: <strong>The</strong> Set <strong>The</strong>orist’s Cosmological Principle cannot be applied to arguefor the truth of (any) large cardinal axioms beyond the axiom “<strong>The</strong>re exists onesupercompact cardinal”. Of course, it could be th<strong>at</strong> solution to the Inner ModelProblem for the axiom “<strong>The</strong>re exists one supercompact cardinal” involves theconstruction of an inner model which is not of the form L[Ẽ] where Ẽ is asequence of partial extenders etc. But this alone would not suffice to resolvethe issue raised by the preceding theorem. <strong>The</strong> reason is th<strong>at</strong> st<strong>at</strong>ement (3) ofthe theorem makes no assumption th<strong>at</strong> the inner model N is constructed froman extender sequence. Further, the necessity of the closeness of N to V (whichis the only issue here) does not require th<strong>at</strong> N ⊆ HOD but only requires th<strong>at</strong>for some cardinal δ,N |= ZFC + “δ is a supercompact cardinal”.And this is witnessed by the class of all E ∩ N such th<strong>at</strong> E is an extender andE ∩ N ∈ N. A solution which solves the inner model problem for the axiom“<strong>The</strong>re exists one supercompact cardinal” and yet involves only the constructionof inner models for which this fails, would be completely unlike all the currentsolutions to the inner model problem for the various large cardinal axioms wherea solution exists.One can correctly specul<strong>at</strong>e th<strong>at</strong> the difficulty is in the requirement th<strong>at</strong>for the inner model N the relevant large cardinal axiom is witnessed to holdby extenders of N which are of the form E ∩ N for some extender E ∈ V .But extenders are the witnesses for large axioms and therefore any genuineconstruction of an inner model N should arguably s<strong>at</strong>isfy this requirement.Moreover, to avoid “trivial” solutions, one has to require th<strong>at</strong> the associ<strong>at</strong>edinner models s<strong>at</strong>isfy some form of being close to V . More precisely, one hasto require th<strong>at</strong> the large cardinals of the inner models N which constitute thesolution have some form of ancestry in the large cardinals of V .<strong>The</strong>re is a silver lining to this dark cloud. Suppose th<strong>at</strong> the Inner ModelProblem <strong>at</strong> the level of one supercompact cardinal can be solved and th<strong>at</strong> thesolution does involve defining inner models which are of the form L[Ẽ] where Ẽ isa sequence of (partial) extenders. <strong>The</strong>n it is possible to analyze the rel<strong>at</strong>ionshipof the inner models L[Ẽ] to V without knowing how the corresponding extender
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 26sequences Ẽ are actually constructed. <strong>The</strong> results to d<strong>at</strong>e in [18] have gre<strong>at</strong>lyclarified the axioms which assert the existence of ω-huge cardinals and revealeda new hierarchy of such axioms. <strong>The</strong> emerging structure theory for these axiomscould well develop to the point where it serves as a surrog<strong>at</strong>e for the existenceof an inner model theory in a revised version of the Set <strong>The</strong>orist’s CosmologicalPrinciple. On general grounds one can argue th<strong>at</strong> if these axioms are consistent,then L[Ẽ] must provide a structure theory for these axioms; because, if theaxioms can hold in V , then they can hold in L[Ẽ] (by the closeness of L[Ẽ]to V ). <strong>The</strong>refore, the revision of the Set <strong>The</strong>orist’s Cosmological Principleth<strong>at</strong> is required may actually not be so severe. <strong>The</strong> revision would be in wh<strong>at</strong>constitutes an inner model theory for those large cardinal axioms beyond thelarge cardinal axiom th<strong>at</strong> asserts the existence of one supercompact cardinal.My point is th<strong>at</strong> simply requiring th<strong>at</strong> there be a generaliz<strong>at</strong>ion of the Axiomof Consructibility which is comp<strong>at</strong>ible with the large cardinal axiom may (forevident reasons) not be sufficient for these axioms. In addition, one may have toimpose much stronger requirements, perhaps more in line with the often quotedspecul<strong>at</strong>ion of Gödel (1947):<strong>The</strong>re might exist axioms so abundant in their verifiable consequences,shedding so much light upon a whole discipline, andfurnishing such powerful methods for solving given problems (andeven solving them, as far as possible, in a constructivistic way)th<strong>at</strong> quite irrespective of their intrinsic necessity they would haveto be assumed <strong>at</strong> least in the same sense as any established physicaltheory.To summarize the point I am <strong>at</strong>tempting to make:<strong>The</strong> found<strong>at</strong>ional basis for asserting th<strong>at</strong> large cardinal axioms beyondthe level of one supercompact cardinal are “true” might liein the structural consequences for L[Ẽ] th<strong>at</strong> their existence implies.Moreover, this claim of truth may require (and reinforce) some versionof the claim th<strong>at</strong> V = L[Ẽ].This specul<strong>at</strong>ion is grounded in a number of preliminary results, [18].1.8 Summary<strong>The</strong>re is now a body of m<strong>at</strong>hem<strong>at</strong>ical evidence th<strong>at</strong> if there is a supercompactcardinal then there is transcendent version, say L Ω , of Gödel’s inner model L:in brief, there is an ultim<strong>at</strong>e L. This development if realized will yield a muchdeeper understanding of the large cardinal axioms:
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 28would provide substantial evidence th<strong>at</strong> there is a single axiom for V whichyields a conception of the <strong>Universe</strong> of Sets which is in fact (for the reasonsarticul<strong>at</strong>ed above) more unambiguous then our present conception of numbertheory. Moreover, the successful construction of L Ω would provide a startingpoint for discovering th<strong>at</strong> axiom.This development would be a significant milestone in our understanding ofthe <strong>Transfinite</strong> <strong>Universe</strong>. I make this claim completely independently of anyspecul<strong>at</strong>ion th<strong>at</strong> there are number theoretic problems which are “orthogonal” toall large cardinal axioms, such as is the case for the problem of the ContinuumHypothesis.
Bibliography[1] A. Beller, R. B. Jensen, and P. Welch. Coding the universe, volume No. 47of London M<strong>at</strong>hem<strong>at</strong>ical Society Lecture Note Series. Cambridge UniversityPress, Cambridge-New York, 1982.[2] A. Dodd and R. Jensen. <strong>The</strong> core model. Ann. M<strong>at</strong>h. <strong>Logic</strong>, 20(1):43–75,1981.[3] Kurt Gödel. Consistency-Proof for the Generalized Continuum-Hypothesis. Proc. N<strong>at</strong>. Acad. Sci. U.S.A, 25:220–224, 1938.[4] Kurt Gödel. <strong>The</strong> Consistency of the Continuum Hypothesis. Annals ofM<strong>at</strong>hem<strong>at</strong>ics Studies, no. 3. Princeton University Press, Princeton, N. J.,1940.[5] Akihiro Kanamori. <strong>The</strong> higher infinite. Perspectives in M<strong>at</strong>hem<strong>at</strong>ical<strong>Logic</strong>. Springer-Verlag, Berlin, 1994. Large cardinals in set theory fromtheir beginnings.[6] Kenneth Kunen. Some applic<strong>at</strong>ions of iter<strong>at</strong>ed ultrapowers in set theory.Ann. M<strong>at</strong>h. <strong>Logic</strong>, 1:179–227, 1970.[7] Kenneth Kunen. Elementary embeddings and infinitary combin<strong>at</strong>orics. J.Symbolic <strong>Logic</strong>, 36:407–413, 1971.[8] Azriel Lévy. Definability in axiom<strong>at</strong>ic set theory. I. In <strong>Logic</strong>, Methodologyand Philos. Sci. (Proc. 1964 Intern<strong>at</strong>. Congr.), pages 127–151. North-Holland, Amsterdam, 1965.[9] William J. Mitchell and John R. Steel. Fine structure and iter<strong>at</strong>ion trees.Springer-Verlag, Berlin, 1994.[10] Jan Mycielski and H. Steinhaus. A m<strong>at</strong>hem<strong>at</strong>ical axiom contradicting theaxiom of choice. Bull. Acad. Polon. Sci. Sér. Sci. M<strong>at</strong>h. Astronom. Phys.,10:1–3, 1962.29
Gödel Book—input: <strong>Woodin</strong> (rev. 2009 Oct 04) 30[11] Itay Neeman. Inner models in the region of a <strong>Woodin</strong> limit of <strong>Woodin</strong>cardinals. Ann. Pure Appl. <strong>Logic</strong>, 116(1-3):67–155, 2002.[12] William N. Reinhardt. Ackermann’s set theory equals ZF. Ann. M<strong>at</strong>h.<strong>Logic</strong>, 2(2):189–249, 1970.[13] Dana Scott. Measurable cardinals and constructible sets. Bull. Acad.Polon. Sci. Sér. Sci. M<strong>at</strong>h. Astronom. Phys., 9:521–524, 1961.[14] S. Shelah. Around classific<strong>at</strong>ion theory of models, volume 1182 of LectureNotes in M<strong>at</strong>hem<strong>at</strong>ics. Springer–Verlag, 1986.[15] John R. Steel. <strong>The</strong> core model iterability problem. Springer-Verlag, Berlin,1996.[16] W. <strong>Hugh</strong> <strong>Woodin</strong>. <strong>The</strong> universe constructed from a sequence of ordinals.Arch. M<strong>at</strong>h. <strong>Logic</strong>, 35(5-6):371–383, 1996.[17] W. <strong>Hugh</strong> <strong>Woodin</strong>. <strong>The</strong> Axiom of Determinacy, forcing axioms, and thenonst<strong>at</strong>ionary ideal. Walter de Gruyter & Co., Berlin, 1999.[18] W. <strong>Hugh</strong> <strong>Woodin</strong>. Suitable extender sequences. Preprint, pages 1–676,2009.[19] Ernst Zermelo. Über Grenzzahlen und Mengenbereiche: Neue UntersuchungenuberdieGrundlagen der Mengenlehre. Fundamenta M<strong>at</strong>hem<strong>at</strong>icae,16:29–47, 1930.