Large cardinals and L-like universes

LARGE CARDINALS AND L-LIKE UNIVERSES SY D. FRIEDMAN There are many different ways to extend the axioms of ZFC. One way is to adjoin the axiom V=L, asserting that every set is constructible. This axiom has many attractive consequences, such as the generalised continuum hypothesis (GCH), the existence of a definable wellordering of the class of all sets, as well as strong combinatorial principles, such as ♦,¤and the existence of morasses. However V=Ladds no consistency strength to ZFC. As many interesting set-theoretic statements have consistency strength beyond ZFC, it is common in set theory to assume at least the existence of inner models of V which contain large cardinals. Can we simultaneously have the advantages of both the axiom of constructibility and the existence of large cardinals? Unfortunately even rather modest large cardinal hypotheses, such as the existence of a measurable cardinal, refute V=L. We can however hope for the following compromise: Vis an “L-like” model containing large cardinals. In this article we explore the possibilities for this assertion, for various notions of “L-like” and for various types of large cardinals. There are two approaches to this problem. The first approach is via the Inner model program. Show that any universe with large cardinals has an L-like inner model with large cardinals. The inner model program, through use of fine structure theory and the theory of iterated ultrapowers, has succeeded in producing very L-like inner models containing many Woodin cardinals. An alternative approach is given by the The author wishes to thank the Austrian Science Fund (FWF) for its generous support and the Centre de Recerca Matem`atica (CRM), where this article was written in the month of September 2005. 1 2 SY D. FRIEDMAN Outer model program. Show that any universe with large cardinals has an L-like outer model with large cardinals. We will show that L-like outer models with extremely large cardinals can be obtained using the method of iterated forcing. Large cardinals A cardinal κis inaccessible iff it is uncountable, regular and larger than the power set of any smaller cardinal. It is measurable iff there is a κcomplete, nonprincipal ultrafilter on κ. Measurability is equivalent to a property expressed in terms of embeddings, and stronger large cardinal properties are also expressed in this way. As usual, Vdenotes the universe of all sets. Let Mbe an inner model, i.e., a transitive proper class that satisfies the axioms of ZFC. A class function j:V→Mis an embedding iff it preserves the truth of formulas with parameters in the language of set theory and is not the identity. If jis an embedding then there is a least ordinal κsuch that j(κ)6=κ, called the critical point of j, which is a measurable cardinal. For an ordinal α,j:V→Mis α-strong iff Vαis contained in M. A cardinal κis α-strong iff there is an α-strong embedding with critical point κ.Strong means α-strong for all α. Kunen ([7]) showed that no embedding is strong. However a cardinal can be strong, as embeddings witnessing its α-strength can vary with α. Stronger properties are obtained by requiring j:V→Mto have strength depending on the image under jof its critical point. For example, κis superstrong iff there is a nontrivial elementary embedding j:V→Mwith critical point κwhich is j(κ)-strong. An important weakening of superstrength is the property that for each f:κ→κthere is a ¯κ < κ closed under fand a nontrivial elementary embedding j:V→Mwith critical point ¯κwhich is j(f)(¯κ)-strong; such κare known as Woodin cardinals. The consistency strength of the existence of a Woodin cardinal is strictly between that of a strong cardinal and a superstrong cardinal. We can demand more than superstrength. A cardinal κis hyperstrong iff it is the critical point of an embedding j:V→Mwhich is j(κ) + 1- strong. For a finite n > 0, n-superstrength is obtained by requiring jto be jn(κ)-strong, where j1=j,jk+1 =j◦jk. Finally, κis ω-superstrong iff it is the critical point of an embedding j:V→Mwhich is n-superstrong for LARGE CARDINALS AND L-LIKE UNIVERSES 3 all n. Kunen’s result [7] shows that no embedding jwith critical point κis jω(κ) + 1-strong, where jω(κ) is the supremum of the jn(κ) for finite n. The inner model program If κis inaccessible, then κis also inaccessible in L, the most L-like model of all. This is not the case for measurability, however if κis measurable then κis measurable in an inner model L[U], where Uis an ultrafilter on κ, which has a definable wellordering and in which GCH, ♦,¤hold and gap 1 morasses exist. For a strong cardinal κthere is a similarly L-like inner model L[E] in which κis strong, where Enow is not a single ultrafilter, but rather a sequence of generalised ultrafilters, called extenders. More recent work yields similar results for Woodin cardinals, and even for Woodin limits of Woodin cardinals (see [9]). However, progress beyond that has been impeded by the so-called iterability problem. The outer model program How can we obtain L-like outer models with large cardinals? For inaccessibles one has the following result of Jensen (see [1]): Theorem 1. (L-coding) There is a generic extension V[G]of Vsuch that a. ZFC holds in V[G]. b. V[G] = L[R]for some real R. c. Every inaccessible cardinal of Vremains inaccessible in V[G]. There are similar L[U] and L[E] coding theorems (see [4] for the former), providing outer models of the form L[U][R] and L[E][R], Ra real, which are just as L-like as L[U] and L[E], preserving measurability and strength, respectively. However the approach via coding is limited in its use. Obtaining L-like outer models via coding depends on the existence of L-like inner models, such as L[U] or L[E], which, as we have observed, are not known to exist beyond Woodin limits of Woodin cardinals. And there are problems with the coding method itself which arise already just past a strong cardinal. A more promising approach is to use iterated forcing. To illustrate this, consider the problem of making the GCH true in an outer model. Begin with an arbitrary model Vof ZFC. Using forcing, we can add a function from ℵ1onto 2ℵ0without adding reals, thereby making CH true. By forcing again, we add a function from (the possibly new) ℵ2onto (the possibly new) 4 SY D. FRIEDMAN 2ℵ1without adding subsets of ℵ1, thereby obtaining 2ℵ1=ℵ2. Continue this indefinitely (via a reverse Easton iteration) and the result is a model of the GCH. Do we preserve large cardinal properties if we make GCH true in this way? The answer is Yes. Theorem 2. (Large cardinals and the GCH) If κis superstrong then there is an outer model in which κis still superstrong and the GCH holds. The same holds for hyperstrong, n-superstrong for finite nand ω-superstrong. Proof. First we describe in more detail the above iteration to make GCH true. By induction on αwe define the iteration Pαof length α:P0is the trivial forcing. For limit λ,Pλis the inverse limit of the Pα,α < λ, if λ is singular and is the direct limit of the Pα,α < λ, if λis regular. For successor α+ 1, Pα+1 =Pα∗Qα, where Qαis the forcing that collapses 2ℵαto ℵα+1 using conditions of size at most ℵα. For any cardinal κof the form iα+1, the entire iteration Pcan be factored as Pκ∗Pκ, where Pκ has a dense subset of size κand Pκis κ+-closed. In particular, strongly inaccessible cardinals remain strongly inaccessible after forcing with P. Now suppose that κis superstrong, witnessed by the embedding j:V→ M, and that Gis P-generic. Let P∗denote M’s version of P. To show that κis superstrong in V[G], it suffices to find a P∗-generic G∗which contains j[G], the pointwise image of Gunder j, as a subclass. Now P∗ αis the same as Pαfor α < j(κ), as jis a superstrong embedding. The first difference between P∗and Pis at j(κ): P∗ j(κ)is the direct limit of the Pα,α < j(κ), as j(κ) is inacessible in M; but j(κ) is not necessarily regular in Vand therefore it is possible that Pj(κ)is the inverse limit of the Pα,α < j(κ). So we cannot simply choose G∗ j(κ)to be Gj(κ), as the latter is generic for the wrong forcing. But this problem is easily fixed: As j(κ) is in fact Mahlo in M, it follows that P∗ j(κ)has the j(κ)-cc in M. So any G∗ j(κ)contained in P∗ j(κ)whose intersection with each Pα,α < j(κ), is Pα-generic must also be P∗ j(κ)- generic. It follows that we can take G∗ j(κ)to simply be the intersection of Gj(κ)with P∗ j(κ). Notice that G∗ j(κ)trivially contains the pointwise image of Gκunder jas jis the identity below κ. Finally we must define a generic G∗, j(κ)for the “upper part” P∗, j(κ)of the P∗iteration, which starts at j(κ) and is defined in the ground model M[G∗ j(κ)]. In addition, G∗, j(κ)must contain the pointwise image of Gκ LARGE CARDINALS AND L-LIKE UNIVERSES 5 under j∗, where j∗is the lifting of jto V[Gκ] and Gκis generic for Pκ, an iteration starting at κdefined over the ground model V[Gκ]. In fact this latter requirement completely determines G∗, j(κ): Lemma 3. j∗[Gκ] generates aP∗, j(κ)-generic over M[G∗ j(κ)], i.e., each predense subclass of P∗, j(κ)which is definable over M[G∗ j(κ)]has an element which is extended by a condition in j∗[Gκ]. Proof. We only consider predense subsets of P∗, j(κ)in M[G∗ j(κ)]; a similar argument works for predense subclasses. We can assume that j:V→Mis given as an extender ultrapower embedding. This means that each element of Mis of the form j(f)(a), where abelongs to VM j(κ)=Vj(κ)and fis a function (in V) with domain Vκ. In particular, Dis of the form σG∗ j(κ)where the name σcan be written as j(f)(a) with fand aas above. Now using the κ+-closure of Pκ, choose a condition pin Gκwhich extends an element of f(¯a) whenever ¯abelongs to Vκand f(¯a)Gκis predense on P∗, κ. Then j∗(p) obviously belongs to j∗[Gκ] and extends an element of j(f)(a)G∗ j(κ)=σG∗ j(κ)=D, as desired. ¤ (Lemma 3) This completes the construction of G∗and therefore the proof that P preserves superstrong cardinals. Now suppose that κis hyperstrong. Again we need to find a P∗-generic G∗containing j[G] as a subclass. The forcings Pj(κ)+1 =Pj(κ)∗Qj(κ)and P∗ j(κ)+1 agree as j(κ) is regular in Vand Mcontains Vj(κ)+1. Also, j∗[gκ], where j∗is the lifting of jto V[Gκ] and gκis the QGκ κ-generic chosen by Gat stage κof the iteration, is a set of conditions in QGj(κ) j(κ)which belongs to M[Gj(κ)] and has size 2κthere; therefore j∗[gκ] has a lower bound in QGj(κ) j(κ). By choosing our generic Gso that gj(κ)includes this lower bound, we can succeed in lifting jto V[Gκ+1]. We may assume that j:V→Mis given by a hyperextender; this means that each element of Mis of the form j(f)(a) where fis a function in Vwith domain Vκ+1 and ais an element of Vj(κ)+1. Then we can use the argument of Lemma 3 to generate the entire generic G∗containing j[G]. The case of n-superstrongs raises a new difficulty. We first treat the case n= 2. As in the superstrong case, Pand P∗may take different limits at j2(κ), as the latter may be singular in V. As in that case, we can obtain aP∗ j2(κ)-generic by intersecting Gj2(κ)with P∗ j2(κ). However we must also 6 SY D. FRIEDMAN ensure that G∗ j2(κ)contain j[Gj(κ)] as a subset. Write Pj(κ)as Pκ∗Pκ j(κ); it suffices to have that G∗ j2(κ)contains j∗[Gκ j(κ)] as a subset, where j∗is the lifting of jto V[Gκ] and Gκ j(κ)is Pκ j(κ)-generic over V[Gκ]. For the latter we need only know that j2(κ) has cofinality unequal to j(κ), for then j∗[Gκ j(κ)] is a set of conditions bounded in j2(κ) and therefore by 2-superstrength belongs to Mand has a lower bound. Lemma 4. Assume that jhas been chosen so that j2(κ)is as small as possible. Then j2(κ)has cofinality κ+. Proof. We may assume that the given 2-superstrong embedding j:V→M has the property that each element of Mis of the form j(f)(a) where fis a function in Vwith domain Vj(κ)and abelongs to Vj2(κ). Now let Hconsist of all elements of Mof the form j(f)(a) where fis a function in Vwith domain Vκand abelongs to Vj(κ). Then after transitive collapse, the inclusions Range f⊆H⊆Mgive rise to embeddings k:V→ Nand l:N→M. The first embedding kis a superstrong embedding with critical point κ(the “superstrong embedding derived from j”) and the second embedding lis an “external” superstrong embedding with critical point j(κ) (“external” in the sense that it is not definable in N). Also k(κ) = j(κ) and l(j(κ)) = j2(κ). Now we argue as follows. Let hbe generic over Nfor the collapse of 2j(κ) to j(κ)+(using conditions of size at most j(κ)). Then hadds a function g from (j(κ)+)Nonto {f∈N|f:j(κ)→j(κ)}such that g¹αbelongs to Nfor each α < (j(κ)+)N. Now for each α < (j(κ)+)Nlet καbe the least cardinal less than j2(κ) closed under all functions l(f) for fin g[α]. Then as j2(κ) is regular in M, each καis less than j2(κ) and there is an “external” superstrong embedding l∗with critical point j(κ) such that l∗(j(κ)) equals κ∗, the supremum of the κα’s. But then (l∗)−1◦j=j∗is a 2-superstrong embedding with critical point κsuch that (j∗)2(κ) = κ∗. By the minimality of j2(κ) it follows that κ∗=j2(κ), so j2(κ) has cofinality (j(κ)+)Nin N[h] and therefore cofinality (j(κ)+)Nin N. As Nis the ultrapower of Vvia an extender with critical point κ, it follows that (j(κ)+)Nhas cofinality κ+, as desired. ¤(Lemma 4) The previous lemma allows us to lift jto V[Gj(κ)]. Then the method of Lemma 3 can be used to generate the entire generic G∗. For the case n > 2 the argument is similar; the general version of Lemma 4 states that jn(κ) has cofinality (jn−2(κ))+when it is chosen minimally for an n-superstrong κ. LARGE CARDINALS AND L-LIKE UNIVERSES 7 Finally we consider ω-superstrength. Again we must choose G∗to be P∗-generic over Mand to contain the pointwise image of Gunder j. Let jω(κ) denote the supremum of the jn(κ), n∈ω. As in Lemma 3 it suffices to find G∗ jω(κ)which is P∗ jω(κ)-generic and contains j[Gjω(κ)] as a subset. Note that j[Gκ] = Gκis trivially contained in Gjω(κ)and j∗[Gκ jω(κ)] has a lower bound in Pκ jω(κ)(as defined in V[Gκ]); by choosing Gjω(κ)to contain this lower bound we may arrange that Gjω(κ)contain j[Gjω(κ)]. It remains to show: Lemma 5. Gjω(κ)∩P∗ jω(κ)is P∗ jω(κ)-generic over M. Proof. Suppose that D∈Mis dense on P∗ jω(κ)and write Das j(f)(a) where fhas domain Vjω(κ)and abelongs to Vjn+1(κ)for some n. (We may assume that every element of Mis of this form.) Choose pin Gjω(κ)such that preduces f(¯a) below jn(κ) whenever ¯abelongs to Vjn(κ)and f(¯a) is open dense on Pjω(κ), in the sense that if qextends pthen qcan be further extended into f(¯a) without changing qat or above jn(κ). Such a p exists using the jn(κ)+-closure of Pjn(κ) jω(κ)in V[Gjn(κ)]. Then j(p) belongs to j[Gjω(κ)] and reduces Dbelow jn+1(κ). As Gjn+1(κ)is Pjn+1(κ)-generic and P,P∗agree below jn+1(κ), it follows that Gjω(κ)∩P∗ jω(κ)intersects D, as desired. ¤(Lemma 5) This completes the proof of Theorem 2. ¤ Another important property of Lis the existence of a definable wellordering of the universe. Theorem 6. (Large cardinals and definable wellorderings) If κis superstrong then there is an outer model in which κis still superstrong and there is a definable wellordering of the universe. The same holds for hyperstrong, n-superstrong for finite nand ω-superstrong. Proof. By Theorem 2 we may assume the GCH. Let κhave one of the large cardinal properties mentioned in the theorem, as witnessed by the embedding j:V→M. Choose λto be a cardinal greater than jω(κ). By the method of L-coding (see Theorem 1), we can enlarge Vwithout adding subsets of λto be of the form L[A], Aa subset of λ+. By the argument of Lemma 3 the embedding jlifts to L[A] and therefore κretains its large cardinal properties. Now we introduce a definable wellordering. Perform a reverse Easton iteration of length λ+, indexed by successor cardinals greater than λ+, where at the i-th successor cardinal, an i+-Cohen set is added iff ibelongs to A. 8 SY D. FRIEDMAN The result is that ibelongs to Aiff not every subset of the successor of the i-th successor cardinal is constructible from a subset of the i-th successor cardinal. Now the result of this iteration is a model of the form L[B] where Bis a subset of λ(λ+). Repeat this to code Busing the next interval of successor cardinals. Continuing this indefinitely yields a model with a wellordering definable from the parameter λ. To eliminate the parameter λ, use a pairing function f: Ord×Ord →Ord on the ordinals and arrange that the universe is of the form L[C] where C is a class of ordinals and for any i,iis in Ciff some subset of the successor to the f(i, j)-th successor cardinal is not constructible from a subset of the f(i, j)-th successor cardinal, for all sufficiently large j.¤ Jensen’s (global) ¤principle asserts the existence of a sequence hCα|α singularisuch that Cαhas ordertype less than αfor each αand C¯α=Cα∩¯α whenever ¯α∈Lim Cα. The following strengthens a result of Doug Burke [2]: Theorem 7. (Superstrong cardinals and ¤) If κis superstrong then there is an outer model in which κis still superstrong and ¤holds. Proof. By Theorem 2 we may assume the GCH. Consider now the reverse Easton iteration Pwhere at the regular stage α,Qαis a Pα-name for the forcing which adds a ¤-sequence on the singular limit ordinals less than α. A condition in Qαis a sequence hCβ|β≤γ,βsingulari,γ < α, such that Cβhas ordertype less than βfor each βand C¯ β=Cβ∩¯ βwhenever ¯ β belongs to Lim Cβ. Using the fact that Pαforces ¤-sequences of any regular length less than α, it is easy to verify by induction that any condition in Qαcan be extended to have arbitrarily large length less than α. Also Qα, and indeed the entire iteration from stage αon, is α-distributive. Let P∗denote M’s version of P. We want to construct G∗to be P∗- generic over Mand to contain j[G] as a subclass. As in earlier arguments, P and P∗agree strictly below j(κ) but not necessarily at j(κ), which is regular in Mbut may be singular in V; as before we take G∗ j(κ)to be Gj(κ)∩P∗ j(κ). Our new task is to define a Qj(κ)-generic gover M[G∗ j(κ)]. Assume that j:V→Mis chosen with j(κ) minimal. Then by the argument of Lemma 4, j(κ) has cofinality κ+. We can assume that jis given by an ultrapower, and therefore that (j(κ)+)Malso has cofinality κ+. Now we can build gin κ+steps, using j(κ)-distributivity to meet fewer than LARGE CARDINALS AND L-LIKE UNIVERSES 9 j(κ) open dense sets at each step. We must also ensure that gextend gκ; but this is easy to arrange as the latter is a condition in the forcing Qj(κ). Finally the rest of G∗can be generated from j[G] as in Lemma 3. ¤ The proof of the previous theorem does not work for hyperstrong κ, and there is a good reason for this. κis subcompact iff for any B⊆Hκ+there are µ < κ,A⊆Hµ+and an elementary embedding j: (Hµ+, A)→(Hκ+, B) with critical point µ. (Note that by elementarity, jmust send µto κ.) Proposition 8. (a) If κis hyperstrong then κis subcompact. (b) (Jensen) If there is a subcompact cardinal then ¤(even when restricted to ordinals between κand κ+) fails. Proof. (a) Suppose that j:V→Mwitnesses hyperstrength. Then for all subsets Bof j(κ)+in the range of j,jgives an elementary embedding of (Hκ+, A) into (Hj(κ)+, B), where j(A) = B; moreover this embedding belongs to Mas jis hyperstrong and j¹Hκ+belongs to Hj(κ)+. As the range of jis an elementary submodel of M, it follows that there is an elementary embedding of some (Hµ+, A) into (Hj(κ)+, B) (sending µto j(κ)) which belongs to the range of j. So j(κ) is subcompact in Range j and therefore by elementarity subcompact in M. As jis elementary, κis subcompact in V. (b) Suppose that κis subcompact and ~ C=hCα|κ<α<κ+,αsingulari has the properties of a ¤-sequence. By thinning out the Cα’s we can ensure that each has ordertype at most κ. Let jbe an embedding from (Hµ+,~ ¯ C) to (Hκ+,~ C), sending µto κ. Let αbe the supremum of the ordinals in the range of j. Then αhas cofinality µ+. The ordinals in the range of jform a < µ-closed and therefore ω-closed unbounded subset of α. And Lim Cαis a closed unbounded subset of α. Therefore the intersection Dof these two sets is unbounded in α. By the coherence property of ~ C, the ordertype of Cβfor sufficiently large βin Dis at least µ. But as the ordertype of Cαis at most κ(in fact less than κ), the ordertype of Cβfor all βin Dis strictly less than κ. Thus there are βin D⊆Range jwith Cβof ordertype not in Range j, contradicting the elementarity of j.¤ For uncountable, regular κ,♦κsays that there exists hDα|α < κi such that Dαis a subset of αfor each αand for every subset Dof κ, {α < κ |Dα=D∩α}is stationary in κ.♦asserts that ♦κholds for every uncountable, regular κ. Theorem 9. (Large cardinals and ♦) If κis superstrong then there is an outer model in which κis still superstrong and ♦holds. The same holds for hyperstrong, n-superstrong for finite nand ω-superstrong.