aa r X i v : . [ m a t h . L O ] M a r INCOMPATIBILITY OF GENERIC HUGENESS PRINCIPLES
MONROE ESKEW
Abstract.
We show that the weakest versions of Foreman’s minimal generichugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover,conventional forcing techniques cannot produce a model of one of these axioms. Introduction
In [5–8], Foreman proposed generic large cardinals as new axioms for mathemat-ics. These principles are similar to strong kinds of traditional large cardinal axiomsbut speak directly about small uncountable objects like ω , ω , etc. Because of this,they are able to answer many classical questions that are not settled by ZFC plustraditional large cardinals. For example, if ω is minimally generically huge, thenthe Continuum Hypothesis holds and there is a Suslin line [8].If κ is the successor of an infinite cardinal µ , then we say that κ is minimallygenerically huge if the poset h <µ κ, ⊇i , the functions from initial segments of µ into κ ordered by end-extension, forces that there is an elementary embedding j : V → M ⊆ V [ G ] with critical point κ , where M is a transitive class closed under j ( κ )-sequences from V [ G ]. We say that κ is minimally generically n -huge whenthe requirement on M is strengthened to closure under j n ( κ )-sequences (where j n is the composition of j with itself n times), and we say κ is minimally genericallyalmost-huge if the requirement is weakened to closure under Suppose κ is a regular cardinal that is P -generically huge to λ , where P is nontrivial and strongly λ -c.c. Then κ + is not Q -generically measurable for a κ -closed Q . Here, “nontrivial” means that forcing with P necessarily adds a new set. Usuba[12] introduced the strong λ -chain condition (strong λ -c.c.), which means that P has no antichain of size λ and forcing with P does not add branches to λ -Suslintrees. As Usuba observed, it is implied by the µ -c.c. for µ < λ and by P × P havingthe λ -c.c. We remark that κ -closure can be weakened to κ -strategic-closure withoutchange to the arguments. The author wishes to thank the Austrian Science Fund (FWF) for the generous support throughgrant number START Y1012-N35 (PI: Vera Fischer). Regarding the history: Woodin proved that it is inconsistent for ω to be min-imally generically 3-huge while ω is minimally generically 1-huge. Subsequently,the author [3] improved this to show the inconsistency of a successor cardinal κ be-ing minimally generically n -huge while κ + m is minimally generically almost-huge,where 0 < m < n . The weakening of the hypothesis to κ being only generically1-huge uses an idea from the author’s work with Cox [1].In contrast to Theorem 1, Foreman [4] exhibited a model where for all n > ω n is P -generically almost-huge to ω n +1 for some ω n − -closed, strongly ω n +1 -c.c.poset P . A simplified construction was given by Shioya [11].We prove Theorem 1 in § § ω is generically huge to ω by a strongly ω -c.c. poset. Ournotations and terminology are standard. We assume the reader is familiar with thebasics of forcing and elementary embeddings.2. Generic huge embeddings and approximation The relevance of the strong κ -c.c. is its connection to the approximation propertyof Hamkins [9]. Suppose F ⊆ P ( λ ). We say that a set X ⊆ λ is approximated by F when X ∩ z ∈ F for all z ∈ F . If V ⊆ W are models of set theory, then we say thatthe pair ( V, W ) satisfies the κ -approximation property for a V -cardinal κ when forall λ ∈ V and all X ⊆ λ in W , if X is approximated by P κ ( λ ) V , then X ∈ V . Wesay that a forcing P has the κ -approximation property when this property is forcedto hold of the pair ( V, V [ G ]). Theorem 2 (Usuba [12]) . If P is a nontrivial κ -c.c. forcing and ˙ Q is a P -namefor a κ -closed forcing, then P ∗ ˙ Q has the κ -approximation property if and only if P has the strong κ -c.c. Theorem 1 follows from the more general lemma below, by setting κ = κ = κ and λ = λ = λ . Lemma 3. The following hypotheses are jointly inconsistent:(1) κ , κ , λ , λ are regular cardinals.(2) P is a nontrivial strongly λ -c.c. poset that forces an elementary embedding j : V → M ⊆ V [ G ] with j ( κ ) = λ , j ( κ ) = λ , P ( λ ) V ⊆ M , and M <λ ∩ V [ G ] ⊆ M .(3) κ +1 is Q -generically measurable for a κ -closed Q .Proof. First note that (3) implies κ ≤ κ and thus by (2), λ ≤ λ . We will needa first-order version of (3). Replace it by the hypothesis that Q is a κ -closed posetand for some θ ≫ λ , Q forces an elementary embedding j : H Vθ → N with criticalpoint κ +1 , where N ∈ V Q is a transitive set. Claim 4. κ <κ = κ .Proof. Let G ⊆ Q be generic over V , and let j : H Vθ → N be an elementaryembedding with critical point κ +1 , where N ∈ V [ G ] is a transitive set. By <κ -distributivity, P κ ( κ ) N ⊆ P κ ( κ ) V , so the cardinality of P κ ( κ ) V must be belowthe critical point of j . (cid:3) Claim 5. λ <λ = λ . NCOMPATIBILITY OF GENERIC HUGENESS PRINCIPLES 3 Proof. Let G ⊆ P be generic over V , and let j : V → M be as hypothesized in(2). By the closure of M , P λ ( λ ) M = P λ ( λ ) V [ G ] . By elementarity and Claim 4, M | = λ <λ = λ . Thus M has a surjection f : λ → P λ ( λ ) V [ G ] ⊇ P λ ( λ ) V . If λ <λ > λ in V , then f would witnesses a collapse of λ +1 , contrary to the λ -c.c. (cid:3) Now let F = P λ ( λ ) V . Let j : V → M ⊆ V [ G ] be as in hypothesis (2). Claim5 implies that F is coded by a single subset of λ in V , so F ∈ M . In M , let A bethe collection of subsets of λ that are approximated by F .Since P ( λ ) V ⊆ M , for each α < λ +1 , there exists an X α ∈ A that codes asurjection from λ to α in some canonical way. Working in M , choose such an X α for each α < λ +1 .By elementarity, λ +1 is j ( Q )-generically measurable in M , witnessed by genericembeddings with domain H Mj ( θ ) . By the closure of M , j ( Q ) is λ -closed in V [ G ].Let H ⊆ j ( Q ) be generic over V [ G ]. Let i : H Mj ( θ ) → N ∈ V [ G ][ H ] be given by j ( Q )-generic measurability, with crit( i ) = δ = λ +1 .Let h X ′ α : α < i ( δ ) i = i ( h X α : α < δ i ). By elementarity, X ′ δ is approximatedby i ( F ) = F . Since P ∗ j ( ˙ Q ) is a nontrivial strongly λ -c.c. forcing followed bya λ -closed forcing, it has the λ -approximation property by Usuba’s Theorem.Therefore, X ′ δ ∈ V . But this is a contradiction, since X ′ δ codes a surjection from λ to ( λ +1 ) V . (cid:3) Remark 6. Suppose ω is P -generically almost-huge and ω is Q -generically mea-surable, where P is strongly ω -c.c. and Q is countably closed. Let j : V → M bean embedding witnessing the P -generic almost-hugeness of ω . Put κ = κ = ω and λ = λ = ω . The only hypothesis of Lemma 3 that fails is P ( ω ) V ⊆ M . On the consistency of generic hugeness It is not known whether any successor cardinal can be minimally generically huge.Moreover, it is not known whether ω can be P -generically huge to ω for an ω -c.c.forcing P . But we do not think that Theorem 1 is evidence that this hypothesisby itself is inconsistent, since there are other versions of generic hugeness for ω that satisfy the hypothesis of Theorem 1 and are known to be consistent relativeto huge cardinals. Magidor [10] showed that if there is a huge cardinal, then ina generic extension, ω is P -generically huge to ω , where P is strongly ω -c.c.Shioya [11] observed that if κ is huge with target λ , then Magidor’s result can beobtained from a two-step iteration of Easton collapses, E ( ω, κ ) ∗ ˙ E ( κ + , λ ). An easierargument shows that after the first step of the iteration, or even in the extensionby the Levy collapse Col( ω, <κ ), ω is P -generically huge to λ by a strongly λ -c.c.forcing P .Theorem 1 shows that in these models, ω is not Q -generically measurable fora countably closed Q . It also shows that if it is consistent for ω to be genericallyhuge to ω by a strongly ω -c.c. forcing, then this cannot be demonstrated by astandard method resembling Magidor’s: Corollary 7. Suppose κ is a huge cardinal with target λ . Suppose P is such that:(1) P is λ -c.c. and contained in V λ .(2) P preserves κ and collapses λ to become κ + .(3) For all sufficiently large α < λ (for example, all Mahlo α beyond a certainpoint), P ∼ = ( P ∩ V α ) ∗ ˙ Q α , where ˙ Q α is forced to be κ -closed. MONROE ESKEW Then in any generic extension by P , κ is not generically huge to λ by a strongly λ -c.c. forcing.Furthermore, suppose λ is supercompact in V , and (3) is strengthened to:(4) For all sufficiently large α < β < λ , P ∼ = ( P ∩ V α ) ∗ ˙Col( κ, β ) ∗ ˙ Q α , where ˙ Q α is forced to be κ -closed.Then κ is not generically huge to λ by a strongly λ -c.c. forcing in any λ -directed-closed forcing extension of V P .Proof. Let j : V → M witness that κ is huge with target λ . By elementarity andthe fact that P ( λ ) ⊆ M , λ is measurable in V . Let U be a normal ultrafilter on λ ,and let i : V → N be the ultrapower embedding.Since the decomposition of (3) holds for all “sufficiently large” α , N | = i ( P ) ∼ = P ∗ ˙ Q , where ˙ Q is forced to be κ -closed. By the closure of N , V also believes that˙ Q is forced by P to be κ -closed. Thus if we take G ⊆ P generic over V , thenthe embedding i can be lifted by forcing with Q . This means that in V [ G ], λ is Q -generically measurable, Q is κ -closed, and λ = κ + . Theorem 1 implies that in V [ G ], κ cannot be generically huge to λ by a strongly λ -c.c. forcing.For the final claim, suppose λ is supercompact in V , and let ˙ R be a P -name fora λ -directed-closed forcing. Let γ be such that (cid:13) P | ˙ R | ≤ γ . By [2, Theorem 14.1],Col( κ, γ ) ∼ = Col( κ, γ ) × R in V P . Let i : V → N be an elementary embedding suchthat crit( i ) = λ , i ( λ ) > γ , and N γ ⊆ N . There is a complete embedding of P ∗ ˙ R into i ( P ) that lies in N . By the closure of N , the quotient forcing is κ -closed in V P ∗ ˙ R .Let G ∗ H ⊆ P ∗ ˙ R be generic. Further κ -closed forcing yields a generic G ′ ⊆ i ( P )that projects to G ∗ H . We can lift the embedding to i : V [ G ] → N [ G ′ ]. Byelementarity, i ( R ) is i ( λ )-directed-closed in N [ G ′ ]. Thus i [ H ] has a lower bound r ∈ i ( R ). By the closure of N , i ( R ) is at least κ -closed in V [ G ′ ]. Forcing below r yields a generic H ′ ⊆ i ( R ) and a lifted embedding i : V [ G ∗ H ] → N [ G ′ ∗ H ′ ].Hence in V [ G ∗ H ], λ is generically measurable via a κ -closed forcing. Theorem 1implies that κ cannot be generically huge to λ by a strongly λ -c.c. forcing. (cid:3) References 1. Sean Cox and Monroe Eskew, Compactness versus hugeness at successor cardinals , arXive-prints (2020), arXiv:2009.14245.2. James Cummings, Iterated forcing and elementary embeddings , Handbook of set theory. Vols.1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883. MR 27686913. Monroe Eskew, Generic large cardinals as axioms , Rev. Symb. Log. (2020), no. 2, 375–387.MR 40922544. Matthew Foreman, More saturated ideals , Cabal seminar 79-81 (Berlin) (A. S. Kechris, D. A.Martin, and Y. N. Maschovakis, eds.), Lec. Notes in Math., vol. 1019, Springer-Verlag, 1983,pp. 1–27.5. , Potent axioms , Trans. Amer. Math. Soc. (1986), no. 1, 1–28.6. , Generic large cardinals: new axioms for mathematics? , Doc. Math. 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