INCOMPATIBILITY OF GENERIC HUGENESS PRINCIPLES

Abstract

We show that the weakest versions of Foreman’s minimal generic hugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover, conventional forcing techniques cannot produce a model of one of these axioms.

1 Introduction

In [5–8], Foreman proposed generic large cardinals as new axioms for mathematics. These principles are similar to strong kinds of traditional large cardinal axioms but speak directly about small uncountable objects like $\omega _1,\omega _2$ , etc. Because of this, they are able to answer many classical questions that are not settled by ZFC plus traditional large cardinals. For example, if $\omega _1$ is minimally generically huge, then the Continuum Hypothesis holds and there is a Suslin line [8].

For a poset $\mathbb {P}$ , let us say that a cardinal $\kappa $ is $\mathbb {P}$ -generically huge if $\mathbb {P}$ forces that there is an elementary embedding $j : V \to M \subseteq V[G]$ with critical point $\kappa $ , where M is a transitive class closed under $j(\kappa )$ -sequences from $V[G]$ . If $\mathbb {P}$ forces that $j(\kappa ) = \lambda $ , we call $\lambda $ the target. We say that $\kappa $ is $\mathbb {P}$ -generically n-huge when the requirement on M is strengthened to closure under $j^n(\kappa )$ -sequences (where $j^n$ is the composition of j with itself n times), and we say $\kappa $ is $\mathbb {P}$ -generically almost-huge if the requirement is weakened to closure under ${<}j(\kappa )$ -sequences. We say that a cardinal $\kappa $ is $\mathbb {P}$ -generically measurable if $\mathbb {P}$ forces that there is an elementary embedding $j : V \to M \subseteq V[G]$ with critical point $\kappa $ , where M is transitive.

If $\kappa $ is the successor of an infinite cardinal $\mu $ , we say that $\kappa $ is minimally generically n-huge if it is $\operatorname {\mathrm {Col}}(\mu ,\kappa )$ -generically n-huge, where $\operatorname {\mathrm {Col}}(\mu ,\kappa )$ is the poset of functions from initial segments of $\mu $ into $\kappa $ ordered by end-extension. The main result of this note is that for a successor cardinal $\kappa $ , it is inconsistent for both $\kappa $ and $\kappa ^+$ to be minimally generically huge.

Theorem 1. Suppose $0<m\leq n$ and $\kappa $ is a regular cardinal that is $\mathbb {P}$ -generically n-huge with target $\lambda $ , where $\mathbb {P}$ is nontrivial and strongly $\lambda $ -c.c. Then $\kappa ^{+m}$ is not $\mathbb {Q}$ -generically measurable for any $\kappa $ -closed $\mathbb {Q}$ .

Here, “nontrivial” means that forcing with $\mathbb {P}$ necessarily adds a new set. Usuba [12] introduced the strong $\lambda $ -chain condition (strong $\lambda $ -c.c.), which means that $\mathbb {P}$ has no antichain of size $\lambda $ and forcing with $\mathbb {P}$ does not add branches to $\lambda $ -Suslin trees. As Usuba observed, $\mathbb {P}$ having the strong $\lambda $ -c.c. is implied by $\mathbb {P}$ having the $\mu $ -c.c. for $\mu <\lambda $ and by $\mathbb {P} \times \mathbb {P}$ having the $\lambda $ -c.c. In particular, if $\theta = \kappa ^{<\mu }$ , then $\operatorname {\mathrm {Col}}(\mu ,\kappa )$ collapses $\theta $ to $\mu $ and is strongly $\theta ^+$ -c.c. Let us also remark that in Theorem 1, $\kappa $ -closure can be weakened to $\kappa $ -strategic-closure without change to the arguments.

Regarding the history: Woodin proved, in unpublished work mentioned in [8, p. 1126], that it is inconsistent for $\omega _1$ to be minimally generically 3-huge while $\omega _3$ is minimally generically 1-huge. Subsequently, the author [3] improved this to show the inconsistency of a successor cardinal $\kappa $ being minimally generically n-huge while $\kappa ^{+m}$ is minimally generically almost-huge, where $0 < m < n$ . The weakening of the hypothesis to $\kappa $ being only generically 1-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 $n>0$ , $\omega _n$ is $\mathbb {P}$ -generically almost-huge with target $\omega _{n+1}$ for some $\omega _{n-1}$ -closed, strongly $\omega _{n+1}$ -c.c. poset $\mathbb {P}$ . A simplified construction was given by Shioya [11].

We prove Theorem 1 in Section 2 via a generalization that is less elegant to state. In Section 3, we discuss what is known about the consistency of generic hugeness by itself and present a corollary of Theorem 1 showing that the usual forcing strategies cannot produce models where $\omega _1$ is generically huge with target $\omega _2$ by a strongly $\omega _2$ -c.c. poset. Our notations and terminology are standard. We assume the reader is familiar with the basics of forcing and elementary embeddings.

2 Generic huge embeddings and approximation

The relevance of the strong $\kappa $ -c.c. is its connection to the approximation property of Hamkins [9]. Suppose $\mathcal {F} \subseteq \mathcal {P}(\lambda )$ . We say that a set $X \subseteq \lambda $ is approximated by $\mathcal {F}$ when $X \cap z \in \mathcal {F}$ for all $z \in \mathcal {F}$ . If $V \subseteq W$ are models of set theory, then we say that the pair $(V,W)$ satisfies the $\kappa $ -approximation property for a V-cardinal $\kappa $ when for all $\lambda \in V$ and all $X \subseteq \lambda $ in W, if X is approximated by $\mathcal {P}_\kappa (\lambda )^V$ , then $X \in V$ . We say that a forcing $\mathbb {P}$ has the $\kappa $ -approximation property when the $\kappa $ -approximation property is forced to hold of the pair $(V,V[G])$ . The following result appears as Lemma 1.5 and Note 1.11 in [12]:

Theorem 2 (Usuba).

If $\mathbb {P}$ is a nontrivial $\kappa $ -c.c. forcing and $\dot {\mathbb {Q}}$ is a $\mathbb {P}$ -name for a $\kappa $ -closed forcing, then $\mathbb {P} * \dot {\mathbb {Q}}$ has the $\kappa $ -approximation property if and only if $\mathbb {P}$ has the strong $\kappa $ -c.c.

Theorem 1 will follow from the more general lemma below.

Lemma 3. The following hypotheses are jointly inconsistent:

1. (1) $\kappa _0\leq \kappa _1$ and $\lambda _0\leq \lambda _1$ are regular cardinals.

2. (2) $\mathbb {P}$ is a nontrivial strongly $\lambda _0$ -c.c. poset that forces an elementary embedding $j : V \to M \subseteq V[G]$ with $j(\kappa _0) = \lambda _0$ , $j(\kappa _1) = \lambda _1$ , $\mathcal {P}(\lambda _1)^V \subseteq M$ , and $M^{<\lambda _0} \cap V[G] \subseteq M$ .

3. (3) $\kappa _1^+$ is $\mathbb {Q}$ -generically measurable for a $\kappa _0$ -closed $\mathbb {Q}$ .

Proof We will need a first-order version of (3) that can be carried through the embedding of (2). Replace it by the (possibly weaker) hypothesis that $\mathbb {Q}$ is a $\kappa _0$ -closed poset and for some $\theta \gg \lambda _1$ , $\mathbb {Q}$ forces an elementary embedding $j : H_\theta ^V \to N$ with critical point $\kappa _1^+$ , where $N \in V^{\mathbb {Q}}$ is a transitive set.

Claim 4. $\kappa _1^{<\kappa _0} = \kappa _1$ .

Proof Let $G \subseteq \mathbb {Q}$ be generic over V, and let $j : H_\theta ^V \to N$ be an elementary embedding with critical point $\kappa _1^+$ , where $N \in V[G]$ is a transitive set. By ${<}\kappa _0$ -distributivity, $\mathcal {P}_{\kappa _0}(\kappa _1)^{N} \subseteq \mathcal {P}_{\kappa _0}(\kappa _1)^{V}$ , so the cardinality of $\mathcal {P}_{\kappa _0}(\kappa _1)^V$ must be below the critical point of j.

Claim 5. $\lambda _1^{<\lambda _0} = \lambda _1$ .

Proof Let $G \subseteq \mathbb {P}$ be generic over V, and let $j : V \to M$ be as hypothesized in (2). By the closure of M, $\mathcal {P}_{\lambda _0}(\lambda _1)^M = \mathcal {P}_{\lambda _0}(\lambda _1)^{V[G]}$ . By elementarity and Claim 4, $M \models \lambda _1^{<\lambda _0} = \lambda _1$ . Thus M has a surjection $f : \lambda _1 \to \mathcal {P}_{\lambda _0}(\lambda _1)^{V[G]} \supseteq \mathcal {P}_{\lambda _0}(\lambda _1)^V$ . If $\lambda _1^{<\lambda _0}> \lambda _1$ in V, then f would witnesses a collapse of $\lambda _1^+$ , contrary to the $\lambda _0$ -c.c.

Now let $\mathcal {F} = \mathcal {P}_{\lambda _0}(\lambda _1)^V$ . Let $j : V \to M \subseteq V[G]$ be as in hypothesis (2). Claim 5 implies that $\mathcal {F}$ is coded by a single subset of $\lambda _1$ in V, so $\mathcal {F} \in M$ . In M, let $\mathcal {A}$ be the collection of subsets of $\lambda _1$ that are approximated by $\mathcal {F}$ . Since $\mathcal {P}(\lambda _1)^V \subseteq M$ , it is clear that $\mathcal {P}(\lambda _1)^V \subseteq \mathcal {A}$ .

For each $\alpha <\lambda _1^+$ , there exists an $X \in \mathcal {A} \cap V$ that codes a surjection from $\lambda _1$ to $\alpha $ in some canonical way. Working in M, choose for each $\alpha <\lambda _1^+$ an $X_\alpha \in \mathcal {A}$ that codes a surjection from $\lambda _1$ to $\alpha $ .

By elementarity, $\lambda _1^+$ is $j(\mathbb {Q})$ -generically measurable in M, witnessed by generic embeddings with domain $H^M_{j(\theta )}$ . By the closure of M, $j(\mathbb {Q})$ is $\lambda _0$ -closed in $V[G]$ . Let $H \subseteq j(\mathbb {Q})$ be generic over $V[G]$ . Let $i : H^M_{j(\theta )} \to N \in M[H]\subseteq V[G][H]$ be given by the $j(\mathbb {Q})$ -generic measurability of $\lambda _1^+$ in M, with $\operatorname {\mathrm {crit}}(i) = \delta = \lambda _1^+$ .

Let $\langle X^{\prime }_\alpha : \alpha < i(\delta ) \rangle = i(\langle X_\alpha : \alpha < \delta \rangle )$ . By elementarity, $X^{\prime }_\delta $ is approximated by $i(\mathcal {F}) = \mathcal {F}$ . Since $\mathbb {P} * j(\dot {\mathbb {Q}})$ is a nontrivial strongly $\lambda _0$ -c.c. forcing followed by a $\lambda _0$ -closed forcing, it has the $\lambda _0$ -approximation property by Usuba’s theorem. Therefore, $X^{\prime }_\delta \in V$ . But this is a contradiction, since $X^{\prime }_\delta $ codes a surjection from $\lambda _1$ to $(\lambda _1^+)^V$ .

Let us now complete the proof of Theorem 1. Suppose $n\geq 1$ , $\kappa <\lambda $ , $\mathbb {P}$ is strongly $\lambda $ -c.c., and $\mathbb {P}$ forces an embedding $j : V \to M \subseteq V[G]$ such that $j(\kappa ) = \lambda $ and M is closed under $j^n(\kappa )$ -sequences from $V[G]$ . By the $\lambda $ -c.c. of $\mathbb {P}$ and the $\lambda $ -closure of M, $(\lambda ^+)^M = (\lambda ^+)^V$ . Suppose inductively that $i<n$ and $(\lambda ^{+i})^M = (\lambda ^{+i})^V \leq j^{i+1}(\kappa )$ . Again, by the chain condition and the $j^{i+1}(\kappa )$ -closure of M, $(\lambda ^{+i+1})^M = (\lambda ^{+i+1})^V$ . Since $\kappa ^{+i}<\lambda ^{+i} = j(\kappa ^{+i})$ , $j(\lambda ^{+i})$ must be an M-cardinal greater than $\lambda ^{+i}$ , so $\lambda ^{+i+1} \leq j(\lambda ^{+i})$ . By elementarity applied to the induction hypothesis, $j(\lambda ^{+i}) \leq j^{i+2}(\kappa )$ . Thus the induction hypothesis carries through up to n. Now suppose $0<m\leq n$ and set $\kappa _0=\kappa $ , $\lambda _0 = \lambda $ , $\kappa _1 = \kappa ^{+m-1}$ , and $\lambda _1 = \lambda _0^{+m-1}$ . Then we have $j(\kappa _0)=\lambda _0$ and $j(\kappa _1) = \lambda _1 \leq j^n(\kappa )$ . If $\kappa ^{+m}$ is also generically measurable by a $\kappa $ -closed forcing, then this assignment of variables satisfies the hypotheses of the lemma, which we have shown to be inconsistent.

Remark 6. Suppose $\omega _1$ is $\mathbb {P}$ -generically almost-huge and $\omega _2$ is $\mathbb {Q}$ -generically measurable, where $\mathbb {P}$ is strongly $\omega _2$ -c.c. and $\mathbb {Q}$ is countably closed. This holds, for example, in Foreman’s model [4]. Let $j : V \to M$ be an embedding witnessing the $\mathbb {P}$ -generic almost-hugeness of $\omega _1$ . Put $\kappa _0=\kappa _1=\omega _1$ and $\lambda _0=\lambda _1=\omega _2$ . The only hypothesis of Lemma 3 that fails is $\mathcal {P}(\omega _2)^V \subseteq M$ .

3 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 $\omega _1$ can be $\mathbb {P}$ -generically huge with target $\omega _2$ for an $\omega _2$ -c.c. forcing $\mathbb {P}$ . But we do not think that Theorem 1 is evidence that this hypothesis by itself is inconsistent, since there are other versions of generic hugeness for $\omega _1$ that satisfy the hypothesis of Theorem 1 and are known to be consistent relative to huge cardinals. Magidor [10] showed that if there is a huge cardinal, then in a generic extension, $\omega _1$ is $\mathbb {P}$ -generically huge with target $\omega _3$ , where $\mathbb {P}$ is strongly $\omega _3$ -c.c. Shioya [11] observed that if $\kappa $ is huge with target $\lambda $ , then Magidor’s result can be obtained from a two-step iteration of Easton collapses, $\mathbb {E}(\omega ,\kappa ) * \dot {\mathbb {E}}(\kappa ^+,\lambda )$ . An easier argument shows that after the first step of the iteration, or even in the extension by the Levy collapse $\operatorname {\mathrm {Col}}(\omega ,{<}\kappa )$ , $\omega _1$ is $\mathbb {P}$ -generically huge with target $\lambda $ by a strongly $\lambda $ -c.c. forcing $\mathbb {P}$ .

Theorem 1 shows that in these models, $\omega _2$ is not $\mathbb {Q}$ -generically measurable for a countably closed $\mathbb {Q}$ . It also shows that if it is consistent for $\omega _1$ to be generically huge with target $\omega _2$ by a strongly $\omega _2$ -c.c. forcing, then this cannot be demonstrated by a standard method resembling Magidor’s:

Corollary 7. Suppose $\kappa $ is a huge cardinal with target $\lambda $ . Suppose $\mathbb {P}$ is such that $:$

1. (1) $\mathbb {P}$ is $\lambda $ -c.c. and contained in $V_\lambda $ .

2. (2) $\mathbb {P}$ preserves $\kappa $ and collapses $\lambda $ to become $\kappa ^+$ .

3. (3) For all sufficiently large $\alpha <\lambda $ (for example, all Mahlo $\alpha $ beyond a certain point), $\mathbb {P} \cong (\mathbb {P} \cap V_\alpha ) * \dot {\mathbb {Q}}_\alpha $ , where $\dot {\mathbb {Q}}_\alpha $ is forced to be $\kappa $ -closed.

Then in any generic extension by $\mathbb {P}$ , $\kappa $ is not generically huge with target $\lambda $ by a strongly $\lambda $ -c.c. forcing.

Furthermore, suppose $\lambda $ is supercompact in V, and (3) is strengthened to:

1. (4) For all sufficiently large $\alpha <\beta <\lambda $ , $\mathbb {P} \cong (\mathbb {P} \cap V_\alpha ) * \dot {\mathrm {Col}}(\kappa ,\beta )* \dot {\mathbb {Q}}_{\alpha ,\beta }$ , where $\dot {\mathbb {Q}}_ {\alpha ,\beta }$ is forced to be $\kappa $ -closed.

Then $\kappa $ is not generically huge with target $\lambda $ by a strongly $\lambda $ -c.c. forcing in any $\lambda $ -directed-closed forcing extension of $V^{\mathbb {P}}$ .

Proof Let $j : V \to M$ witness that $\kappa $ is huge with target $\lambda $ . By elementarity and the fact that $\mathcal {P}(\lambda ) \subseteq M$ , $\lambda $ is measurable in V. Let $\mathcal {U}$ be a normal ultrafilter on $\lambda $ , and let $i : V \to N$ be the ultrapower embedding.

Since the decomposition of (3) holds for all “sufficiently large” $\alpha $ , $N \models i(\mathbb {P}) \cong \mathbb {P} * \dot {\mathbb {Q}}$ , where $\dot {\mathbb {Q}}$ is forced to be $\kappa $ -closed. By the closure of N, V also believes that $\dot {\mathbb {Q}}$ is forced by $\mathbb {P}$ to be $\kappa $ -closed. Thus if we take $G \subseteq \mathbb {P}$ generic over V, then the embedding i can be lifted by forcing with $\mathbb {Q}$ . This means that in $V[G]$ , $\lambda $ is $\mathbb {Q}$ -generically measurable, $\mathbb {Q}$ is $\kappa $ -closed, and $\lambda = \kappa ^+$ . Theorem 1 implies that in $V[G]$ , $\kappa $ cannot be generically huge with target $\lambda $ by a strongly $\lambda $ -c.c. forcing.

For the final claim, suppose $\lambda $ is supercompact in V, and let $\dot {\mathbb {R}}$ be a $\mathbb {P}$ -name for a $\lambda $ -directed-closed forcing. Let $\gamma $ be such that $\Vdash _{\mathbb {P}} |\dot {\mathbb {R}}| \leq \gamma $ . By [2, Theorem 14.1], $\operatorname {\mathrm {Col}}(\kappa ,\gamma ) \cong \operatorname {\mathrm {Col}}(\kappa ,\gamma ) \times \mathbb {R}$ in $V^{\mathbb {P}}$ . Let $i : V \to N$ be an elementary embedding such that $\operatorname {\mathrm {crit}}(i) = \lambda $ , $i(\lambda )> \gamma $ , and $N^\gamma \subseteq N$ . By applying (4) in N, there is in N a complete embedding of $\mathbb {P} * \dot {\mathbb {R}}$ into $i(\mathbb {P})$ , such that the quotient forcing is equivalent to something of the form $\operatorname {\mathrm {Col}}(\kappa ,\gamma )*\dot {\mathbb {Q}}_{\lambda ,\gamma }$ , where $\dot {\mathbb {Q}}_{\lambda ,\gamma }$ is forced to be $\kappa $ -closed in $N^{\mathbb {P}*\dot {\mathbb {R}} * \dot {\mathrm {Col}}(\kappa ,\gamma )}$ . By the closure of N, the quotient is forced to be $\kappa $ -closed in $V^{\mathbb {P}*\dot {\mathbb {R}}}$ .

Let $G * H \subseteq \mathbb {P}*\dot {\mathbb {R}}$ be generic. Further $\kappa $ -closed forcing yields a generic $G' \subseteq i(\mathbb {P})$ that projects to $G*H$ . We can lift the embedding to $i : V[G] \to N[G']$ . By elementarity, $i(\mathbb {R})$ is $i(\lambda )$ -directed-closed in $N[G']$ . Thus $i[H]$ has a lower bound $r \in i(\mathbb {R})$ . By the closure of N, $i(\mathbb {R})$ is at least $\kappa $ -closed in $V[G']$ . Forcing below r yields a generic $H' \subseteq i(\mathbb {R})$ and a lifted embedding $i : V[G*H] \to N[G'*H']$ . Hence in $V[G*H]$ , $\lambda $ is generically measurable via a $\kappa $ -closed forcing. Theorem 1 implies that $\kappa $ cannot be generically huge with target $\lambda $ by a strongly $\lambda $ -c.c. forcing.