1 Introduction
In [Reference Foreman5–Reference Foreman8], 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 [Reference Foreman8].
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 [Reference Usuba12] 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 [Reference Foreman8, 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 [Reference Eskew3] 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 [Reference Cox and Eskew1].
In contrast to Theorem 1, Foreman [Reference Foreman4] 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 [Reference Shioya11].
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 [Reference Hamkins9]. 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 [Reference Usuba12]:
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)
$\kappa _0\leq \kappa _1$
and
$\lambda _0\leq \lambda _1$
are regular cardinals. -
(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)
$\kappa _1^+$
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 [Reference Foreman4]. 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 [Reference Magidor10] 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 [Reference Shioya11] 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)
$\mathbb {P}$
is
$\lambda $
-c.c. and contained in
$V_\lambda $
. -
(2)
$\mathbb {P}$
preserves
$\kappa $
and collapses
$\lambda $
to become
$\kappa ^+$
. -
(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:
-
(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 [Reference Cummings2, 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.
Funding
The author wishes to thank the Austrian Science Fund (FWF) for the generous support through grants P34603 and START Y1012-N35 (PI: Vera Fischer).
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