1 Introduction
In [Reference Bagaria and Magidor1, Reference Bagaria and Magidor2], Bagaria and Magidor introduced the notion of
$\lambda $
-strong compactness (see Definition 2.1), which generalized the well-known notion of strong compactness.
$\lambda $
-strong compactness shares some similarities with strong compactness. For example,
$\lambda $
-strong compactness can be characterized in terms of compactness properties of infinitary languages, elementary embeddings, ultrafilters, etc. (see [Reference Bagaria and Magidor1, Reference Bagaria and Magidor2, Reference Usuba13]).
It turns out that the notion of
$\lambda $
-strong compactness, especially for the case
$\lambda =\omega _1$
, provides a weaker large cardinal strength, which can be used to prove various results known to follow from strong compactness. For example, the
$\mathrm {SCH}$
holds above the least
$\omega _1$
-strongly compact cardinal (see [Reference Bagaria and Magidor1]). Using this fact, recently Goldberg [Reference Goldberg8] proved a celebrated conjecture of WoodinFootnote
1
by showing that if the conjecture fails, then there exists an
$\omega _1$
-strongly compact cardinal, and the conjecture holds if the
$\mathrm {SCH}$
holds above some cardinal, and in particular, if there is an
$\omega _1$
-strongly compact cardinal. Besides these consequences,
$\lambda $
-strong compactness also corresponds to the exact large cardinal strength of some natural properties of interest in different areas (see [Reference Bagaria and Magidor1, Reference Bagaria and Magidor2]).
The least
$\lambda $
-strongly compact cardinal is of particular interest, because it may have very odd properties. Bagaria and Magidor [Reference Bagaria and Magidor1] showed that this cardinal must be a limit cardinal. But surprisingly, it may not be weakly inaccessible. Namely, they showed that the first
$\omega _1$
-strongly compact cardinal can be singular (see [Reference Bagaria and Magidor2]). Also recently Gitik [Reference Gitik7] constructed a model of
$\mathrm {ZFC}$
, relative to the existence of a supercompact cardinal, in which the least
$\lambda $
-strongly compact cardinal is not strongly compact, but stays regular. He also constructed a model of
$\mathrm {ZFC}$
, relative to the existence of two supercompact cardinals, in which the least
$\lambda $
-strongly compact cardinal is not a strong limit cardinal.
However, there are some limitations about the cofinality of the least
$\lambda $
-strongly compact cardinal. By a standard argument in [Reference Bagaria and Magidor1], one can see the cofinality must be greater than or equal to the least measurable cardinal.
Following the results of Bagaria–Magidor, we became very curious about the exact limitations of the cofinality of the least
$\lambda $
-strongly compact cardinal. This is how our work got started.
In this paper, Theorem 4.2 extends the consistency result of Bagaria–Magidor [Reference Bagaria and Magidor2, Theorem 6.1] to
$\lambda $
-measurable cardinals, and Proposition 4.7 shows this result is optimal. As a corollary (Corollary 4.6), we show that relative to the existence of two supercompact cardinals, for any regular cardinal
$\delta $
between them, it is consistent that
$\delta $
is the cofinality of the least
$\omega _1$
-strongly compact cardinal.
1.1 The structure of the paper
In Section 2 we cover some basic technical preliminaries about
$\lambda $
-strongly compact cardinals, Radin forcing and iterated ultrapowers. We give the main idea of the proof of our consistency result in Section 3. Finally, in Section 4 we prove the consistency result and show that it is optimal.
2 Preliminaries
We use V to denote the ground model in which we work. For any ordinals
$\alpha <\beta $
, we use
$[\alpha , \beta ], [\alpha , \beta ),(\alpha , \beta ]$
, and
$(\alpha , \beta )$
for the corresponding standard interval notation. Let
$\mathrm {id}_{M}$
denote the class identity function from M to M, and we will simply write
$\mathrm {id}$
when M is clear from the context. For a sequence u, let
$\mathrm {lh}(u)$
denote the length of u. For an elementary embedding
$j : V \rightarrow M$
with M transitive,
$\mathrm {crit}(j)$
denotes the critical point of j.
For every
$\gamma $
with Cantor normal form
$\gamma =\omega ^{\gamma _1}+\dots +\omega ^{\gamma _n}$
, where
$\gamma _1 \geq \dots \geq \gamma _n$
, we let
$\beta _{\gamma }:=1+\gamma _n$
. By induction, one may easily show that if
$\gamma $
is a successor ordinal, then
$\beta _{\gamma }=1$
; and if
$\gamma $
is a limit ordinal, then
$\beta _{\gamma }=\limsup _{\alpha <\gamma }(\beta _{\alpha }+1)$
.
For a cardinal
$\theta $
, a sequence
$\langle \mathcal {C}_{\alpha } \ | \ \alpha <\theta ^+ \rangle $
is a
$\square _{\theta ,\omega }$
-sequence if and only if whenever
$\alpha $
is a limit ordinal with
$\theta <\alpha <\theta ^+$
,
-
(1)
$1 \leq |\mathcal {C}_{\alpha }| \leq \omega $
, and -
(2) for all
$C \in \mathcal {C}_{\alpha }$
:-
(a) C is a club subset of
$\alpha $
. -
(b) C has order type at most
$\theta $
. -
(c) If
$\eta $
is a limit point of C, then
$C \cap \eta \in \mathcal {C}_{\eta }$
.
-
For every A with
$|A| \geq \kappa $
, let
$\mathcal {P}_{\kappa }(A)=\{x \subseteq A \ | \ |x|<\kappa \}$
. A set
$U \subseteq \mathcal {P}_{\kappa }(A)$
is a measure if it is a non-principal
$\kappa $
-complete ultrafilter on
$\mathcal {P}_{\kappa }(A)$
. A measure U on
$\mathcal {P}_{\kappa }(A)$
is fine if for every
$x \in \mathcal {P}_{\kappa }(A)$
,
$\{y \in \mathcal {P}_{\kappa }(A) \ | \ x\subseteq y \} \in U$
. A measure U on
$\mathcal {P}_{\kappa }(A)$
is normal if for any function
$f:\mathcal {P}_{\kappa }(A) \rightarrow A$
with
$\{x \in \mathcal {P}_{\kappa }(A) \ | \ f(x) \in x \} \in U$
, there is a set in U on which f is constant.
A cardinal
$\kappa $
is
$\alpha $
-supercompact if there exists an elementary embedding
$j:V \rightarrow M$
with M transitive such that
$\mathrm {crit}(j)=\kappa $
,
$j(\kappa )>\alpha $
and M is closed under sequences of length
$\alpha $
. A cardinal
$\kappa $
is supercompact if it is
$\alpha $
-supercompact for every
$\alpha $
. Equivalently,
$\kappa $
is supercompact if and only if for every
$\alpha \geq \kappa $
, there is a normal fine measure on
$\mathcal {P}_{\kappa }(\alpha )$
(see [Reference Kanamori10, Section 22]).
2.1
$\lambda $
-strongly compact cardinals
Definition 2.1 [Reference Bagaria and Magidor1, Reference Bagaria and Magidor2]
Suppose
$\delta \geq \lambda $
are uncountable cardinals.
-
(1) For every
$\alpha \geq \delta $
,
$\delta $
is
$\lambda $
-strongly compact up to
$\alpha $
if there exists a definable elementary embedding
$j:V \rightarrow M$
with M transitive, such that
$\mathrm {crit}(j) \geq \lambda $
and there exists a
$D \in M$
such that
$j''\alpha \subseteq D$
and
$M \models |D|<j(\delta )$
. -
(2)
$\delta $
is
$\lambda $
-strongly compact if
$\delta $
is
$\lambda $
-strongly compact up to
$\alpha $
for every
$\alpha \geq \delta $
.
It is easy to see that if
$\delta $
is
$\lambda $
-strongly compact, then it is
$\lambda '$
-strongly compact for every uncountable cardinal
$\lambda '<\lambda $
, and any cardinal greater than
$\delta $
is also
$\lambda $
-strongly compact.
We say that
$\delta $
is
$\lambda $
-measurable if and only if it is
$\lambda $
-strongly compact up to
$\delta $
.
Usuba gave a characterization of
$\lambda $
-strongly compact cardinals in terms of
$\lambda $
-complete uniform ultrafilters [Reference Usuba13, Theorem 1.2], which generalized a result of Ketonen. The following proposition is a simple local version of the characterization.
Proposition 2.2. Suppose
$\delta \geq \lambda $
are uncountable regular cardinals. Then
$\delta $
is
$\lambda $
-measurable if and only if
$\delta $
carries a
$\lambda $
-complete uniform ultrafilter, i.e., there is a
$\lambda $
-complete ultrafilter U over
$\delta $
such that every
$A \in U$
has cardinality
$\delta $
.
Proof If
$\delta $
is
$\lambda $
-measurable, then there exists a definable elementary embedding
$j:V \rightarrow M$
with M transitive, such that
$\mathrm {crit}(j) \geq \lambda $
and there exists a
$D \in M$
so that
$j''\delta \subseteq D$
and
$M \vDash |D|<j(\delta )$
. Since
$\delta $
is regular, we have
$j(\delta )$
is regular in M. Hence
$\sup (j''\delta )<j(\delta )$
. Now we may define a
$\lambda $
-complete uniform ultrafilter U over
$\delta $
by
$X \in U$
if and only if
$X \subseteq \delta $
and
$\sup (j''\delta ) \in j(X)$
.
Conversely, if
$\delta $
carries a
$\lambda $
-complete uniform ultrafilter, say U, then the canonical embedding
$j_U:V \rightarrow M_U \cong \mathrm {Ult}(V,U)$
satisfies
$\mathrm {crit}(j_U) \geq \lambda $
and
$\sup (j''\delta ) \leq [\mathrm {id}]_U<j(\delta )$
. Thus
$j_U$
witnesses that
$\delta $
is
$\lambda $
-measurable.
Theorem 2.3 [Reference Bagaria and Magidor1]
The least
$\lambda $
-strongly compact cardinal is a limit cardinal.
2.2 Radin forcing
We will generally follow [Reference Bagaria and Magidor2, Section 6.1] for the presentation of Radin forcing. For the sake of completeness, we also review its definition and some related basic properties, including the coherence of measure sequences (Lemma 2.5), the characterization of Radin generic objects via the geometric conditions (Theorem 2.9), and the construction of Radin generic objects via iterated ultrapowers (Theorem 2.12). For the readers’ convenience, we also give proofs for some of these properties. Readers who are familiar with Radin forcing may skip these details.
We first define measure sequences, which are the building blocks of the Radin forcing.
Definition 2.4. A non-empty sequence
$u=\langle u(\alpha ) \ | \ \alpha <\mathrm {lh}(u) \rangle $
is a measure sequence if there exists a definable elementary embedding
$j:V \rightarrow M$
with M transitive such that
$u(0)=\mathrm {crit}(j)$
, and for each
$\alpha $
with
$0<\alpha <\mathrm {lh}(u)$
,
$u \upharpoonright \alpha \in M$
and
$u(\alpha )=\{A \subseteq V_{u(0)} \ | \ u \upharpoonright \alpha \in j(A)\}$
.
For simplicity of notation, we write
$\kappa (u)$
for
$u(0)$
,
$\mathcal {F}(u)$
for
$\bigcap _{0<\alpha <\mathrm {lh}(u)}u(\alpha )$
if the length of u is greater than
$1$
, and
$\mathcal {F}(u)$
for
$\{\emptyset \}$
otherwise.
The following lemma is a mild modification of a lemma of Cummings and Woodin [Reference Gitik, Foreman and Kanamori6, Lemma 5.1], which shows that every measure sequence u with
$\mathrm {lh}(u)<\kappa (u)$
is coherent.
Lemma 2.5. Suppose
$u=\langle u(\alpha ) \ | \ \alpha <\mathrm {lh}(u) \rangle $
is a measure sequence with
$1<\mathrm {lh}(u)<\kappa (u)$
. For every
$\alpha $
with
$0< \alpha <\mathrm {lh}(u)$
, let
$j_{\alpha }:V \rightarrow N_{\alpha } \cong \mathrm {Ult}(V,u(\alpha ))$
be the canonical embedding. Then
$u \upharpoonright \alpha \in N_{\alpha }$
and the measure sequence of length
$\alpha +1$
given by
$j_{\alpha }$
is exactly
$u \upharpoonright (\alpha +1)$
.
Proof Let
$\kappa :=\kappa (u)$
. Since u is a measure sequence, we may find a definable elementary embedding
$j:V \rightarrow N$
with N transitive such that for every
$\alpha $
with
$0<\alpha <\mathrm {lh}(u)$
,
$u \upharpoonright \alpha \in N$
and
$u(\alpha )=\{A \subseteq V_{\kappa } \ | \ u \upharpoonright \alpha \in j(A)\}$
.
Fix any
$\alpha $
with
$0< \alpha <\mathrm {lh}(u)$
. For any
$x \in N_{\alpha }$
, we can find a function
$f:V_{\kappa } \rightarrow V$
representing it, and we denote x by
$[f]_{\alpha }$
. Now we will define an embedding
$k:N_{\alpha } \rightarrow N$
. Let
$k([f]_{\alpha })=j(f)(u \upharpoonright \alpha )$
for every
$[f]_{\alpha } \in N_{\alpha }$
. Then it is easy to see k is well-defined and elementary, and
$j=k \circ j_{\alpha }$
.
Claim 2.6.
$u \upharpoonright \alpha =[\mathrm {id}]_{\alpha } \in N_{\alpha }$
.
Proof By the definition of k,
$k([\mathrm {id}]_{\alpha })=j(\mathrm {id})(u \upharpoonright \alpha )=u \upharpoonright \alpha $
. So we only need to prove that
$k(u \upharpoonright \alpha )=u \upharpoonright \alpha $
.
We will first prove
$k(\kappa )=\kappa $
. Note that for every
$\beta <\kappa $
,
$k(\beta )=k(j_{\alpha }(\beta ))=j(\beta )=\beta $
, so
$\mathrm {crit}(k)\geq \kappa $
. Meanwhile,
$k([\mathrm {id}]_{\alpha }(0))=u(0)=\kappa $
since
$k([\mathrm {id}]_{\alpha })=u \upharpoonright \alpha $
with
$\alpha>0$
. But since
$k(\beta )=\beta <\kappa $
for every
$\beta <\kappa $
, it follows that
$[\mathrm {id}]_{\alpha }(0) \geq \kappa $
. Thus
$[\mathrm {id}]_{\alpha }(0) = \kappa $
. Consequently,
$k(\kappa )=\kappa $
and
$\mathrm {crit}(k)>\kappa $
.
Now let us prove
$k(u\upharpoonright \alpha )=u \upharpoonright \alpha $
. It is easy to see
$N_{\alpha } \cap V_{\kappa +1}=V_{\kappa +1}= N \cap V_{\kappa +1}$
and
$$ \begin{align} \forall X \in N_{\alpha} \cap V_{\kappa+1}(k(X)=X). \end{align} $$
Take any
$\eta <\alpha $
. Then
$u(\eta )=k''u(\eta ) \subseteq k(u(\eta ))$
by (1). Since
$N_{\alpha } \cap V_{\kappa +1}=V_{\kappa +1}= N \cap V_{\kappa +1}$
, and by the maximality of
$u(\eta )$
as a filter, we have
$k(u(\eta ))=u(\eta )$
. Note also that as k fixes the length of
$u \upharpoonright \alpha $
, we have
$k(u\upharpoonright \alpha )=u \upharpoonright \alpha $
.
Now we can prove that the measure sequence of length
$\alpha +1$
obtained from
$j_{\alpha }$
, say v, is exactly
$u \upharpoonright (\alpha +1)$
by induction on
$\beta $
with
$\beta \leq \alpha $
. Obviously,
$v(\beta )=\kappa =u(\beta ) \in N_{\alpha }$
if
$\beta =0$
. Now suppose inductively that
$v \upharpoonright \beta =u \upharpoonright \beta \in N_{\alpha }$
. For every
$X \in V_{\kappa +1}$
,
$$\begin{align*}X \in v(\beta) \Leftrightarrow u \upharpoonright \beta=v \upharpoonright \beta \in j_{\alpha}(X) \Leftrightarrow u \upharpoonright \beta & = k(u \upharpoonright \beta) \in k(j_{\alpha}(X))\\ &=j(X) \Leftrightarrow X \in u(\beta). \end{align*}$$
The first and last
$“\Leftrightarrow ”$
hold by definition, the first equality holds by induction, the second
$“\Leftrightarrow ”$
holds by elementarity of k, and the second equality was proved above. Hence
$v=u \upharpoonright (\alpha +1)$
.
We define next the class
$U_{\infty }$
of measure sequences as follows. Let
$U_0=\{ u \ | \ u$
is a measure sequence
$\}$
, and for every
$n<\omega $
, let
$U_{n+1}=\{ u \in U_n \ | \ U_n \cap V_{\kappa (u)}\in \mathcal {F}(u)\}$
. Finally, set
$U_{\infty }=\bigcap _{n<\omega }U_n$
. The point is that if
$u \in U_{\infty }$
, then for every
$\alpha $
with
$0<\alpha <\mathrm {lh}(u)$
,
$u(\alpha )$
concentrates on
$U_{\infty } \cap V_{\kappa (u)}$
.
The class
$U_{\infty }$
is non-empty if there exists a
$j:V \rightarrow M$
with
$\mathrm {crit}(j)=\kappa $
,
$V_{\kappa +2} \subseteq M$
, and M is closed under sequences of length
$\kappa $
, for then we can get a measure sequence u from j with
$\mathrm {lh}(u) \geq (2^{\kappa })^+$
and for every
$\alpha <(2^{\kappa })^+$
,
$u \upharpoonright \alpha \in U_{\infty }$
(see [Reference Cummings and Woodin5] or [Reference Gitik, Foreman and Kanamori6] for details). In particular, this also holds for every
$\alpha $
-supercompact embedding with
$\alpha \geq |V_{\kappa +2}|$
.
In the sequel, if we say u is a measure sequence, we mean that u is in
$U_{\infty }$
. Given a measure sequence u of length at least
$2$
, we may now define the Radin forcing
$R_u$
.
Definition 2.7.
$R_u$
consists of finite sequences
$p=\langle (u_0,A_0),\ldots , (u_n,A_n)\rangle $
, where:
-
(1) For every
$i \leq n$
,
$u_i \in U_{\infty }$
,
$A_i \in \mathcal {F}(u_i)$
, and
$A_i \subseteq U_{\infty }$
. -
(2) For every
$i<n$
,
$(u_i,A_i) \in V_{\kappa (u_{i+1})}$
. -
(3)
$u_n=u$
.
(We say
$u_0,\ldots ,u_n$
occur in p.)
The ordering on
$R_u$
is defined as follows. If
$p = \langle (u_0, A_0),\ldots ,(u_n, A_n)\rangle $
and
$q = \langle (v_0, B_0),\ldots ,(v_m, B_m)\rangle $
are in
$R_u$
, then
$p \leq q$
if and only if:
-
(1)
$\{v_0,\ldots ,v_m\}\subseteq \{u_0,\ldots ,u_n\}$
. -
(2) For each
$j \leq m$
and
$i \leq n$
, if
$v_j = u_i$
, then
$A_i \subseteq B_j$
. -
(3) If
$i \leq n$
is such that
$u_i \notin \{v_0,\ldots ,v_m\}$
and if
$j \leq m$
is the least such that
$u_i(0) < v_j (0)$
, then
$u_i \in B_j$
and
$A_i \subseteq B_j$
.
Given an
$R_u$
-generic filter G over V, let
$g_G:=\langle g_{\alpha } \ | \ \alpha <\mathrm {lh}(g_G) \rangle $
be the generic sequence given by G. Namely,
$g_G$
is the unique sequence consisting of all measure sequences w, such that
$w \neq u$
and w occurs in some
$p\in G$
; and if
$\alpha <\beta <\mathrm {lh}(g_G)$
, then
$\kappa (g_{\alpha })<\kappa (g_{\beta })$
. Also let
$C_G=\{ \kappa (g_{\alpha }) \ | \ \alpha <\mathrm {lh}(g_G) \}$
. Then
$C_G$
is a club subset of
$\kappa (u)$
. In addition, if
$\mathrm {lh}(u)<\kappa (u)$
, then there is a condition
$p \in R_u$
such that p forces that the order type of
$C_G$
is
$\omega ^{-1+\mathrm {lh}(u)}$
. (See [Reference Gitik, Foreman and Kanamori6] for details.)
It is not hard to see that G can be recovered from
$g_G$
, so we may view
$g_G$
as the generic object. Indeed, G consists of all
$p \in R_u$
such that:
-
(1) If v occurs in p and
$v \neq u$
, then
$v=g_{\alpha }$
for some
$\alpha <\mathrm {lh}(g_G)$
. -
(2) For every
$\alpha < \mathrm {lh}(g_G)$
,
$g_{\alpha }$
occurs in some
$q \leq p$
.
Definition 2.8. Suppose M is an inner model of
$\mathrm {ZFC}$
and
$\delta $
is a limit ordinal. Let
$w(\delta )$
be a measure sequence in M, and let
$w=\langle w(\alpha ) \ | \ \alpha <\delta \rangle $
be a sequence of measure sequences in M. Then w is geometric with respect to
$w(\delta )$
and M if and only if the following holds:
-
(1) The sequence
$\langle \kappa (w(\alpha )) \ | \ \alpha \leq \delta \rangle $
is increasing continuous. -
(2) For every limit
$\alpha \leq \delta $
and every
$A \in M \cap V_{\kappa (w(\alpha ))+1}$
,
$A \in \mathcal {F}(w(\alpha ))$
if and only if
$w \upharpoonright \alpha $
is eventually contained in A, i.e., there exists an
$\alpha _{A}<\alpha $
such that for every
$\gamma $
with
$\alpha _A<\gamma <\alpha $
,
$w(\gamma ) \in A$
.
The following theorem, due to W. Mitchell, characterizes Radin generic sequences in terms of the geometric condition. We follow the notation of Definition 2.8 in the statement of the next theorem.
Theorem 2.9 [Reference Mitchell11]
A sequence w is geometric with respect to
$w(\delta )$
and M if and only if w is a Radin generic sequence given by some
$R_{w(\delta )}$
-generic filter over M.
According to (2) of Definition 2.8, for the case
$\delta <\kappa (w(0))$
, if w is geometric w.r.t.
$w(\delta )$
and M, i.e., w is a Radin generic sequence given by some
$R_{w(\delta )}$
-generic filter over M, then
$\mathcal {F}(w(\alpha ))$
concentrates on measure sequences of length less than
$\mathrm {lh}(w(\alpha ))$
for every
$\alpha \leq \delta $
. Hence, it is easily seen that
$\mathrm {lh}(w(\alpha ))=1$
if
$\alpha <\delta $
is a successor ordinal, and
$\mathrm {lh}(w(\alpha ))=\limsup _{\gamma <\alpha }(\mathrm {lh}(w(\gamma ))+1)$
if
$\alpha <\delta $
is a limit ordinal. In other words,
$\mathrm {lh}(w(\alpha ))=\beta _{\alpha }$
for every
$\alpha \leq \delta $
, where
$\beta _{\alpha }$
is defined at the beginning of this section.
Now we may define u-iterated ultrapowers as follows.
Definition 2.10. Suppose
$\mathrm {lh}(u)<\kappa (u)$
, and
$\delta \leq \omega ^{-1+\mathrm {lh}(u)}$
is a limit ordinal.
-
(1)
$\langle M_{\alpha }, \pi _{\alpha ,\alpha '} \ | \ \alpha \leq \alpha ' \leq \delta \rangle $
is an iterated ultrapower if and only if:-
(a)
$M_0=V$
and
$\pi _{\alpha ,\alpha }=\mathrm {id}_{M_{\alpha }}$
for every
$\alpha \leq \delta $
. -
(b)
$M_{\alpha +1} \cong \mathrm {Ult}(M_{\alpha },W_{\alpha })$
is a transitive class, where
$W_{\alpha } \in M_{\alpha }$
is a
$\kappa _{\alpha }$
-complete ultrafilter over
$\kappa _{\alpha }$
(or
$M_{\alpha }\cap V_{\kappa _{\alpha }}$
) for some
$\kappa _{\alpha }$
, and the ultrapower is constructed in
$M_{\alpha }$
;
$\pi _{\alpha ,\alpha +1}:M_{\alpha } \rightarrow M_{\alpha +1} \cong \mathrm {Ult}(M_{\alpha },W_{\alpha })$
is the canonical embedding, and for every
$\gamma < \alpha $
,
$\pi _{\gamma ,\alpha +1} = \pi _{\alpha ,\alpha +1} \circ \pi _{\gamma ,\alpha }$
. -
(c) If
$\gamma \leq \delta $
is a limit ordinal, then
$M_{\gamma }$
is the direct limit of
$\langle M_{\alpha }, \pi _{\alpha ,\alpha '} \ | \ \alpha \leq \alpha ' < \gamma \rangle $
, and for every
$\alpha < \gamma $
,
$\pi _{\alpha ,\gamma }:M_{\alpha } \rightarrow M_{\gamma }$
is the corresponding embedding.
-
-
(2)
$\langle M_{\alpha }, \pi _{\alpha ,\alpha '} \ | \ \alpha \leq \alpha ' \leq \delta \rangle $
is a u-iterated ultrapower if and only if it is an iterated ultrapower, and in (1b),
$\kappa _{\alpha }=\pi _{0,\alpha }(\kappa (u))$
and
$W_{\alpha }=\pi _{0,\alpha }(u)(\beta _{\alpha })$
.
For simplicity of notation, we write
$\pi _{\alpha }$
for
$\pi _{0,\alpha }$
for every
$\alpha \leq \delta $
,
$\pi $
for
$\pi _{\delta }$
and M for
$M_{\delta }$
. Here we require that the length
$\delta $
of a u-iterated ultrapower is less than or equal to
$\omega ^{-1+\mathrm {lh}(u)}$
, because
$\beta _{\alpha }$
should be less than
$\mathrm {lh}(u)$
for every
$\alpha <\delta $
.
Next, following the notation of the definition above, let
$\langle M_{\alpha }, \pi _{\alpha ,\alpha '} \ | \ \alpha \leq \alpha ' \leq \delta \rangle $
be the u-iterated ultrapower of length
$\delta $
, let
$w=\langle \pi _{\alpha }(u) \upharpoonright \beta _{\alpha } \ | \ \alpha <\delta \rangle $
, and let
$w(\delta )=\pi _{\delta }(u) \upharpoonright \beta _{\delta }$
.
We will use u-iterated ultrapowers to construct a Radin generic sequence over some target model. We first prove the following lemma.
Lemma 2.11. Suppose
$\theta \leq \delta $
is a limit ordinal. If
$\eta <\theta $
satisfies that
$\beta _{\alpha }<\beta _{\theta }$
for every
$\alpha $
with
$\eta \leq \alpha <\theta $
, then for every
$\bar {A} \in \mathcal {F}(\pi _{\eta }(u) \upharpoonright \beta _{\theta }) $
, we have
$$ \begin{align} \{ w(\alpha) \ | \ \eta \leq \alpha <\theta \} \subseteq \pi_{\eta,\theta}(\bar{A}). \end{align} $$
In particular, for every limit
$\theta \leq \delta $
, if
$A \in \mathcal {F}(\pi _{\theta }(u) \upharpoonright \beta _{\theta })=\mathcal {F}(w(\theta ))$
, then
$$ \begin{align} w \upharpoonright \theta \text{ is eventually contained in } A. \end{align} $$
Proof For every
$\alpha $
with
$\eta \leq \alpha <\theta $
, since
$\bar {A} \in \mathcal {F}(\pi _{\eta }(u) \upharpoonright \beta _{\theta })$
and
$\pi _{\eta ,\alpha }$
is elementary, we have
$M_{\alpha } \models \pi _{\eta ,\alpha }(\bar {A})\in \mathcal {F}(\pi _{\alpha }(u) \upharpoonright \beta _{\theta })$
. By our assumption,
$\beta _{\alpha }<\beta _{\theta }$
, so
$M_{\alpha } \models \pi _{\eta ,\alpha }(\bar {A})\in \pi _{\alpha }(u)(\beta _{\alpha })$
. Meanwhile, by the definition of the u-iterated ultrapower
$\langle M_{\alpha '}, \pi _{\alpha ',\alpha ''} \ | \ \alpha ' \leq \alpha '' \leq \delta \rangle $
, we have
$\pi _{\alpha ,\alpha +1}:M_{\alpha } \rightarrow M_{\alpha +1} \cong \mathrm {Ult}(M_{\alpha },\pi _{\alpha }(u)(\beta _{\alpha }))$
. Note also that
$\pi _{\alpha }(u)$
is a measure sequence in
$M_{\alpha }$
, and the measure sequence of length
$\beta _{\alpha }+1$
obtained from
$\pi _{\alpha ,\alpha +1}$
is exactly
$\pi _{\alpha }(u)\upharpoonright (\beta _{\alpha }+1)$
by Lemma 2.5. Hence,
$$\begin{align*}M_{\alpha+1} \models w(\alpha)=\pi_{\alpha}(u)\upharpoonright \beta_{\alpha} \in \pi_{\alpha,\alpha+1}(\pi_{\eta,\alpha}(\bar{A}))=\pi_{\eta,\alpha+1}(\bar{A}). \end{align*}$$
Since
$\pi _{\alpha +1,\theta }$
is elementary, and
$w(\alpha )$
is fixed by
$\pi _{\alpha +1,\theta }$
, i.e.,
$\pi _{\alpha +1,\theta }(w(\alpha ))=w(\alpha )$
, we have
$M_{\theta } \models w(\alpha ) \in \pi _{\eta ,\theta }(\bar {A})$
. Hence
$w(\alpha ) \in \pi _{\eta ,\theta }(\bar {A})$
. So (2) holds.
Now take any limit
$\theta \leq \delta $
, and we prove that (3) holds. Since
$A \in \mathcal {F}(\pi _{\theta }(u) \upharpoonright \beta _{\theta })$
, we can pick a sufficiently large
$\bar {\theta }<\theta $
, so that
$\beta _{\alpha }<\beta _{\theta }$
for every
$\alpha $
with
$\bar {\theta } \leq \alpha <\theta $
, and there exists an
$\bar {A} \in M_{\bar {\theta }}$
such that
$\pi _{\bar {\theta },\theta }(\bar {A})=A$
. Then
$\bar {A} \in \mathcal {F}(\pi _{\bar {\theta }}(u) \upharpoonright \beta _{\theta })$
. Hence
$\{ w(\alpha ) \ | \ \bar {\theta } \leq \alpha <\theta \} \subseteq A$
, which means
$w \upharpoonright \alpha $
is eventually contained in A.
The point of the lemma above is that by Lemma 2.5, for every
$\alpha < \delta $
, the measure sequence of length
$\beta _{\alpha }+1$
obtained from
$\pi _{\alpha ,\alpha +1}$
is exactly
$\pi _{\alpha }(u)\upharpoonright (\beta _{\alpha }+1)$
. So for any iterated ultrapower
$\langle M_{\alpha }, \pi _{\alpha ,\alpha '} \ | \ \alpha \leq \alpha ' \leq \delta \rangle $
, if the measure sequence of length
$\beta _{\alpha }$
+ 1 obtained from
$\pi _{\alpha ,\alpha +1}$
is
$\pi _{\alpha }(u) \upharpoonright (\beta _{\alpha } + 1)$
for every
$\alpha <\delta $
, then the lemma above also holds. For example, we may let
$\pi _{\alpha ,\alpha +1}$
be the ultrapower map given by
$\pi _{\alpha }(u)(\theta )$
for every
$\alpha < \delta $
in (1b) if there is a
$\theta $
with
$\delta < \theta < \mathrm {lh}(u)$
. Then the lemma above also holds for this new iterated ultrapower.
The following theorem is essentially due to Radin [Reference Radin12] (see also [Reference Cummings and Woodin5, Theorem 6.7.1]).
Theorem 2.12 [Reference Radin12]
The sequence w is geometric with respect to
$w(\delta )$
and M(=
$M_{\delta }$
), and we have an
$R_{w(\delta )}$
-generic filter over M given by w.
Proof We only prove (2) of the geometric condition here, i.e., for every limit
$\alpha \leq \delta $
and every
$A \in M \cap V_{\kappa (w(\alpha ))+1}$
,
$A \in \mathcal {F}(w(\alpha ))$
if and only if
$w \upharpoonright \alpha \text { is eventually contained in } A$
.
If
$A \in \mathcal {F}(w(\alpha ))$
, then by Lemma 2.11,
$w \upharpoonright \alpha $
is eventually contained in A.
If
$A \notin \mathcal {F}(w(\alpha ))$
, then
$A \notin \pi _{\alpha }(u)(\gamma )$
for some
$\gamma <\beta _{\alpha }$
. Let
$B=M \cap V_{\kappa (w(\alpha ))} \setminus \{ x \in A \ | \ \mathrm {lh}(x)=\gamma \}$
. Then
$B \in \mathcal {F}(w(\alpha ))$
. Hence,
$w \upharpoonright \alpha $
is eventually contained in B. Note also that
$\langle \eta <\alpha \ | \ \beta _{\eta }=\gamma \rangle $
is unbounded in
$\alpha $
, and we have that
$w \upharpoonright \alpha $
is not eventually contained in A.
Therefore, w is geometric w.r.t.
$w(\delta )$
and M, and we have an
$R_{w(\delta )}$
-generic filter over M obtained from w.
3 Main idea of the consistency result (Theorem 4.2)
Suppose
$\kappa $
is a supercompact cardinal and
$\delta <\kappa $
is a
$\lambda $
-measurable cardinal. Let
$j:V \rightarrow M$
be a suitable supercompact ultrapower map, let
$i:M \rightarrow N$
be an ultrapower map given by some
$\lambda $
-complete uniform ultrafilter in M, and let
$\pi =i \circ j$
. Let u be the measure sequence of length
$\delta $
obtained from j, and let G be a suitable
$R_u$
-generic filter over V.
For the purpose of making this paper easier to read, we next give the idea of the proof of Bagaria–Magidor from [Reference Bagaria and Magidor2, Theorem 6.1], as well as our idea of the proof of Theorem 4.2.
In the proof of Bagaria–Magidor they only consider the case
$\lambda =\delta $
, i.e.,
$\delta $
is measurable, and take the Radin forcing
$R_u$
to turn
$\kappa $
into a
$\lambda $
-strongly compact cardinal.
To prove the
$\lambda $
-strong compactness of
$\kappa $
, they lift the composite embedding
$\pi =i \circ j$
in a
$\pi (R_{u})/ \dot {G}_{R_{i(u)\upharpoonright \delta }}$
-generic extension of
$V[G]$
. Here, j grants the
$\delta $
-strong compactness of
$\kappa $
at the end, and i makes
$\langle (i(u) \upharpoonright \delta ,i(A)) \rangle $
addible to
$\pi (\langle (u,A)\rangle )$
for every
$\langle (u,A)\rangle \in G$
. Namely,
$\langle (i(u) \upharpoonright \delta ,i(A)),(\pi (u),\pi (A) \rangle \in \pi (R_u)$
for every
$\langle (u,A)\rangle \in G$
. In addition, since
$R_u$
has a particular closure property,
$i''g_G$
can generate an
$R_{i(u)\upharpoonright \delta }$
-generic filter by a variation of the transfer argument (see [Reference Bagaria and Magidor2], also [Reference Cummings, Foreman and Kanamori3, Proposition 15.1]). Thus by Silver’s criterion (see [Reference Cummings, Foreman and Kanamori3, Proposition 9.1]), a lifting embedding of
$\pi $
can be obtained. However, this embedding is not definable in
$V[G]$
. To remedy this, Bagaria–Magidor use a closure argument, which relies not only on the closure of the Radin forcing itself, but also on the closure of N, to show that the filter generated by the lifted embedding is
$\lambda $
-complete. Thus
$\kappa $
is
$\lambda $
-strongly compact in
$V[G]$
(here and next, actually
$\kappa $
is
$\lambda $
-strongly compact up to
$\kappa '$
for some
$\kappa '$
, but a simple trick can solve this problem by lifting class-many embeddings with the same u).
In our proof, we also take the Radin forcing
$R_u$
to turn
$\kappa $
into a
$\lambda $
-strongly compact cardinal, but handle the general case (i.e.,
$\lambda $
may not equal to
$\delta $
). The major difference, or the novelty of the proof, is the lifting argument. The argument is more complicated in the general case, because
$\delta $
is
$\lambda $
-measurable, and so it may have stronger consistency strength than measurability (however, we don’t know if under the existence of a supercompact cardinal, it is possible to make some regular cardinal, for example,
$\lambda ^+$
, into a
$\lambda $
-measurable cardinal). We next give more details about the lifting argument.
We start with the strategy of Bagaria–Magidor, and we still consider to lift the composite embedding
$\pi $
by using Silver’s criterion. But there is a problem. There may exist unboundedly many
$\alpha <\delta $
with
$\sup (i''\alpha )<i(\alpha )$
below
$\delta $
, because
$\delta $
may surpass the critical point of
$\pi $
. Then for every such
$\alpha $
, there will be a gap below
$\pi (g_{\alpha })$
to be filled in, namely, a generic object in a similar sense of
$\pi (R_{u})/ \dot {G}_{R_{i(u)\upharpoonright \delta }}$
. But there is no place to fill in the gap below
$\pi (g_{\alpha })$
, since there are unboundedly many
$\pi (g_{\gamma })$
below
$\pi (g_{\alpha })$
. Hence, we can’t lift the embedding
$\pi $
.
To overcome the problem, we invoke Theorem 2.12, which states that a Radin generic object can be generated by some iterated ultrapower, to fill in a gap. However, we may need to fill in many gaps. Hence, a sequence of iterated ultrapowers should be taken to fill in all these gaps (an iterated ultrapower above
$\kappa $
is also taken to fill in the gap above
$\kappa $
, i.e., the counterpart of
$\pi (R_{u})/ \dot {G}_{R_{i(u)\upharpoonright \delta }}$
, so that we can avoid the closure argument).
So in the proof of Theorem 4.2, we may get an iteration given by a composition of embeddings: an ultrapower map given by a
$\lambda $
-complete uniform ultrafilter over
$\delta $
, a sequence of iterated ultrapowers for filling in gaps below
$\kappa $
, a supercompact embedding, and an iterated ultrapower for filling in gaps above
$\kappa $
. This iteration lives in
$V[G]$
since it is guided by the
$R_u$
-generic object G over V. Then we can build a
$\pi (R_{u})$
-generic object H over the target model in
$V[G]$
. In addition,
$\pi ''G \subseteq H$
. So we may obtain a lifting embedding of
$\pi $
in
$V[G]$
by using Silver’s criterion, which witnesses the
$\lambda $
-strong compactness of
$\kappa $
.
4 Main results
The following proposition shows that Radin generic sequence above an
$\omega _1$
-strongly compact cardinal destroys the
$\omega _1$
-strong compactness of the smaller cardinal.
Proposition 4.1. Suppose u is a measure sequence of length at least
$2$
. Then in an
$R_u$
-generic extension of V, there is no
$\omega _1$
-strongly compact cardinal below
$\kappa (u)$
.
Proof Let
$\kappa :=\kappa (u)$
and let G be an
$R_u$
-generic filter. Suppose, towards a contradiction, that there is a
$\gamma <\kappa $
such that
$\gamma $
is
$\omega _1$
-strongly compact. Note that there is a Prikry sequence contained in
$C_G \setminus \gamma $
. Then there is a
$\square _{\theta ,\omega }$
-sequence, say
$\vec {\mathcal {C}}=\langle \mathcal {C}_{\alpha } \ | \ \alpha <\theta ^+ \rangle $
, for some
$\theta \in C_G \setminus \gamma $
(see [Reference Cummings and Schimmerling4, Theorem 4.2]). By the
$\omega _1$
-strong compactness of
$\gamma $
, there exists an elementary embedding
$k:V[G] \rightarrow M'$
such that
$\sup (k''\theta ^+)<k(\theta ^+)$
. Then
$k(\vec {C})$
is a
$\square _{k(\theta ),\omega }$
-sequence in
$M'$
. Let
$\beta :=\sup (k''\theta ^+)$
, and pick a
$C' \in k(\vec {\mathcal {C}})(\beta )$
. Then we have:
-
(i)
$C'$
is a club subset of
$\beta $
. -
(ii)
$C'$
has order type at most
$k(\theta )$
. -
(iii) If
$\alpha $
is a limit point of
$C'$
, then
$C' \cap \alpha \in \mathcal {C}_{\alpha }$
.
It is easy to see that
$k''\theta ^+$
is a stationary subset of
$\beta $
, so there are unboundedly many
$\alpha <\theta ^+$
, such that
$k(\alpha )$
is a limit point of
$C'$
. For any such
$\alpha $
, by (iii), we have
$C' \cap k(\alpha )=k(C^{\prime }_{\alpha })$
for some
$C^{\prime }_{\alpha } \in \mathcal {C}_{\alpha }$
. Hence, these
$C^{\prime }_{\alpha }$
are pairwise compatible, i.e., for any such
$\alpha <\alpha '$
,
$k(C^{\prime }_{\alpha '})\cap k(\alpha )=k(C^{\prime }_{\alpha })$
. By elementarity of k, we have
$C^{\prime }_{\alpha '}\cap \alpha =C^{\prime }_{\alpha }$
. So the union of these
$C^{\prime }_{\alpha }$
, say C, is a club subset of
$\theta ^+$
, and
$C \cap \alpha =C^{\prime }_{\alpha }$
for any such
$\alpha $
. Hence, C has order type
$\theta ^+$
. However, any such
$C^{\prime }_{\alpha }$
has order type at most
$\theta $
, a contradiction. Hence, there is no
$\omega _1$
-strongly compact cardinal below
$\kappa $
.
Theorem 4.2. Suppose
$\kappa $
is a supercompact cardinal, and
$\delta <\kappa $
is a
$\lambda $
-measurable cardinal for some uncountable cardinal
$\lambda $
. Then in a Radin generic extension of V that preserves the
$\lambda $
-measurability of
$\delta $
,
$\kappa $
is the least
$\lambda $
-strongly compact cardinal and has cofinality
$\delta $
.
Proof First let us find a measure sequence u on
$\kappa $
of length
$\delta $
in order to obtain a suitable Radin forcing
$R_u$
.
For every
$\kappa '>|V_{\kappa +2}|$
, let
$U_{\kappa '}$
be a normal fine measure over
$\mathcal {P}_{\kappa }(\kappa ')$
, and let
$j_{\kappa '}:V \rightarrow M_{\kappa '} \cong \mathrm {Ult}(V,U_{\kappa '})$
be the corresponding supercompact embedding. Let
$u_{\kappa '}$
be the measure sequence of length
$\delta $
obtained from
$j_{\kappa '}$
. Since there are at most
$2^{2^{\kappa }}$
many such measure sequences of length
$\delta $
, there exists a proper class
$\mathcal {S}$
of ordinals and a measure sequence
$u=\langle u(\alpha ) \ | \ \alpha <\delta \rangle $
such that for every
$\kappa ' \in \mathcal {S}$
,
$u=u_{\kappa '}$
.
Let
$R_u$
be the Radin forcing for u. Pick a condition
$\langle (u,C) \rangle \in R_u$
so that
$C \cap V_{\delta +1}=\emptyset $
and it forces
$\mathrm {lth}(g_{\dot {G}})=\delta $
. Then C consists of measure sequences of length less than
$\delta $
. Let G be an
$R_u$
-generic filter over V with
$\langle (u,C) \rangle \in G$
. Then G adds no new subsets of
$\delta $
, and so
$\delta $
is also
$\lambda $
-measurable in
$V[G]$
. Let
$g_G=\langle g_{\alpha } \ | \ \alpha <\delta \rangle $
be the generic sequence given by G. Then
$g_G$
is geometric w.r.t. u and V by Theorem 2.9.
We will define a composite embedding
$\pi $
. Since
$\delta $
is
$\lambda $
-measurable, by Proposition 2.2, there exists a
$\lambda $
-complete uniform ultrafilter over
$\delta $
, say W. Let
$i:V \rightarrow N_0=\mathrm {Ult}(V,W)$
be the canonical map. W.l.o.g., we may assume
$\mathrm {crit}(i)=\lambda $
. Otherwise, let
$\lambda '=\mathrm {crit}(i)$
. Then we can prove that
$\kappa $
is
$\lambda '$
-strongly compact with the same proof. Notice that as
$\lambda ' \geq \lambda $
,
$\kappa $
is also
$\lambda $
-strongly compact.
For simplicity of notation, let
$g_{\delta }:=u$
. For every
$\alpha \leq \delta $
, define
$$ \begin{align} s(\alpha)=\begin{cases} \sup(i''\alpha), & \text{if }\alpha\text{ is a limit ordinal}, \\ i(\alpha), & \text{otherwise}. \end{cases} \end{align} $$
Then the intervals
$[s(\alpha ),i(\alpha )]$
for
$\alpha \leq \delta $
constitute a partition of
$i(\delta )+1$
by (4). So for every
$\theta \leq i(\delta )$
, we may let
$[\theta ]$
be the unique ordinal such that
$s([\theta ])\leq \theta \leq i([\theta ])$
.
Take any
$\kappa ' \in \mathcal {S}$
and let
$U:=U_{\kappa '}$
. Then in
$V[G]$
, we may construct an iterated ultrapower
$\langle N_{\theta },\pi _{\theta ,\theta '} \ | \ \theta \leq \theta '\leq i(\delta ) \rangle $
as follows:
-
(i)
$N_0=V$
. -
(ii)
$\pi _{\theta ,\theta }=\mathrm {id}_{N_{\theta }}$
for every
$\theta \leq i(\delta )$
. -
(iii) If
$\theta \leq i(\delta )$
is a limit ordinal, then
$N_{\theta }$
is the direct limit of
$\langle N_{\theta _0},\pi _{\theta _0,\theta _1} \ | \ \theta _0 \leq \theta _1\leq \theta \rangle $
, together with elementary embeddings
$\pi _{\theta _0,\theta }:N_{\theta _0} \rightarrow N_{\theta }$
for all
$\theta _0 < \theta $
. -
(iv) If
$\theta \leq i(\delta )$
is a successor ordinal with
$s([\theta ]) \leq \theta < i([\theta ])$
, then
$N_{\theta +1}$
is the transitive class isomorphic to
$\mathrm {Ult}(N_{\theta },\pi _{\theta }(U))$
if
$\theta =s(\delta )$
, or isomorphic to
$\mathrm {Ult}(N_{\theta },\pi _{\theta }(g_{[\theta ]})(\beta _{\theta }))$
, otherwise;
$\pi _{\theta ,\theta +1}:N_{\theta } \rightarrow N_{\theta +1}$
is the corresponding ultrapower map, and for every
$\gamma <\theta $
, let
$\pi _{\gamma ,\theta +1}=\pi _{\theta ,\theta +1} \circ \pi _{\gamma ,\theta }$
. -
(v) If
$\theta \leq i(\delta )$
is a successor ordinal with
$\theta =i([\theta ])$
, then
$N_{\theta +1}=N_{\theta }$
;
$\pi _{\theta ,\theta +1}=\mathrm {id}_{N_{\theta }}$
, and for every
$\gamma <\theta $
, let
$\pi _{\gamma ,\theta +1}=\pi _{\gamma ,\theta }$
.
For simplicity of notation, let
$\pi _{\theta }=\pi _{0,\theta } \circ i$
for every
$\theta \leq i(\delta )$
, and let
$\pi =\pi _{i(\delta )}$
and
$N=N_{i(\delta )}$
. Let
$w=\langle \pi _{\theta }(g_{[\theta ]})\upharpoonright \beta _{\theta }\ | \ \theta <i(\delta ) \rangle $
, let
$w(i(\delta ))=\pi _{i(\delta )}(g_{\delta })\upharpoonright \beta _{i(\delta )}=\pi (g_{\delta })$
, and let
$\vec {\kappa }=\langle \kappa (w(\theta )) \ | \ \theta \leq i(\delta ) \rangle $
. Then
$\vec {\kappa }=\langle \pi _{\theta }(\kappa (g_{[\theta ]})) \ | \ \theta \leq i(\delta ) \rangle $
since
$\kappa (w(\theta ))=\kappa (\pi _{\theta }(g_{[\theta ]}))=\pi _{\theta }(\kappa (g_{[\theta ]}))$
for every
$\theta \leq i(\delta )$
.
In the iteration
$\pi $
, these identity class functions in (5) are used for simplicity of notation.
The iterated ultrapower
$\langle N_{\theta },\pi _{\theta ,\theta '} \ | \ \theta \leq \theta ' \leq i(\delta ) \rangle $
is well-founded (see [Reference Jech9, Theorem 19.30]), so
$N_{\theta }$
is transitive for every
$\theta \leq i(\delta )$
.
Claim 4.3. In
$V[G]$
, w is geometric with respect to
$w(i(\delta ))$
and N. That is
$:$
-
(1) The sequence
$\vec {\kappa }$
is increasing continuous. -
(2) For every limit
$\theta \leq i(\delta )$
and every
$A \in N \cap V[G]_{\kappa (w(\theta ))+1}$
,
$A \in \mathcal {F}(w(\theta ))$
if and only if
$w \upharpoonright \theta $
is eventually contained in A.
Proof For every
$\alpha \leq \delta $
with
$s(\alpha )<i(\alpha )$
, note that
$\langle N_{\theta },\pi _{\theta ,\theta '} \ | \ s(\alpha ) < \theta \leq \theta ' \leq i(\alpha ) \rangle $
is the
$\pi _{s(\alpha )+1}(g_{\alpha })$
-iterated ultrapower of length
$i(\alpha )$
over
$N_{s(\alpha )+1}$
, we have
$w \upharpoonright (s(\alpha ),i(\alpha )]$
is geometric w.r.t.
$w(i(\alpha ))$
and
$N_{i(\alpha )}$
by Theorem 2.12. Note also that
$N_{i(\alpha )}\cap V[G]_{\kappa (w(i(\alpha )))+1}= N \cap V[G]_{\kappa (w(i(\alpha )))+1}$
and
$w(i(\alpha )) \in N$
, it follows that
$w \upharpoonright (s(\alpha ),i(\alpha )]$
is also geometric w.r.t.
$w(i(\alpha ))$
and N. Hence for the proof of (1) and (2), we only need to consider the case that
$\theta \leq i(\delta )$
is a limit ordinal and
$\theta = s([\theta ])$
.
Fact 4.4. For every
$\alpha \leq \delta $
, if
$\gamma \leq s(\alpha )$
, then
$\kappa (g_{\alpha })$
is fixed by
$\pi _{\gamma }$
, i.e.,
$\pi _{\gamma }(\kappa (g_{\alpha }))=\kappa (g_{\alpha })$
.
Proof If
$\gamma <s(\alpha )$
, then the iterated ultrapower
$\langle N_{\theta },\pi _{\theta ,\theta '} \ | \ s(\alpha ) < \theta \leq \theta ' \leq \gamma \rangle $
is taken in
$V[w \upharpoonright [\gamma ]]$
. Since
$\kappa (g_{\alpha })>\kappa (\pi _{\gamma }(g_{[\gamma ]}))$
is inaccessible in
$V[w \upharpoonright [\gamma ]]$
, it is fixed by
$\pi _{\gamma }$
.
If
$\gamma =s(\alpha )$
, then
$\gamma =\lim _{\xi <\alpha }i(\xi )$
. So for every
$\eta <\pi _{\gamma }(\kappa (g_{\alpha }))$
, there is a
$\xi <\alpha $
and an
$\bar {\eta }<\pi _{i(\xi )}(\kappa (g_{\alpha }))$
such that
$\pi _{i(\xi ),\gamma }(\bar {\eta })=\eta $
. Meanwhile, since
$C_G$
is a club subset of
$\kappa $
and
$\pi _{i(\xi )}(\kappa (g_{\alpha '}))=\kappa (g_{\alpha '})$
for every
$\alpha '$
with
$\xi <\alpha '\leq \alpha $
, we have
$\{\pi _{i(\xi )}(\kappa (g_{\alpha '})) \ | \ \xi <\alpha '<\alpha \}=\{\kappa (g_{\alpha '}) \ | \ \xi <\alpha '<\alpha \}$
is also a club subset of
$\pi _{i(\xi )}(\kappa (g_{\alpha }))=\kappa (g_{\alpha })$
. Hence,
$\bar {\eta }<\pi _{i(\xi )}(\kappa (g_{\alpha '}))$
for some
$\alpha '<\alpha $
. Then by elementarity of
$\pi _{i(\xi ),\gamma }$
, we have
$\eta =\pi _{i(\xi ),\gamma }(\bar {\eta })<\pi _{\gamma }(\kappa (g_{\alpha '}))= \pi _{i(\alpha ')}(\kappa (g_{\alpha '}))<\pi _{i(\alpha ')}(\kappa (g_{\alpha }))=\kappa (g_{\alpha })$
. Hence,
$\pi _{\gamma }(\kappa (g_{\alpha }))=\kappa (g_{\alpha })$
.
By the fact above, the sequence
$\langle \kappa (w(s(\alpha ))) \ | \ \alpha \leq \delta \rangle =\langle \kappa (g_{\alpha })) \ | \ \alpha \leq \delta \rangle $
is increasing continuous. Meanwhile, for every
$\alpha < \delta $
with
$s(\alpha )<i(\alpha )$
,
$\vec {\kappa } \upharpoonright (s(\alpha ),i(\alpha )]$
is between
$\kappa (w(s(\alpha )))=\kappa (g_{\alpha })$
and
$\kappa (w(s(\alpha +1)))=\kappa (w(i(\alpha )+1))=\kappa (g_{\alpha +1})$
, and
$\vec {\kappa } \upharpoonright (s(\delta ),i(\delta )]$
is above
$\kappa (g_{\delta })$
. Notice also that
$\vec {\kappa } \upharpoonright (s(\alpha ),i(\alpha )]$
is increasing continuous, it follows that (1) holds.
Take any limit
$\theta \leq i(\delta )$
with
$\theta =s([\theta ])$
.
Lemma 4.5. For every
$\eta \leq \theta $
, if
$B \in \mathcal {F}(\pi _{\eta }(g_{[\theta ]}))$
, then there is an
$m<[\theta ]$
such that:
-
(i) For every
$\alpha $
with
$\ m \leq \alpha <[\theta ]$
, we have
$\pi _{\eta }(g_{\alpha }) \in B$
. -
(ii) For every limit
$\alpha $
with
$m<\alpha <[\theta ]$
, we have
$B \cap V[G]_{\pi _{\eta }(\kappa (g_{\alpha }))} \in \mathcal {F}(\pi _{\eta }(g_{\alpha }))$
.
Proof We prove (i) and (ii) by induction on
$\eta $
. If
$\eta = 0$
, then since
$\pi _{\eta } = i : V \rightarrow N_0 \cong \mathrm {Ult}(V,W)$
,
$|W| < \kappa (g_0)$
, like in Case 2 below, we easily know that there is an m such that (i) and (ii) hold for B. Now suppose (i) and (ii) hold for every
$\xi <\eta $
. Then there are two cases for
$\eta $
(the case that
$\eta $
is a successor ordinal with
$\eta -1 = i([\eta -1])$
is trivial since
$\pi _{\eta -1}= \mathrm {id}$
, so we omit it):
Case 1.
$\eta $
is a limit ordinal. Then there is an
$\bar {\eta }<\eta $
and a
$\bar {B} \in N_{\bar {\eta }}$
such that
$\pi _{\bar {\eta },\eta }(\bar {B})=B$
. Since
$B \in \mathcal {F}(\pi _{\eta }(g_{[\theta ]}))$
and
$\pi _{\bar {\eta },\eta }$
is elementary, we have
$\bar {B} \in \mathcal {F}(\pi _{\bar {\eta }}(g_{[\theta ]}))$
. Thus by induction, there is an
$m<[\theta ]$
such that (i) and (ii) hold for
$\bar {\eta }$
and
$\bar {B}$
. Since
$\pi _{\bar {\eta },\eta }$
is elementary, it follows that for such an m, (i) and (ii) hold for
$\eta $
and B.
Case 2.
$\eta $
is a successor ordinal with
$s([\eta -1]) \leq \eta -1<i([\eta -1])$
. Then
$\eta <\theta $
,
$[\eta ]<[\theta ]$
and
$\pi _{\eta -1,\eta }$
is the ultrapower map given by
$\pi _{\eta -1}(g_{[\eta -1]})(\beta _{\eta -1})$
. Since
$B \in \mathcal {F}(\pi _{\eta }(g_{[\theta ]}))$
is in
$N_{\eta }$
, it can be represented by a function
$f \in N_{\eta -1}$
with domain
$N_{\eta -1} \cap V[G]_{\pi _{\eta -1}(\kappa (g_{[\eta -1]}))}$
, and for every
$x \in \mathrm {dom}(f)$
,
$f(x) \in \mathcal {F}(\pi _{\eta -1}(g_{[\theta ]}))$
. Now work in
$N_{\eta -1}$
, and let
$\bar {B}=\bigcap _{x \in \mathrm {dom}(f)}f(x)$
. Then
$\bar {B} \in \mathcal {F}(\pi _{\eta -1}(g_{[\theta ]}))$
since
$\mathcal {F}(\pi _{\eta -1}(g_{[\theta ]}))$
is
$\pi _{\eta -1}(\kappa (g_{[\theta ]}))$
-complete and
$\pi _{\eta -1}(\kappa (g_{[\theta ]}))>\pi _{\eta -1}(\kappa (g_{[\eta -1]}))$
. By induction, there is an
$m<[\theta ]$
such that (i) and (ii) hold for
$\eta -1$
and
$\bar {B}$
.
Also,
$\pi _{\eta -1,\eta }(\bar {B})\subseteq B$
since
$\bar {B} \subseteq f(x)$
for each
$x \in \mathrm {dom}(f)$
. Hence, since
$\pi _{\eta -1,\eta }$
is elementary, it follows that for such an m, (i) and (ii) hold for
$\eta $
and B as well.
So, in any case, (i) and (ii) hold. Hence by induction, the lemma holds.
Now we will prove that (2) holds. If
$A \in \mathcal {F}(w(\theta ))=\mathcal {F}(\pi _{\theta }(g_{[\theta ]})\upharpoonright \beta _{\theta })$
, then w.l.o.g., we may assume
$A \in \mathcal {F}(\pi _{\theta }(g_{[\theta ]}))$
since for every sufficient large
$\xi <\theta $
, the length of
$w(\xi )$
is less than
$\beta _{\theta }$
. Since
$\theta $
is a limit ordinal, there is a
$\bar {\theta }<\theta $
and an
$\bar {A} \in N_{\bar {\theta }}$
such that
$\pi _{\bar {\theta },\theta }(\bar {A})=A$
. Note that
$A \in \mathcal {F}(\pi _{\theta }(g_{[\theta ]}))$
and
$\pi _{\bar {\theta },\theta }$
is elementary, we have
$\bar {A} \in \mathcal {F}(\pi _{\bar {\theta }}(g_{[\theta ]}))$
.
Now by Lemma 4.5, there is an m with
$[\bar {\theta }]<m<[\theta ]$
such that the following holds:
-
(i) For every
$\alpha $
with
$m \leq \alpha <[\theta ]$
,
$\pi _{\bar {\theta }}(g_{\alpha }) \in \bar {A}$
. -
(ii) For every limit
$\alpha $
with
$m<\alpha <[\theta ]$
,
$\bar {A} \cap V[G]_{\kappa (\pi _{\bar {\theta }}(g_{\alpha }))} \in \mathcal {F}(\pi _{\bar {\theta }}(g_{\alpha }))$
.
For every
$\alpha $
with
$m \leq \alpha <[\theta ]$
, since (i) holds and
$\pi _{\bar {\theta },\theta }$
is elementary, it follows that
$w(i(\alpha ))=\pi _{i(\alpha )}(g_{\alpha })=\pi _{\theta }(g_{\alpha }) \in \pi _{\bar {\theta },\theta }(\bar {A})=A$
. Hence,
$$ \begin{align} \{ w(i(\alpha)) \ | \ m \leq \alpha<[\theta] \} \subseteq A. \end{align} $$
For every limit
$\alpha $
with
$m < \alpha < [\theta ]$
, since (ii) holds and
$\pi _{\bar {\theta },s(\alpha )}$
is elementary, we have
$$\begin{align*}\pi_{\bar{\theta},s(\alpha)}(\bar{A}) \cap V[G]_{\kappa(\pi_{s(\alpha)}(g_{\alpha}))} \in \mathcal{F}(\pi_{s(\alpha)}(g_{\alpha})). \end{align*}$$
If
$s(\alpha )<i(\alpha )$
, then
$\beta _{\eta }<\beta _{i(\alpha )}$
for every
$\eta $
with
$s(\alpha ) \leq \eta <i(\alpha )$
. Hence by Lemma 2.11,
$$ \begin{align} \{ w(\eta) \ | \ s(\alpha) \leq \eta < i(\alpha) \} \subseteq \pi_{\bar{\theta},i(\alpha)}(\bar{A}) \cap V[G]_{\pi_{i(\alpha)}(\kappa(g_{\alpha}))} \subseteq A. \end{align} $$
Therefore, we have
$\{ w(\eta ) \ | \ i(m) < \eta <\theta \} \subseteq A$
by (5) and (6) for some
$m<[\theta ]$
. In other words,
$w \upharpoonright \theta $
is eventually contained in A.
If
$A \notin \mathcal {F}(w(\theta ))$
, then
$A \notin w(\theta )(\eta )$
for some
$0<\eta <\beta _{\theta }$
. Hence
$\{ \gamma <\theta \ | \ w(\gamma ) \notin A$
and
$\beta _{\gamma }=\eta \}$
is unbounded in
$\theta $
, which means that
$w \upharpoonright \theta $
is not eventually contained in A.
In
$V[G]$
, let
$H\subseteq \pi (R_{u})$
be the filter given by w. Namely, H is the set of all
$p \in \pi (R_{u})$
such that:
-
(i) If v occurs in p, then
$v=w(\theta )$
for some
$\theta \leq i(\delta )$
. -
(ii) For any
$\theta \leq i(\delta )$
,
$w(\theta )$
occurs in some
$q \leq p$
.
It follows from Claim 4.3 that H is
$\pi (R_u)$
-generic over N. Now we will prove that
$\pi '' G \subseteq H$
. Take any condition
$p=\langle (g_{\alpha _1},A_1),\ldots ,(g_{\alpha _n},A_n) \rangle \in G$
, where
$\alpha _n=\delta $
. We need to prove that
$\pi (p) \in H$
. Let
$\alpha _0=-1$
for simplicity. Note that
$\pi (g_{\alpha _j})=\pi _{i(\alpha _j)}(g_{\alpha _j})=w(i(\alpha _j))$
and
$\pi (A_j)=\pi _{i(\alpha _j)}(A_j)$
for every
$1 \leq j \leq n$
, so we have
$$\begin{align*}\pi(p)=\langle (w(i(\alpha_1)),\pi_{i(\alpha_1)}(A_1)),\ldots,(w(i(\alpha_n)),\pi_{i(\alpha_n)}(A_n)) \rangle. \end{align*}$$
Then by the definition of H, we only need to prove that for every
$m \leq n$
,
$w(\theta ) \in \pi _{i(\alpha _{m+1})}(A_{m+1})$
for every
$\theta $
with
$i(\alpha _m) <\theta < i(\alpha _{m+1})$
. Take any such an m and a
$\theta $
. Since
$p \in G$
, we have
$$ \begin{align} \{ g_{\eta} \ | \ \alpha_m<\eta<\alpha_{m+1} \} \subseteq A_{m+1}. \end{align} $$
There are two cases:
Case 1.
$\theta =i([\theta ])$
. Then
$[\theta ]<\alpha _{m+1}$
, and
$g_{[\theta ]} \in A_{m+1}$
. Since
$\pi _{i(\alpha _{m+1})}$
is elementary,
$$\begin{align*}w(\theta)=\pi_{i([\theta])}(g_{[\theta]})=\pi_{i(\alpha_{m+1})}(g_{[\theta]}) \in \pi_{i(\alpha_{m+1})}(A_{m+1}). \end{align*}$$
Case 2.
$s([\theta ])\leq \theta <i([\theta ])$
. Then
$[\theta ]$
is a limit ordinal. By (7) and the characterization of genericity, i.e., Theorem 2.9,
$A_{m+1} \cap V[G]_{\kappa (g_{[\theta ]})} \in \mathcal {F}(g_{[\theta ]})$
. Since
$\pi _{\theta }$
is elementary, it follows that
$\pi _{\theta }(A_{m+1} \cap V[G]_{\kappa (g_{[\theta ]})}) \in \mathcal {F}(\pi _{\theta }(g_{[\theta ]}))$
. Note also that
$\beta _{\theta }<\beta _{i([\theta ])}$
, by Lemma 2.11 (for the case
$\theta =s(\delta )$
, since the measure sequence
$\pi _{s(\delta )}(u)$
is obtained from
$\pi _{s(\delta ),s(\delta )+1}$
, this lemma is also true), we have
$$\begin{align*}w(\theta) \in \pi_{i([\theta])}(A_{m+1} \cap V[G]_{\kappa(g_{[\theta]})})=\pi_{i(\alpha_{m+1})}(A_{m+1} \cap V[G]_{\kappa(g_{[\theta]})}) \subseteq \pi_{i(\alpha_{m+1})}(A_{m+1}). \end{align*}$$
Hence in any case,
$w(\theta ) \in \pi _{i(\alpha _{m+1})}(A_{m+1})$
. So
$\pi ''G \subseteq H$
, and therefore, we may lift
$\pi $
and obtain an elementary embedding
$\pi ^+:V[G] \rightarrow N[H]$
by using Silver’s criterion.
Let
$D=\pi _{s(\delta )+1,i(\delta )}(\pi_{s(\delta ),s(\delta )+1}^{\prime\prime}(\pi _{s(\delta )}(\kappa ')))$
. Since
$\pi _{s(\delta ),s(\delta )+1}$
witnesses that
$\pi _{s(\delta )}(\kappa )$
is
$\pi _{s(\delta )}(\kappa ')$
-supercompact, we have
$\pi _{s(\delta ),s(\delta )+1}^{\prime\prime}(\pi _{s(\delta )}(\kappa ')) \in N_{s(\delta )+1}$
. Note also that
$\pi _{s(\delta )+1,i(\delta )}$
is elementary, we have
$D=\pi _{s(\delta )+1,i(\delta )}(\pi _{s(\delta ),s(\delta )+1}^{\prime\prime}(\pi _{s(\delta )}(\kappa '))) \in N$
. Meanwhile,
$\pi ''\kappa ' \subseteq D$
and
$N \models |D|<\pi (\kappa )$
. So
$\pi ^+$
witnesses
$\kappa $
is
$\lambda $
-strongly compact up to
$\kappa '$
.
Since
$\mathcal {S}$
is a proper class and
$\kappa ' \in \mathcal {S}$
is arbitrary,
$\kappa $
is
$\lambda $
-strongly compact in
$V[G]$
.
Now by Fact 4.1, we can see that
$\kappa $
is the least
$\lambda $
-strongly compact cardinal in
$V[G]$
(actually,
$\kappa $
is the least
$\omega _1$
-strongly compact cardinal).
This concludes the proof of Theorem 4.2.
Corollary 4.6. Suppose
$\lambda <\kappa $
are supercompact cardinals. Then for every regular cardinal
$\delta $
with
$\lambda \leq \delta <\kappa $
, there exists a generic extension of V, in which
$\kappa $
is the least
$\lambda $
-strongly compact cardinal and has cofinality
$\delta $
.
Proof Since
$\lambda $
is a supercompact cardinal and
$\delta $
is regular, it follows that
$\delta $
is
$\lambda $
-measurable. By Theorem 4.2, in some Radin generic extension that adds a Radin generic sequence of length
$\delta $
,
$\kappa $
is the least
$\lambda $
-strongly compact cardinal and has cofinality
$\delta $
.
The following proposition generalizes [Reference Bagaria and Magidor1, Theorem 2.3] and shows that our consistency result (Theorem 4.2) is optimal.
Proposition 4.7. Suppose
$\kappa $
is the least
$\lambda $
-strongly compact cardinal and has cofinality
$\delta $
. Then
$\delta $
is
$\lambda $
-measurable. Namely,
$\delta $
carries a
$\lambda $
-complete uniform ultrafilter over
$\delta $
.
Proof We first prove that there exists a definable elementary embedding
$j:V \rightarrow M$
with M transitive such that
$j(\kappa )>\sup (j''\kappa )$
. Since
$\kappa $
is the least
$\lambda $
-strongly compact cardinal, it follows that
$\kappa $
is a limit cardinal by Theorem 2.3, and for every
$\gamma <\kappa $
, there is an
$\alpha _{\gamma }>\kappa $
such that
$\gamma $
is not
$\lambda $
-strongly compact up to
$\alpha _{\gamma }$
. Let
$\alpha =\sup (\{\alpha _{\gamma } \ | \ \gamma <\kappa \})^+$
. Then
$\eta $
is not
$\lambda $
-strongly compact up to
$\alpha $
for every
$\eta <\kappa $
.
By the
$\lambda $
-strong compactness of
$\kappa $
, there is a definable elementary embedding
$j:V \rightarrow M$
with M transitive, so that
$\mathrm {crit}(j) \geq \lambda $
and there is a
$D \in M$
such that
$j''\alpha \subseteq D$
and
$M \models |D|<j(\kappa )$
. If
$\sup (j''\kappa )=j(\kappa )$
, then there is a cardinal
$\eta <\kappa $
such that
$M \models |D|<j(\eta )$
. Thus j witnesses that
$\eta $
is
$\lambda $
-strongly compact up to
$\alpha $
, a contradiction. Hence,
$\sup (j''\kappa )<j(\kappa )$
.
For such an embedding j, we claim that
$\sup (j''\delta )<j(\delta )$
. If not,
$j(\delta )=\sup (j''\delta )$
. Since
$\kappa $
is a limit cardinal and has cofinality
$\delta $
, there is an increasing cofinal sequence
$\vec {\kappa }=\langle \kappa _{\alpha } \ | \ \alpha <\delta \rangle $
of cardinals converging to
$\kappa $
. By elementarity,
$j(\vec {\kappa })$
is an increasing cofinal sequence on
$j(\kappa )$
in M. Then in V,
$\langle j(\vec {\kappa })(j(\alpha )) \ | \ \alpha <\delta \rangle $
is an increasing cofinal sequence on
$j(\kappa )$
, since
$j(\delta )=\sup (j''\delta )$
. By elementarity,
$j(\vec {\kappa })(j(\alpha ))=j(\vec {\kappa }(\alpha ))=j(\kappa _{\alpha })$
. So
$\langle j(\kappa _{\alpha }) \ | \ \alpha <\delta \rangle $
is an increasing cofinal sequence on
$j(\kappa )$
, which means that
$j(\kappa )=\sup (j''\kappa )$
, a contradiction. Therefore,
$j(\delta )>\sup (j''\delta )$
.
Hence,
$\delta $
carries a
$\lambda $
-complete uniform ultrafilter by the proof of Proposition 2.2, and it is also
$\lambda $
-measurable.
Acknowledgments
We would like to thank Professor Joan Bagaria and Professor Liuzhen Wu for making useful suggestions, which improved clarity and readability of the paper. This work started from 2019. The paper was submitted in January 2022. And we received the revision request in November 2022.
Funding
During the work, the first author was supported by NSFC No. 11871464 until July 2022 and the China Scholarship Council (Grant No. 202004910742) from October 2020 to October 2021. The first author was also supported by the European Research Council (grant agreement ERC-2018-StG 802756) from July 2022. In November 2022, the paper was revised according to reviewer’s comments and suggestions. The second author was supported by UKRI Future Leaders Fellowship (MR/T021705/2).
