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% THEOREMS
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{question}{Question}
\newtheorem{remark}{Remark}
\thanks{The first author's research was
partially supported by PSC-CUNY grants.}
\subjclass[2000]{03E25, 03E35, 03E45, 03E55}
\keywords{Supercompact cardinal, supercompact
Prikry forcing, GCH, symmetric inner model.}
\date{December 6, 2011
(revised April 5, 2012)}
%\date{\today}
\begin{document}
\title[Consec. singular cardinals and the continuum function]{Consecutive singular cardinals and the continuum function}
\author[Arthur W. Apter]{Arthur W. Apter}
\address[Arthur W. Apter]{
Department of Mathematics,
Baruch College of CUNY,
One Bernard Baruch Way,
New York, New York 10010 USA, and
Department of Mathematics,
The Graduate Center of CUNY,
365 Fifth Avenue,
New York, NY 10016 USA}
\email[A. W. ~Apter]{awapter@alum.mit.edu}
\urladdr{http://faculty.baruch.cuny.edu/aapter/}
\author[Brent Cody]{Brent Cody}
\address[Brent Cody]{
Department of Mathematics,
The Graduate Center of CUNY,
365 Fifth Avenue,
New York, NY 10016 USA}
\email[B. ~Cody]{bcody@gc.cuny.edu}
\urladdr{https://wfs.gc.cuny.edu/BCody/www/}
\maketitle
\begin{abstract}
We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\ZF+\lnot\AC$ holds, $\kappa$ and $\kappa^+$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\ZF+\lnot\AC_\omega$ in which either (1) $\aleph_1$ and $\aleph_2$ are both singular and the continuum function at $\aleph_1$ can be precisely controlled, or (2) $\aleph_\omega$ and $\aleph_{\omega+1}$ are both singular and the continuum function at $\aleph_\omega$ can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals $\kappa$ and $\kappa^+$ in a model of $\ZF$. Some open questions concerning the continuum function in models of $\ZF$ with consecutive singular cardinals are posed.
\end{abstract}
\section{Introduction}
In this paper we will be motivated by the question: Are there models of $\ZF$ with consecutive singular cardinals $\kappa$ and $\kappa^+$ such that ``$\GCH$ fails at $\kappa$'' in the sense that there is a sequence of distinct subsets of $\kappa$ of length greater than $\kappa^+$? Let us start by considering some known models of $\ZF$ that have consecutive singular cardinals.
Gitik showed in \cite{Gitik:AllUncountableCardinalsCanBeSingular} that from a proper class of strongly compact cardinals, $\langle\kappa_\alpha\mid\alpha\in\ORD\rangle$, there is a model of $\ZF+\lnot\AC_\omega$ in which all uncountable cardinals are singular. Essentially he uses a certain type of generalized Prikry forcing that simultaneously singularizes and collapses each $\kappa_\alpha$, thereby resulting in a model in which the class of uncountable well-ordered cardinals consists of the previously strongly compact $\kappa_\alpha$'s and their limits. In this model, every uncountable cardinal is singular, and for each $\alpha\in\ORD$ and for each limit ordinal $\lambda$, all cardinals in the open intervals $(\kappa_\alpha,\kappa_{\alpha+1})$ and $(\sup_{\beta<\lambda}\kappa_\beta,\kappa_\lambda)$ have been collapsed to have size $\kappa_\alpha$ and $\sup_{\beta<\lambda}\kappa_\beta$ respectively. Since each $\kappa_\alpha$ is a strong limit cardinal in the ground model, it follows that in Gitik's final model there is no cardinal $\kappa$ that has a sequence of distinct subsets of length greater than---or even equal to---$\kappa^+$.
(Of course, trivially, in any model of $\ZF$, for any cardinal $\kappa$,
there is always a $\kappa$-sequence of distinct subsets of $\kappa$ given
by the sequence of intervals $\la [\alpha,\kappa)\mid\alpha<\kappa\ra$.
This trivially also implies that,
for any $\beta \in (\kappa, \kappa^+)$,
there is a $\beta$-sequence of distinct subsets
of $\kappa$ as well.)
For similar reasons, the models constructed in \cite{Gitik:RegularCardinalsInModelsOfZF} and \cite{ApterDimitiouKoepke:TheFirstMeasurableCardinalCanBe} will also not have consecutive singular cardinals $\kappa$ and $\kappa^+$ with a sequence of distinct subsets of $\kappa$ of length even $\kappa^+$.
%Essentially he starts with a proper class of strongly compact cardinals $\langle\kappa_\alpha\mid\alpha\in\ORD\rangle$, singularizes each of them using a certain type of generalize Prikry forcing, and then uses a collapse forcing to construct a model such that the class of previously strongly compact cardinals becomes the class of cardinals, all of which are singular. At the beginning of the construction each $\kappa_\alpha$ is strongly compact and is hence also a strong limit. Furthermore, since the generalized Prikry forcing does not change this and the remaining stage in the construction collapses every cardinal in each interval $(\kappa_\alpha,\kappa_{\alpha+1})$ to have size $\kappa_\alpha$, one can see that in Gitik's final model there is no cardinal $\kappa$ that has a sequence of distinct subsets of length greater than---or even equal to---$\kappa^+$. For similar reasons, the models constructed in \cite{Gitik:RegularCardinalsInModelsOfZF} and \cite{ApterDimitiouKoepke:TheFirstMeasurableCardinalCanBe} will also not have consecutive singular cardinals $\kappa$ and $\kappa^+$ with a sequence of distinct subsets of $\kappa$ of length even $\kappa^+$.
%There has been a great deal of work in which, by forcing over models of $\AD$, models are constructed having consecutive singular cardinals, as exemplified by \cite{Apter:ADAndPatterns}. \marginpar{\tiny Why not just say ``models of $\AD$ are constructed.'' I'm assuming the forcing extensions of models of $\AD$ are themselves models of $\AD$.} However, in a model of $\AD$, no cardinal $\kappa<\Theta$ can have a sequence of subsets of length $\kappa^+$ let alone of longer length (see \cite{Steel:OutlineOfInnerModelTheory}).
There has been a great deal of work, involving forcing over models of $\AD$, in which models are constructed having consecutive singular cardinals, as exemplified by \cite{Apter:ADAndPatterns}. However, in any model of $\AD$, no cardinal $\kappa<\Theta$ has a sequence of distinct subsets of length $\kappa^+$ let alone of longer length (see \cite{Steel:OutlineOfInnerModelTheory}). Thus, forcing over a model of $\AD$ does not seem to yield, in any obvious way, a model containing consecutive singular cardinals, $\kappa$ and $\kappa^+$, in which there is a sequence of distinct subsets of $\kappa$ of length $\kappa^+$.
In this article, we will show that from a supercompact cardinal, there are models of $\ZF+\lnot\AC$ that have consecutive singular cardinals, say $\kappa$ and $\kappa^+$, such that there is a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. Indeed, we will prove the following.
%In this article, we will show that from a supercompact cardinal, there are models of $\ZF+\lnot\AC$ that have consecutive singular cardinals, say $\kappa$ and $\kappa^+$, such that the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. Indeed, we will prove the following.
%Let us now make a few comments about what we mean by precise control of the continuum function at $\kappa$. Suppose $\kappa$ is supercompact and $\theta>\kappa$ is a cardinal and that we obtain
%Let us summarize our method. Suppose $\kappa$ is supercompact and $\theta>\kappa$ is a cardinal. We want to address the question, is there a model of $\ZF$ in which $\kappa$ and $\kappa^+$ are both singular and there is a sequence of distinct subsets of $\kappa$ of length $\theta$? As indicated by the above discussion, to make this question more interesting we will add an additional requirement, which is that in the final model in which $\kappa$ and $\kappa^+$ are both singular we also have $\theta>\kappa^+$.
%\begin{remark}
%Let $\lambda>\kappa$ be singular with $\cf(\lambda)<\kappa$ and let $\P_U$ be the supercompact Prikry forcing defined using a normal measure $U$ on $P_\kappa\lambda$ (see Section \ref{sectionpreliminaries} for a discussion of supercompact Prikry forcing). In the extension $V[G]$ by $\P$, $\kappa$ has cofinality $\omega$ and cardinals in $(\kappa,\lambda]$ are collapsed to have size $\kappa$. There is a particular symmetric inner model $N\subseteq V[G]$, which we will define below, such that $\cf(\kappa)^N=\omega$ and $(\kappa^+)^N=\lambda$---that is, $\lambda$ becomes uncollapsed in moving from $V[G]$ to $N$.
%\end{remark}
\begin{theorem}\label{maintheorem}
Suppose $\kappa$ is supercompact, $\GCH$ holds, and $\theta$ is an ordinal. Then there is a forcing extension $V[G]$ that has a symmetric inner model $N\subseteq V[G]$ of $\ZF+\lnot\AC$ in which the following hold.
\begin{enumerate}
\item $\kappa$ and $\kappa^+$ are both singular with $\cf(\kappa)^N=\omega$ and $\cf(\kappa^+)^N<\kappa$.
\item $\kappa$ is a strong limit cardinal that is a limit of inaccessible cardinals.
\item There is a sequence of distinct subsets of $\kappa$ of length $\theta$.
\end{enumerate}
\end{theorem}
Let us remark here that property (3) in Theorem \ref{maintheorem} makes this result interesting, since none of the previously known models with consecutive singular cardinals discussed above satisfies it when $\theta\geq\kappa^+$. Since the definitions of ``strong limit cardinal'' and ``inaccessible cardinal'' generally do not make sense in models of $\lnot\AC$, let us explain why the assertion in Theorem \ref{maintheorem}
that (2) holds in $N$ makes sense.
It will be the case that $N$ and $V$ will have the same bounded subsets
of $\kappa$, and from this it follows that the usual definitions of
``$\kappa$ is a strong limit cardinal'' and ``$\delta<\kappa$ is an inaccessible cardinal'' make sense in $N$.
%For $\delta<\kappa$, the usual definition of $``\delta$ is inaccessible'' makes sense in $N$ because $N$ and $V$ will have the same bounded subsets of $\kappa$.
%need to explain what we mean by saying, (2) holds in $N$. For $\delta<\kappa$, the usual definition of $``\delta$ is inaccessible'' makes sense in $N$ because $N$ and $V$ will have the same bounded subsets of $\kappa$. Hence, saying that $\kappa$ is a limit of inaccessible cardinals in $N$ makes sense. Similaraly, saying that ``$\kappa$ is a strong limit cardinal'' holds in $N$ makes sense.
%For $\delta<\kappa$, we say that $\delta$ is an inaccessible cardinal in $N$ if and only if $\delta$ is an inaccessible cardinal in $V$. This makes sense because $N$ and $V$ will have the same bounded subsets of $\kappa$. Thus the assertion in Theorem \ref{maintheorem}(2) above, that $\kappa$ is a limit of inaccessible cardinals, makes sense in this context. Similarly, the usual definition for ``$\kappa$ is a strong limit cardinal'' makes sense in $N$ because $N$ and $V$ have the same bounded subsets of $\kappa$.
%To describe what (2) means in the model $N\models\lnot\AC$, it will suffice to explain what we mean by ``a strong limit cardinal'' below $\kappa$, in the model $N$. For $\delta<\kappa$, we say that $\delta$ is a strong limit cardinal in $N$ if and only if $\delta$ is a strong limit cardinal in $V$. This makes sense because $N$ and $V$ will have the same bounded subsets of $\kappa$.
Using the methods of \cite{Bull:SuccessiveLargeCardinals}, \cite{Apter:SomeResultsOnConsecutiveLargeCardinals}, and \cite{ApterHenle:RelativeConsistencyResultsViaStrongCompactness} we also obtain the following two results.
\begin{theorem}\label{theoremcollapse1}
Suppose $\kappa$ is supercompact, $\GCH$ holds, and $\theta$ is an ordinal. Then there is a model of $\ZF+\lnot \AC_\omega$ in which $\cf(\aleph_1)=\cf(\aleph_2)=\omega$, and there is a sequence of distinct subsets of $\aleph_1$ of length $\theta$.
%If it is consistent with $\ZFC$ that there is a supercompact cardinal, then for any ordinal $\theta$ there is a model of $\ZF+\lnot\AC_\omega$ in which $\aleph_1$ and $\aleph_2$ are both singular and there is a sequence of distinct subsets of $\aleph_1$ of length $\theta$.
\end{theorem}
\begin{theorem}\label{theoremcollapseomega}
Suppose $\kappa$ is supercompact, $\GCH$ holds, and $\theta$ is an ordinal. Then there is a model of $\ZF+\lnot\AC_\omega$ in which $\aleph_\omega$ and $\aleph_{\omega+1}$ are both singular with $\omega\leq\cf(\aleph_{\omega+1})<\aleph_\omega$, and there is a sequence of distinct subsets of $\aleph_\omega$ of length $\theta$.
%If it is consistent with $\ZFC$ that there is a supercompact cardinal, then for any ordinal $\theta$ there is a model of $\ZF+\lnot\AC_\omega$ in which $\aleph_\omega$ and $\aleph_{\omega+1}$ are both singular and there is a sequence of distinct subsets of $\aleph_\omega$ of length $\theta$. \marginpar{\tiny In Theorem \ref{theoremcollapseomega} we could say}
\end{theorem}
We note that in Theorem \ref{theoremcollapse1}, $\aleph_1$ and $\aleph_2$ can be replaced with $\delta$ and $\delta^+$ respectively, where $\delta$ is the successor of any ground model regular cardinal less than $\kappa$. Also, in Theorem \ref{theoremcollapseomega}, we note that $\aleph_\omega$ and $\aleph_{\omega+1}$ can be replaced by $\eta$ and $\eta^+$ respectively, where $\eta<\kappa$ can be any reasonably defined singular limit cardinal of cofinality $\omega$. We will return to these issues later.
Let us now give a brief outline of the rest of the paper. In Section \ref{sectionpreliminaries}, we include a definition of the basic forcing notion we will use and outline its important properties. In Section \ref{sectionmaintheorem}, we give a detailed proof of Theorem \ref{maintheorem}. In Section \ref{sectioncollapse}, we sketch the proofs of Theorem \ref{theoremcollapse1} and Theorem \ref{theoremcollapseomega}. In Section \ref{sectionadditional}, we discuss a result in which we separate the lengths of distinct subsets of consecutive singular cardinals, and we also pose some open questions.
\begin{comment}
Discuss Gitik's model in which every cardinal is singular as well as some other background results on consecutive singular/large cardinals and $\lnot AC$ models.
The following might be useful references for the introduction:
\begin{enumerate}
\item Levy proved in \cite{Levy:Independence} that it is consistent with $\ZF$ that $\aleph_1$ is singular. (Maybe I should only mention more recent results on consecutive singular cardinals?)
\item Gitik proved in \cite{Gitik:AllUncountableCardinalsCanBeSingular} that the consistency of the theory ``$\ZFC$ + there is a proper class of measurable cardinals'' implies the consistency of ``$\ZF$ + every uncountable cardinal is sinuglar.''
\item \cite{Bull:SuccessiveLargeCardinals}? Should we cite Bull's thesis?
\item \cite{Apter:SomeResultsOnConsecutiveLargeCardinals}
\end{enumerate}
\begin{theorem}\label{maintheoremconsistency}
The consistency of the theory ``$\ZFC$ + $\exists\kappa$ such that $\kappa$ is supercompact'' implies the consistency of ``$\ZF$ + $\lnot\AC_\omega$ + $\exists\kappa$ such that $\kappa$ and $\kappa^+$ are singular with cofinality $\omega$ and there is a $\lambda$-sequence of subsets of $\kappa$'' where $\lambda$ is an arbitrary cardinal.
\end{theorem}
\end{comment}
\section{Preliminaries}\label{sectionpreliminaries}
In this section, we will briefly discuss the various forcing notions used. If $\kappa$ is a regular cardinal and $\lambda$ is an ordinal, $\Add(\kappa,\lambda)$ denotes the standard partial order for adding $\lambda$ Cohen subsets to $\kappa$. If $\lambda>\kappa$ is an inaccessible cardinal, $\Coll(\kappa,{<}\lambda)$ is the standard partial order for collapsing $\lambda$ to $\kappa^+$ and all cardinals in the interval $[\kappa,\lambda)$ to $\kappa$.
For further details, we refer the reader to \cite{Jech:Book}. For a given partial order $\P$ and a condition $p\in\P$, we define $\P/p:=\{q\in\P\mid q\leq p\}$. If $\varphi$ is a statement in the forcing language associated with $\P$ and $p\in\P$, we write $p \parallel \varphi$ if and only if $p$ decides $\varphi$.
We will now review the definition and important features of supercompact
Prikry forcing and refer the reader to
\cite{Gitik:Handbook} or \cite{Apter:SuccessorsOfSingularCardinalsAnd}
for details. Suppose $\kappa$ is $\lambda$-supercompact and
that $U$ is a normal fine measure on $P_\kappa\lambda$
satisfying the Menas partition property
(see \cite{Menas:Paper}
for a definition and a proof of
the fact that if $\kappa$ is $2^\lambda$-supercompact,
then $P_\kappa\lambda$ has a normal fine measure
with this property).
For $P,Q\in P_\kappa\lambda$ we say that $P$ is \emph{strongly included} in $Q$ and write $P\strongsubset Q$ if $P\subseteq Q$ and $\ot(P)<\ot(Q\cap\kappa)$. We define \emph{supercompact Prikry forcing} $\P$ to be the set of all ordered tuples of the form $\la P_1,\ldots, P_n, A\ra$ such that
\begin{enumerate}
\item $P_1,\ldots,P_n$ is a finite $\strongsubset$-increasing sequence of elements of $P_\kappa\lambda$,
\item $A\in U$, and
\item for every $Q\in A$, $P_n\strongsubset Q$.
\end{enumerate}
Given $\la P_1,\ldots,P_n,A\ra,\la Q_1,\ldots,Q_m,B\ra\in\P$ we say that $\la P_1,\ldots,P_n,A\ra$ \emph{extends} $\la Q_1,\ldots,Q_m,B\ra$ and write $\la P_1,\ldots,P_n,A\ra\leq \la Q_1,\ldots,Q_m,B\ra$ if and only if
\begin{enumerate}
\item $n\geq m$,
\item for each $k\leq m$, $P_k=Q_k$,
\item $A\subseteq B$, and
\item $\{P_{m+1},\ldots,P_n\}\subseteq B$.
\end{enumerate}
%Furthermore, we say that $\la P_1,\ldots,P_n,A\ra$ is a \emph{direct extension} of $\la Q_1,\ldots,Q_m,B\ra$ and write $\la P_1,\ldots,P_n,A\ra\leq^* \la Q_1,\ldots,Q_m,B\ra$ if and only if $\la P_1,\ldots,P_n\ra=\la Q_1,\ldots,Q_m\ra$ and $A\subseteq B$.
Since any two conditions of the form $\la P_1\ldots,P_n,A\ra$ and $\la P_1,\ldots, P_n, B\ra$ in $\P$ are compatible, one may easily show that $\P$ is $(\lambda^{<\kappa})^+$-c.c. Since $U$ satisfies the Menas partition property, it follows that forcing with $\P$ does not add new bounded subsets to $\kappa$. In the forcing extension by $\P$, $\kappa$ has cofinality $\omega$, and if $\lambda>\kappa$ then certain cardinals will be collapsed according to the following.
\begin{lemma}\label{lemmaprikryforcing}
Every $\gamma\in [\kappa,\lambda]$ of cofinality at least $\kappa$ (in V) changes its cofinality to $\omega$ in $V[G]$. Moreover, in $V[G]$, every cardinal in $(\kappa,\lambda]$ is collapsed to have size $\kappa$.
\end{lemma}
%\noindent For a proof of the facts mentioned above as well as Lemma \ref{lemmaprikryforcing} one may consult Gitik's article in \cite{HandbookOfSetTheoryVolume2}.
\section{The Proof of Theorem \ref{maintheorem}}\label{sectionmaintheorem}
Now we will begin the proof of Theorem \ref{maintheorem}. We note that our proof amalgamates the methods used in \cite{ApterHenle:RelativeConsistencyResultsViaStrongCompactness} with those of \cite{Apter:SuccessorsOfSingularCardinalsAnd}.
\begin{proof}[Proof of Theorem \ref{maintheorem}]
Suppose $\kappa$ is supercompact and $\theta$ is an ordinal in some initial model $V_0$ of $\ZFC+\GCH$. We will show that there is a forcing extension of $V_0$ that has a symmetric inner model $N$ in which $\kappa$ and $\kappa^+$ are both singular with $\cf(\kappa)^N=\omega$ and $\cf(\kappa^+)^N<\kappa$, and there is a $\theta$-sequence of subsets of $\kappa$. By first forcing the supercompactness of $\kappa$ to be Laver indestuctible, as in \cite{Laver:MakingSupercompactnessIndestructible}, and then forcing with $\Add(\kappa,\theta)$, we may assume without loss of generality that $\kappa$ is supercompact and $2^\kappa=\theta$ in a forcing extension $V$ of $V_0$. Let $\lambda$ be a cardinal such that $\kappa<\lambda$ and $\cf(\lambda)^V<\kappa$. In $V$, let $\P$ be the supercompact Prikry forcing relative to some normal fine measure $U$ on $P_\kappa\lambda$ satisfying the Menas partition property. Let $G$ be $V$-generic for $\P$ and let $\langle P_n\mid n<\omega\rangle$ be the supercompact Prikry sequence associated with $G$; that is, $\langle P_n\mid n<\omega\rangle$ is the sequence of elements of $P_\kappa\lambda$ such that for each $n<\omega$, there is an $A\in U$ with $(P_1,\ldots,P_n,A)\in G$.
By Lemma \ref{lemmaprikryforcing}, it follows that in $V[G]$, the cofinality of $\kappa$ is $\omega$, and every ordinal in the interval $(\kappa,\lambda]$ has size $\kappa$. Furthermore, since the supercompact Prikry forcing adds no new bounded subsets to $\kappa$, it follows that $\kappa$ remains a cardinal in $V[G]$. We will now define a symmetric inner model $N\subseteq V[G]$ in which $\kappa^+=\lambda$, and we will argue that the conclusions of Theorem \ref{maintheorem} hold in $N$.
%\marginpar{\tiny Let us make a note concerning the chain condition of $\P$. Since any two conditions with the same stem are compatible it follows that any antichain of $\P$ has size at most $|P_\kappa\lambda|$. We assumed that $\cf^V(\lambda)<\kappa$ and from this it follows that $|P_\kappa\lambda|=\lambda^{<\kappa}=\lambda^+$. Thus $\P$ is $\lambda^{++}$-c.c. and hence may collapse $\lambda^+$. Regardless of this, $\lambda^+$ will be a cardinal in the symmetric inner model $N$ we will define below.}
In order to define $N$, we need to discuss a way of restricting
the forcing conditions in $\P$. First note that,
as in \cite{Apter:SuccessorsOfSingularCardinalsAnd},
for $\delta\in[\kappa,\lambda]$ a regular cardinal,
$U\restrict\delta:=U\cap P(P_\kappa\delta)$ is a normal fine measure on
$P_\kappa\delta$ satisfying the Menas partition property.
%\marginpar{\tiny In \cite[Proof of Theorem 1]{Apter:SuccessorsOfSingularCardinalsAnd}
%you say $\hat{U}$ can be chosen on $P_\kappa(2^\lambda)$
%so that $U=\hat{U}\restrict\lambda$ has the Menas partition proerty.
%So, does our $U\restrict\delta$ really have the Menas partition property?}
Let $\P_{U\restrict\delta}$ denote the supercompact Prikry forcing associated with $U\restrict\delta$. If $p=\la Q_1,\ldots Q_n,A\ra\in \P$ we define $p\restrict\delta:=\la Q_1\cap\delta,\ldots,Q_n\cap\delta,A\cap P_\kappa\delta\ra$ and note that $p\in \P_{U\restrict\delta}$. If $A\in P_\kappa\lambda$ we define $A\restrict\delta:=A\cap P_\kappa\delta$. The Mathias genericity criterion \cite{Mathias:OnSequencesGenericInTheSenseOfPrikry} for supercompact Prikry forcing yields that $r_\delta:=\langle P_n\cap\delta\mid n<\omega\rangle$ generates a $V$-generic filter for $\P_{U\restrict\delta}$. Indeed, $G\restrict\delta:=G\cap \P_{U\restrict\delta}$ is the generic filter for $\P_{U\restrict\delta}$ generated by $r_\delta$. $N$ is now defined informally as the smallest model of $\ZF$ extending $V$ which contains $r_\delta$ for each regular cardinal $\delta\in[\kappa,\lambda)$ but not the full supercompact Prikry sequence $r:=\langle P_n\mid n<\omega\rangle$.
We may define $N$ more formally as follows. Let $\mathcal{L}$ be the forcing language associated with $\P$ and let $\mathcal{L}_1\subseteq\mathcal{L}$ be the ramified sublanguage containing symbols $\check{v}$ for each $v\in V$, a unary predicate $\check{V}$ (interpreted as $\check{V}(\check{v})$ if and only if $v\in V$), and symbols $\dot{r}_\delta$ for each regular cardinal $\delta\in[\kappa,\lambda)$. We define $N$ inductively inside $V[G]$ as follows:
\begin{align*}
N_0&=\emptyset, \\
N_\delta&=\bigcup_{\alpha<\delta}N_\alpha\textrm{ for $\delta$ a limit ordinal}, \\
N_{\alpha+1}&=\{x\subseteq N_\alpha\mid \textrm{$x$ can be defined over $\langle N_\alpha,\in,c\rangle_{c\in N_\alpha}$} \\
& \hspace{1in}\textrm{using a forcing term $\tau\in\mathcal{L}_1$ of rank $\leq\alpha$}\}, \textrm{ and}\\
N&=\bigcup_{\alpha\in\ORD} N_\alpha.
\end{align*}
Standard arguments show that $N\models \ZF$.
As usual, each $\check{v}$ for $v\in V$ may be chosen so
as to be invariant under %any automorphism of $\P_U$.
any isomorphism $\Psi : \P/p \to \P/q$ for
$p, q \in \P$.
Further, terms $\tau$ mentioning only $\dot{r}_\delta$ may be chosen so
as to be invariant under %any automorphism of $\P_U$ which preserves
any isomorphism $\Psi : \P/p \to \P/q$ which preserves
the meaning of $r_\delta$.
The following lemma provides the key to showing that $N$ has the desired features.
\begin{lemma}\label{keylemma}
If $x\in N$ is a set of ordinals, then for some regular cardinal $\delta\in [\kappa,\lambda)$, $x\in V[r_\delta]$.
\end{lemma}
\begin{proof}
Let us note that the following proof of Lemma \ref{keylemma} blends ideas found in the proofs of \cite[Lemma 1.5]{Apter:SuccessorsOfSingularCardinalsAnd} and \cite[Lemma 2.1]{ApterHenle:RelativeConsistencyResultsViaStrongCompactness}. Let $\tau$ be a term in $\mathcal{L}_1$ for $x$. Suppose $\beta$ is an ordinal, $p\forces_{\P,V}\tau\subseteq\beta$, and $p\in G$. Since $\tau\in\mathcal{L}_1$, it follows that $\tau$ mentions finitely many terms of the form $\dot{r}_\delta$. Without loss of generality, we may assume that $\tau$ mentions a single $\dot{r}_\delta$. We will show that $x\in V[r_\delta]$. Let
$$y:=\{\alpha<\beta\mid\exists q\leq p\ (q\restrict\delta\in G\restrict\delta \textrm{ and } q\forces_{\P,V}\alpha\in\tau)\}.$$
We will show that $x=y$. Since it is clear that $y\in V[r_\delta]$, this will suffice. Suppose $\alpha\in x$, and choose $p'\leq p$ with $p'\in G$ such that $p'\forces_{\P,V}\alpha\in \tau$. Since $p'\restrict \delta\in G\restrict \delta$, we conclude that $\alpha\in y$. Thus, $x\subseteq y$. Now suppose $\alpha\in y$, and let $q\leq p$ with $q\restrict\delta\in G\restrict\delta$ and $q\forces_{\P,V}\alpha\in\tau$. There is a $q'\in G$ such that $q'\decides\alpha\in\tau$. If $q'\forces\alpha\in\tau$, then $\alpha\in x$ and we are done; thus we assume that $q'\forces\alpha\notin\tau$. Write $q=\langle Q_1,\ldots,Q_l, A\rangle$ and $q'=\langle Q_1',\ldots, Q'_m, A'\rangle$, where without loss of generality we assume that $l\kappa$ to an uncountable cardinal, and also preserves enough of the original large cardinal properties of $\kappa$ to allow these collapses to occur. As pointed out by the referee of this paper, by the work of Woodin
\cite{Woodin:SuitableExtenderModelsOne} on inner models for supercompact cardinals, it appears as though this is impossible.
% This suggests that one start with two large cardinals, say $\kappa<\lambda$, and then blow up the size of the power set of $\lambda$, while preserving enough of the large cardinal properties of $\kappa$ and $\lambda$ so as to be able to change the cofinality of $\lambda$ to some uncountable cardinal and to change the cofinality of $\kappa$, while simultaneously collapsing all cardinals in the interval $(\kappa,\lambda)$ to $\kappa$.
\begin{thebibliography}{Mat73}
\bibitem[ADK]{ApterDimitiouKoepke:TheFirstMeasurableCardinalCanBe}
Arthur~W. Apter, Ioanna~Matilde Dimitr\'{i}ou, and Peter Koepke.
\newblock The first measurable cardinal can be the first uncountable regular
cardinal at any successor height.
\newblock {\em \emph{Submitted to the}
%The Mathematical Logic Quarterly
Mathematical Logic Quarterly}.
\bibitem[AH91]{ApterHenle:RelativeConsistencyResultsViaStrongCompactness}
Arthur~W. Apter and James~M. Henle.
\newblock Relative consistency results via strong compactness.
\newblock {\em Fundamenta Mathematicae}, 139(2):133--149, 1991.
\bibitem[Apt83]{Apter:SomeResultsOnConsecutiveLargeCardinals}
Arthur~W. Apter.
\newblock Some results on consecutive large cardinals.
\newblock {\em Annals of Pure and Applied Logic}, 25(1):1--17, 1983.
\bibitem[Apt85]{Apter:SuccessorsOfSingularCardinalsAnd}
Arthur~W. Apter.
\newblock Successors of singular cardinals and measurability.
\newblock {\em Advances in Mathematics}, 55(3):228--241, 1985.
\bibitem[Apt96]{Apter:ADAndPatterns}
Arthur~W. Apter.
\newblock {AD} and patterns of singular cardinals below {$\Theta$}.
\newblock {\em Journal of Symbolic Logic}, 61(1):225--235, 1996.
\bibitem[Bul78]{Bull:SuccessiveLargeCardinals}
Everett~L. Bull.
\newblock Successive large cardinals.
\newblock {\em Annals of Mathematical Logic}, 15(2):161--191, 1978.
\bibitem[CFM01]{CumForMag:SquareScales}
James Cummings, Matthew Foreman, and Menachem Magidor.
\newblock Squares, scales and stationary reflection.
\newblock {\em Journal of Mathematical Logic}, 1(1):35--98, 2001.
\bibitem[Git80]{Gitik:AllUncountableCardinalsCanBeSingular}
Moti Gitik.
\newblock All uncountable cardinals can be singular.
\newblock {\em Israel Journal of Mathematics}, 35(1-2):61--88, 1980.
\bibitem[Git85]{Gitik:RegularCardinalsInModelsOfZF}
Moti Gitik.
\newblock Regular cardinals in models of {ZF}.
\newblock {\em Transactions of the American Mathematical Society},
290(1):41--68, 1985.
\bibitem[Git02]{Gitik:BlowingUpPowerOfASingularCardinal}
Moti Gitik.
\newblock Blowing up power of a singular cardinal--wider gaps.
\newblock {\em Annals of Pure and Applied Logic}, 116(1-3):1--38, 2002.
\bibitem[Git10]{Gitik:Handbook}
Moti Gitik.
\newblock Prikry-type forcings.
\newblock In Akihiro Kanamori and Matthew Foreman, editors, {\em Handbook of
Set Theory}, volume~2, chapter~14, pages 1351---1447. Springer, 2010.
\bibitem[Jec03]{Jech:Book}
Thomas Jech.
\newblock {\em Set Theory: The Third Millennium Edition, revised and expanded}.
\newblock Springer, 2003.
\bibitem[Lav78]{Laver:MakingSupercompactnessIndestructible}
Richard Laver.
\newblock Making the supercompactness of $\kappa$ indestructible under
$\kappa$-directed closed forcing.
\newblock {\em Israel Journal of Mathematics}, 29(4):385---388, 1978.
\bibitem[Mat73]{Mathias:OnSequencesGenericInTheSenseOfPrikry}
Adrian~R.~D. Mathias.
\newblock On generic sequences in the sense of {P}rikry.
\newblock {\em Journal of the Australian Mathematical Society}, 15:409--414,
1973.
\bibitem[Men76]{Menas:Paper}
Telis~K. Menas.
\newblock A combinatorial property of $P_\kappa\lambda$.
\newblock {\em Journal of Symbolic Logic}, 41(1):225--234, 1976.
\bibitem[SRK78]{SolovayReinhardtKanamori}
Robert~M. Solovay, William~N. Reinhardt, and Akihiro Kanamori.
\newblock Strong axioms of infinity and elementary embeddings.
\newblock {\em Annals of Mathematical Logic}, 13(1):73--116, 1978.
\bibitem[Ste10]{Steel:OutlineOfInnerModelTheory}
John~R. Steel.
\newblock An outline of inner model theory.
\newblock In Akihiro Kanamori and Matthew Foreman, editors, {\em Handbook of
Set Theory}, volume~3, chapter~19, pages 1595--1684. Springer, 2010.
\bibitem[Woo10]{Woodin:SuitableExtenderModelsOne}
W.~Hugh Woodin.
\newblock Suitable extender models I.
\newblock {\em Journal of Mathematical Logic}, 10(1-2):101--339, 2010.
\end{thebibliography}
\end{document}
%=======================================
% I MIGHT WANT THIS LATER IF WE WANT TO GENERATE A NEW BIBLIOGRAPHY
%=======================================
\bibliography{masterbib}
\bibliographystyle{alpha}
\begin{comment}
\begin{thebibliography}{Mat73}
\bibitem[ADK]{ApterDimitiouKoepke:TheFirstMeasurableCardinalCanBe}
Arthur~W. Apter, Ioanna~Matilde Dimitr\'{i}ou, and Peter Koepke.
\newblock The first measurable cardinal can be the first uncountable regular
cardinal at any successor height.
\newblock {\em \emph{Submitted to the}
%The Mathematical Logic Quarterly
Mathematical Logic Quarterly}.
\bibitem[AH91]{ApterHenle:RelativeConsistencyResultsViaStrongCompactness}
Arthur~W. Apter and James~M. Henle.
\newblock Relative consistency results via strong compactness.
\newblock {\em Fundamenta Mathematicae}, 139(2):133--149, 1991.
\bibitem[Apt83]{Apter:SomeResultsOnConsecutiveLargeCardinals}
Arthur~W. Apter.
\newblock Some results on consecutive large cardinals.
\newblock {\em Annals of Pure and Applied Logic}, 25(1):1--17, 1983.
\bibitem[Apt85]{Apter:SuccessorsOfSingularCardinalsAnd}
Arthur~W. Apter.
\newblock Successors of singular cardinals and measurability.
\newblock {\em Advances in Mathematics}, 55(3):228--241, 1985.
\bibitem[Apt96]{Apter:ADAndPatterns}
Arthur~W. Apter.
\newblock {AD} and patterns of singular cardinals below {$\Theta$}.
\newblock {\em Journal of Symbolic Logic}, 61(1):225--235, 1996.
\bibitem[Bul78]{Bull:SuccessiveLargeCardinals}
Everett~L. Bull.
\newblock Successive large cardinals.
\newblock {\em Annals of Mathematical Logic}, 15(2):161--191, 1978.
\bibitem[Git80]{Gitik:AllUncountableCardinalsCanBeSingular}
Moti Gitik.
\newblock All uncountable cardinals can be singular.
\newblock {\em Israel Journal of Mathematics}, 35(1-2):61--88, 1980.
\bibitem[Git85]{Gitik:RegularCardinalsInModelsOfZF}
Moti Gitik.
\newblock Regular cardinals in models of {ZF}.
\newblock {\em Transactions of the American Mathematical Society},
290(1):41--68, 1985.
\bibitem[Git02]{Gitik:BlowingUpPowerOfASingularCardinal}
Moti Gitik.
\newblock Blowing up power of a singular cardinal--wider gaps.
\newblock {\em Annals of Pure and Applied Logic}, 116(1-3):1--38, 2002.
\bibitem[Git10]{Gitik:Handbook}
Moti Gitik.
\newblock Prikry-type forcings.
\newblock In Akihiro Kanamori and Matthew Foreman, editors, {\em Handbook of
Set Theory}, volume~2, chapter~14, pages 1351---1447. Springer, 2010.
\bibitem[Jec03]{Jech:Book}
Thomas Jech.
\newblock {\em Set Theory: The Third Millennium Edition, revised and expanded}.
\newblock Springer, 2003.
\bibitem[Lav78]{Laver:MakingSupercompactnessIndestructible}
Richard Laver.
\newblock Making the supercompactness of $\kappa$ indestructible under
$\kappa$-directed closed forcing.
\newblock {\em Israel Journal of Mathematics}, 29(4):385---388, 1978.
\bibitem[Mat73]{Mathias:OnSequencesGenericInTheSenseOfPrikry}
Adrian~R.~D. Mathias.
\newblock On generic sequences in the sense of {P}rikry.
\newblock {\em Journal of the Australian Mathematical Society}, 15:409--414,
1973.
\bibitem[Men76]{Menas:Paper}
Telis~K. Menas.
\newblock A combinatorial property of $P_\kappa\lambda$.
\newblock {\em Journal of Symbolic Logic}, 41(1):225--234, 1976.
\bibitem[Ste10]{Steel:OutlineOfInnerModelTheory}
John~R. Steel.
\newblock An outline of inner model theory.
\newblock In Akihiro Kanamori and Matthew Foreman, editors, {\em Handbook of
Set Theory}, volume~3, chapter~19, pages 1595--1684. Springer, 2010.
\end{thebibliography}
\end{comment}
\end{document}