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$\ $ \vskip 1.5in
\centerline{``Some New Upper Bounds in Consistency Strength for
Certain Choiceless Large Cardinal Patterns''}
\vskip .5in
\centerline{by}
\vskip .5in
\centerline{Arthur W. Apter*}
\centerline{Department of Mathematics}
\centerline{Baruch College of CUNY}
\centerline{New York, New York 10010}
\vskip 1.75in
\noindent Abstract: In this paper, we show that certain
choiceless models of $ZF$ originally constructed using an almost
huge cardinal can be constructed using cardinals strictly weaker
in consistency strength. \hfil\break\break
\noindent *The research for this paper was partially supported
by NSF Grant DMS - 8616774. In addition, the author would like to
thank the referee for comments which considerably helped improve
the presentation of the material of this paper. \hfil\break
\vfill\eject
\vskip 1in
\centerline{``Some New Upper Bounds in Consistency Strength for
Certain Choiceless Large Cardinal Patterns''}
\vskip .5in
\centerline{by}
\vskip .5in
\centerline{Arthur W. Apter}
\centerline{Department of Mathematics}
\centerline{Baruch College of CUNY}
\centerline{New York, New York 10010}
\vskip .75in
\indent Ever since the startling and ground breaking results of
Martin, Steel, and Woodin (see [MS1] and [MS2]) that $AD$ is
equiconsistent with $\om$ many Woodin cardinals, a fundamental
goal of those engaged in research in large cardinals and forcing
has been to try and reduce the strength of axioms used in
establishing relative consistency results, thereby in effect
providing a new upper bound in consistency strength for the
conclusion established. Good examples of this are provided by
the work done on the Singular Cardinals Hypothesis
($SCH$). Originally, Magidor showed in [Ma1] and [Ma2] that
from $GCH$ and a cardinal $\kappa$ which is $\kappa^{++}$ supercompact
it is possible to force and construct a model in which $\al_\om$
is a strong limit cardinal and $2^{\al_\om}=\al_\omp2$ and
from cardinals $\kappa < \l$ so that $\kappa$ is supercompact and
$\l$ is huge it is possible to force and construct a model in
which $GCH$ holds below $\al_\om$ and $2^{\al_\om}=
\al_\omp2$. Later, Woodin showed (unpublished; see [FW] and
[C1] for related results) that one could produce models for
both of the above versions of $SCH$ using a cardinal no
stronger than a $p^2(\kappa)$ - hypermeasurable. Thus, Woodin
was able to reduce quite drastically the upper bounds in
consistency strength for both of the previously mentioned
versions of $SCH$. Lower bounds in consistency strength for
these versions of $SCH$ on the order of measurable cardinals of
high Mitchell order had earlier been provided by Mitchell (see
[Mi1] and [Mi2]). \hfil\break
\indent The purpose of this paper is to note that the hypotheses
originally used to produce certain models of $ZF+\neg AC$ can
be reduced using the extender technology expounded in [MS1] and
[MS2]. Specifically, from an almost huge cardinal $\k0$ (where
$\k0$ is almost huge iff there is a \break $j:V \to M$ with crit$(j)
=\k0$ so that $M^{ \om$
is regular iff $\kappa$ is measurable''. \hfil\break
\indent We show that all of the above models can be constructed
using cardinals strictly weaker in consistency strength than an
almost huge cardinal. Specifically, for Theorems 1 and 3, we
use a cardinal $\kappa$ satisfying the following properties.
\hfil\vskip .09in
\noindent 1. $\kappa$ is superstrong via $j$, i.e., there is
a $j:V \to M$ with crit$(j) = \kappa$ so that $V_{j(\kappa)}
\subseteq M$. \hfil\vskip .09in
\noindent 2. $\kappa$ is $ \kappa$ which is $< j(\kappa)$ supercompact. \hfil\vskip .09in
\noindent For Theorem 2, we require in addition to the above
that it is possible to force and preserve these properties to
obtain a model in which the $< j(\kappa)$ supercompactness of
$\kappa$ is indestructible under $\kappa$ - directed closed
forcings $P$ so that $\vert TC(P) \vert < j(\kappa)$. We will
comment more on this property in the conclusions to the paper.
\hfil\break
\indent Note that properties 1 and 2 above imply that the
cardinal $\ks$ exists. To see this, observe that since
$V_{j(\kappa)} \subseteq M$, $M \models ``\kappa$ is $<
j(\kappa)$ supercompact'', so by reflection, $S = \{\d <
\kappa : \d$ is $< \kappa$ supercompact$\}$ is unbounded in
$\kappa$. By elementariness, $\{\d < j(\kappa) : \d$ is
$< j(\kappa)$ supercompact$\}$ is unbounded in $M$ in
$j(\kappa)$, meaning that $M \models ``$There is a least
cardinal $\ks > \kappa$ below $j(\kappa)$ which is $<
j(\kappa)$ supercompact''. The fact that $V_{j(\kappa)}
\subseteq M$ again implies that $\ks$ exists in $V$ and is
the same cardinal as in $M$. \hfil\break
\indent Our proof is divided into two parts, namely first
showing that if $\k0$ is almost huge, then $\k0$ has a normal
measure concentrating on cardinals $\kappa$ with the above
properties, and then showing that such cardinals suffice to
construct the models previously described. The first part of
the proof uses the aforementioned extender technology, for
which our references are [MS1], [MS2], [An], and
[C1]. Specifically, let $j_0 : V \to M$ witness the almost
hugeness of $\k0$, and let $T \subseteq M$ be a transitive
set so that $\k0 \in T$. Fix maps $p:{[T]}^{< \om} \to HF$ and
$\pi : {[T]}^{< \om} \to V$ depending on $T$ so that $\pi(a) :
\la a, \in \ra \to \la p(a), \in \ra$ is an
$\in$ isomorphism, i.e., $x \in a$, $y \in a$, and $x \in
y$ iff $\pi(a)(x) \in \pi(a)(y)$. Further, for any $a \in
{[T]}^{< \om}$, define $\theta(a)$ as the least $\theta
\ge \c$ so that $j(\theta) \ge$ rank($a$). This enables us to
define ultrafilters $E_a$ over ${[V_{\theta(a)}]}^{p(a)}$ by
$X \in E_a$ iff ${(\pi(a))}^{-1} \in j_0(X)$. $E =
\la \la E_a : a \in {[T]}^{< \om} \ra, \pi \ra$ is the
$(\k0, T)$ extender derived from $j_0$. When we compute the
ultrapower by $E$, the model we get is just the transitive
collapse of $\{j(F)(a) : a \in {[T]}^{< \om}$ and dom$(F) =
V^{\vert a \vert}_{\theta(a)}\}$ which is an elementary
substructure of $M$ containing $T$. \hfil\break
\noindent Lemma: Let $\mu$ be the normal measure over $\k0$
generated by $j_0$. Then $\{\c < \k0 :$ There is a
$\jk : V \to M_\c$ with crit$(\jk) = \c$ and $V_{\jk(\c)}
\subseteq M_\c$, $\c$ is a $< \jk(\c)$ supercompact cardinal,
$V_{\jk(\ks + 1)} \subseteq M_\c$, and it is possible to force
and preserve these properties to obtain a model in which the
embedding $\jk$ extends to $j^*_\c$ and in which the
$< j^*_\c(\c)$ supercompactness of $\c$ is indestructible under
$\c$ - directed closed forcings $P$ so that $\vert TC(P) \vert
< j^*_\c(\c)\} \in \mu$. \hfil\break
\indent Proof of Lemma: The proof of this Lemma uses the usual
ideas. First, note that the almost hugeness of $\k0$ implies
that the previous argument giving the existence of $\ks$ works
here as well and gives the existence of $\c^*_0$. Next, since
$j_0 \vert V_{\c^*_0 + 1}$ is a $p(\c^*_0)$ sequence, the
closure properties of $M$ ensure that $j_0 \vert
V_{\c^*_0 + 1} \in M$. Further, for $T = V^M_{j_0(\c^*_0 + 1)}$,
the map $\pi : {[T]}^{< \om} \to M$ can be taken as an element of
$M$. Thus, since $\theta(a) \le \c^*_0 + 2$ for each $a \in
{[T]}^{< \om}$, the closure properties of $M$ ensure that the
definition of $E_a$ for each $a \in {[T]}^{< \om}$ can be given
in $M$, meaning that the definition of the extender $E =
\la \la E_a : a \in {[T]}^{< \om} \ra, \pi \ra$ can be given in
$M$, i.e., $M \models ``E$ is a $(\k0, V_{j_0(\c^*_0 + 1)})$
extender''. \hfil\break
\indent Let now $k:M \to Ult(M,E)$ be the elementary embedding
generated by the ultrapower of $M$ via $E$. $M$ and $Ult(M,E)$
agree through rank $k(\c^*_0 + 1) = j_0(\c^*_0 + 1)$; in
particular, $j_0 \vert V_{\c^*_0 + 1} = k \vert
V_{\c^*_0 + 1}$ so $j_0(\c_0) = k(\c_0)$ and $j_0(\c^*_0) =
k(\c^*_0)$. Hence, since the closure properties of $M$ imply
that $M \models ``\c_0$ is $< j_0(\c_0)$ supercompact'',
working in $M$ we can construct the elementary embedding $k$
from $M$ into some transitive inner model $M'$ so that
$M'$ and $M$ agree through rank $k(\c^*_0 + 1) =
j_0(\c^*_0 + 1)$ and $M \models ``\c_0$ is $< k(\c_0)$
supercompact''. Further, in [Ap2] it is shown using an
argument of Gitik that since $\c_0$ is an almost huge
cardinal there is a notion of forcing $Q$ definable as a
proper class in $V_{j_0(\c_0)}$ so that for $G$ $V$ - generic
over $Q$ there is an $H \in V[G]$ $M$ - generic over
$j_0(Q) - Q$ satisfying that $j_0$ extends to $j^*_0 : V[G]
\to M[G*H]$ where $j^*_0$ witnesses the the almost hugeness of
$\c_0$ in $V[G]$, $M[G*H]$ is closed under $< j^*_0(\c_0)$ ($=
< j_0(\c_0)$) sequences, $V \models ``\d$ is $< j_0(\c_0)$
supercompact'' iff $V[G] \models ``\d$ is $ 0$, $\a < \c^+$ iff $\la \mu_\b : \b < \a \ra
=_{\rm df} \mu_{< \a} \in j(X)$, and $\mu_{< \c^+} =
\la \mu_\a : \a < \c^+ \ra$. Since $\la j(\a) : \a <
\ks \ra \subseteq V_{j(\ks)}$ and $V_{j(\ks + 1)} \subseteq
M$, the definition of $\mu_0$ makes sense, and since
$\la \mu_\b : \b < \a \ra$ for $\a > 0$, $\a < \c^+$ is a
sequence of measures over $V_\ks$, the fact that
$V_{j(\ks + 1 )} \subseteq M$ (much less is actually needed)
implies that the definition of $\mu_{< \a}$ and
$\mu_{< \c^+}$ both make sense. \hfil\break
\indent Let now $R_{< \c^+}$ be supercompact Radin forcing over
$P_\c(\ks)$ of length $\c^+$ defined using $j$. For the
specifics of the definition, we refer the reader to [G]; we
note only that since $M$ contains enough of $V$ to allow
$\ks$ and $R_{< \c^+}$ to be defined, reflection shows that for
each element $\la p, u, a \ra$ of a Radin condition in
$R_{< \c^+}$ where $p \in P_\c(\ks)$, $u$ is the appropriate
Radin sequence of measures, and $A$ is the associated sequence
of measure 1 sets, $\overline p$ (the order type of $p$) is the
least $< \c$ supercompact cardinal $> \overline{p \cap \c}$,
which is itself a $< \c$ supercompact cardinal. This
property of $R_{< \c^+}$ then allows the model of [G] to be
constructed in the same manner and satisfy the conclusions of
Theorem 1. \hfil\break
\indent To see that such a $\c$ suffices in the construction of
the model for Theorem 3, we note that the proof of Theorem 3 in
[Ap1] first lets $\c_1$ be the least element of the set $S =
\{\d < \c : \d$ is $< \c$ supercompact$\}$ and then does the
construction of [G] so that in the model $M$ formed, all
cardinals $\ge$ a certain cardinal $\a$ are singular and the
bounded subsets of $\a$ are the same as those in $V$, thereby
ensuring that $\c_1$ remains $< \a$ supercompact in $M$. As
just observed, a $\c$ with the first three properties described
in the Lemma suffices in the construction of $M$. The proof in
[Ap1] then proceeds by forcing over $M$ and then forming the
appropriate inner model $N$ using the product of a supercompact
Radin forcing present in $V$ with $Col(\om, <\c_1)$ (the
L\'evy collapse which turns $\c_1$ into $\al_1$) to collapse
$\c_1$ to $\al_1$, make $\al_1$ measurable via the club filter,
and collapse $\a$ to $\c^+_1$ (thereby ensuring that all
cardinals $\ge \al_2$ in $N$ are singular). Since all that is
used in [Ap1] for this part of the proof is that $\c_1$
remains $< \a$ supercompact in $M$ via the same measures as in
$V$, this portion of the proof goes through as well, allowing
the model of [Ap1] to be constructed using a $\c$ with the
first three properties of the Lemma. \hfil\break
\indent Let now $\c$ satisfy the fourth property described
earlier as well as the first three. By this property, we can
force and obtain $j^* : V \to M$ so that crit$(j^*) = \c$ and
$V_{j^*(\c)} \subseteq M$, $V_{j^*(\ks + 1)} \subseteq M$ as
well, and $\c$ is a $< j^*(\c)$ supercompact cardinal whose
$< j^*(\c)$ supercompactness is indestructible under
$\c$ - directed closed forcings $P$ so that $\vert
TC(P) \vert < j^*(\c)$. As in [Ap2], this property reflects
so that $C = \{ \d < \c : \d$ is $< \c$ supercompact and the
$< \c$ supercompactness of $\d$ is indestructible under
$\c$ - directed closed forcings $P$ so that $\vert TC(P) \vert
< \c \} \in \mu$ for the $\mu$ on $\c$ generated by $j^*$. Our
earlier arguments show that since the construction of [G] can
be done with this $j^*$, and since this construction together
with $C \in \mu$ ensures the additional large cardinal
hypotheses on the cardinals of $N_A$, such a $\c$ suffices in the
construction of the model for Theorem 2. \hfil\break
\indent In conclusion, we remark that there is nothing a priori
which ensures that a cardinal which satisfies the properties
used to establish Theorems 1 and 3 will in addition automatically
satisfy the property needed to establish Theorem 2. There are,
however, some results concerning weak indestructibility properties
of strong cardinals. The reader is referred to [C2] and [GS]
for further details. \hfil\break\break
\noindent Acknowledgement: The author wishes to thank
Alessandro Andretta for several helpful conversations on the
subject of extenders. \hfil\break\vfill\eject
\frenchspacing\vskip 1in
\centerline{References}
\vskip 1.25in
\item{[An]} Andretta, A.: Handwritten notes on extenders.
\hfil\break
\item{[Ap1]} Apter, A.: On a problem inspired by
Determinacy. {\it Israel J. Math. 61}, 256-270(1988) \hfil\break
\item{[Ap2]} Apter, A.: Some results on consecutive large
cardinals II: Applications of Radin forcing.\break {\it Israel
J. Math. 52}, 273-292(1985) \hfil\break
\item{[C1]} Cummings, J.: A model in which GCH holds at
successors but fails at limits. {\it Transactions
A.M.S.} (to appear) \hfil\break
\item{[C2]} Cummings, J.: Forcing and the arithmetic of
small cardinals. (to appear) \hfil\break
\item{[FW]} Foreman, M. and Woodin, H.: The GCH can fail
everywhere. {\it Annals of Math. 133},\break 1-36(1991) \hfil\break
\item{[G]} Gitik, M.: Regular cardinals in models of
ZF. {\it Transactions A.M.S. 290}, 41-68(1985) \hfil\break
\item{[GS]} Gitik, M. and Shelah, S.: On certain
indestructibility of strong cardinals and a question of
Hajnal. {\it Arkiv. f\"ur Math. Logik 28}, 35-42(1989)
\hfil\break
\item{[Ma1]} Magidor, M.: On the Singular Cardinals Problem
I. {\it Israel J. Math. 28}, 1-31(1977) \hfil\break
\item{[Ma2]} Magidor, M.: On the Singular Cardinals Problem
II. {\it Annals of Math. 106}, 517-549(1977) \hfil\break
\item{[Mi1]} Mitchell, W.: Sets constructed from sequences of
measures revisited. {\it J. Symbolic Logic 48}, 600-609(1983)
\hfil\break
\item{[Mi2]} Mitchell, W.: The core model for sequences of
measures I. {\it Proceedings of the Cambridge Philosophical
Soc. 95}, 229-260(1984) \hfil\break
\item{[MS1]} Martin, D.A. and Steel, J.: A proof of
Projective Determinacy. {\it J.A.M.S. 2}, 71-125(1989)
\hfil\break
\item{[MS2]} Martin, D.A. and Steel, J.: Projective
Determinacy. {\it Proceedings Nat. Acad. Sci. U.S.A. 85},
6582-6586(1988) \hfil\break\vfill\eject\end