\voffset=-.15in
\magnification=1200
\vsize=7.65in
\tolerance = 100000
\def\forces{\hbox{$\|\hskip -2pt \hbox{--}$\hskip 2pt}}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
\hrule height.#2pt}}}}
\def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3}
\def\finpf{$\parindent=4.5in\indent\square\ \parindent=20pt$}
\def\hb{\hfil\break}
\def\a{ \alpha }
\def\b{ \beta }
\def\g{ \gamma }
\def\f{ \varphi }
\def\d{ \delta }
\def\l{ \lambda }
\def\la{\langle}
\def\ra{\rangle}
\def\no{\noindent}
\def\un{\underline}
\def\ov{\overline}
\def\P{\cal P}
\def\U{\cal U}
\def\H{\cal H}
\def\L{\cal L}
\def\K{\cal K}
\def\Ud{{\cal U} \vert \delta}
\def\pud{{\cal P}_{{\cal U} \vert \delta}}
\def\alom{\aleph_\omega}
\def\vrd{V[r \vert \delta]}
\def\vrg{V[r,g]}
\def\vrgd{V[r \vert \delta , g]}
\def\vrgd0{V[r \vert \delta_0, g]}
\def\keqal{\kappa = \aleph_\omega}
\def\pkl{P_\kappa(\lambda)}
\def\kintl{[\kappa, \lambda)}
\def\alomp1{\aleph_{\omega + 1}}
\def\chid{{[\chi]}_{\sim_\delta}}
\def\pvk{p^{V[r \vert \delta, g \vert \d]}(\kappa)}
\long\def\pic{ \la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra}
\def\pic0{\la p_1, \ldots, p_n, f_0, \ldots, f_n, A_0, F_0 \ra}
\def\fordc{\forall n < \omega [{\rm DC}_{\aleph_n}]}
\def\mag{ \subset_{{}_{{}_{\!\!\!\!\!\sim}}} }
$\ $ \vskip .75in
\centerline{``Instances of Dependent Choice and the Measurability of
$\alomp1$''}
\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 .25in
\centerline{and}
\vskip .25in
\centerline{Menachem Magidor}
\centerline{Department of Mathematics}
\centerline{The Hebrew University}
\centerline{Jerusalem, Israel}
\vskip 1in
\noindent Abstract: Starting from cardinals $\kappa < \l$ where
$\kappa$ is $2^\l$ supercompact and $\l>\kappa$ is measurable, we
construct a model for the theory ``ZF + $\fordc$ + $\alomp1$ is
a measurable cardinal''. This is the maximum amount of dependent
choice consistent with the measurability of $\alomp1$. \hfil
\vskip .25in
\noindent *The research of the first author was partially supported
by NSF Grant DMS-8616774 and PSC-CUNY Grant 661374. In
addition, the first author would like to thank the second author
and his family for all of the hospitality shown to him and his
wife during his sabbatical in Israel. \hb
\noindent AMS(MOS) Subject Classifications:
03E25, 03E35, 03E55.\hfil\break
\vfill\eject
\vskip 1in
\centerline{``Instances of Dependent Choice and the Measurability
of $\alomp1$''}
\vskip .5in
\centerline{by}
\vskip .5in
\centerline{Arthur W. Apter}
\vskip .25in
\centerline{and}
\vskip .25in
\centerline{Menachem Magidor}
\vskip .75in
\indent It is a well known theorem of Martin (see [Ke]) that
AD + DC $\vdash$ ``$\alomp1$ is a measurable cardinal''. Thus,
assuming the consistency of $\omega$ many Woodin cardinals, since
the results of [MS1] and [MS2] show that $L[R] \models$ AD + DC,
$L[R]$ becomes a canonical model for the theory ``ZF + DC
+ $\alomp1$ is a measurable cardinal'' under these circumstances.
\hfil\break
\indent A priori, there is no reason to think that
${\rm DC}_\omega$ is the maximum amount of dependent choice
consistent with the measurability of $\alomp1$, even though all
previous forcing constructions [A3], [A4], [AH] yielding models in
which $\alomp1$ is measurable also satisfied $\neg
{\rm AC}_\omega$. Indeed, the first choice principle outright
inconsistent with the measurability of $\alomp1$ is
${\rm AC}_{\alomp1}$. Thus, one wonders if it is possible to
construct a model in which $\alomp1$ is a measurable cardinal and
instances of dependent choice beyond ${\rm DC}_\omega$ are
true. \hfil\break
\indent The purpose of this paper is to show that it is possible to
construct a model in which $\alomp1$ is measurable and in which
the maximum amount of dependent choice not outright inconsistent
with the measurability of $\alomp1$ is true. Specifically, we
prove the following \hfil\vskip .2in\noindent
Theorem: Con(ZFC + There are cardinals $\kappa < \l$ so that
$\kappa$ is $2^\l$ supercompact and $\l$ is a measurable
cardinal) $\Longrightarrow$
Con(ZF + $\fordc$ + $\alomp1$ is a measurable cardinal).
\hfil\vskip .2in\noindent
Note that since $\aleph_\omega$ is a singular cardinal, $\fordc$ is
equivalent to ${\rm DC}_{\aleph_\omega}$ [J]. \hfil\break
\indent It will turn out that our Theorem can be established
using the weaker hypothesis of the existence of cardinals
$\kappa < \l$ so that $\kappa$ is $\l$ supercompact and $\l$
is measurable. Since we do not wish excessive technicalities
to obscure the proof of our Theorem, we will defer until the end
of the paper an explanation of how the Theorem can be
established using this weaker hypothesis.
\hfil\break
\indent The proof of our Theorem uses a modification of the forcing
conditions of [Mag] together with ideas used by
Kafkoulis in his thesis [Ka1]. Before beginning the proof of our
Theorem, however, we briefly mention some preliminary
information. Basically, our notation and terminology will be that
of [Mag], a paper with which we are assuming complete
familiarity. One difference we point out is that for $\g < \d$
regular cardinals, we use the notation ${\rm Col}(\g, <\d)$,
whereas [Mag] uses the notation ${\rm Col}(\g, \d)$, for the
L\'evy collapse of $\d$ to $\g^+$, i.e., for the partial ordering
$\{f:\g \times \d \to \d$ : $f$ is a function so that
$\vert {\rm dom}(f) \vert < \g$ and $f(\la \a, \b \ra)
< \b\}$ ordered by inclusion. For a condition
$p \in {\rm Col}(\g, <\d)$ and $\g<\sigma<\d$,
$p \vert \sigma = \{\la\la \a,\b \ra, \rho \ra \in
p : \b < \sigma\}$ is a condition in ${\rm Col}(\g, <\sigma)$,
and if $G$ is $V$-generic over ${\rm Col}(\g, <\delta)$, then
$G \vert \sigma = \{p \vert \sigma : p \in G\}$ is $V$-generic
over ${\rm Col}(\g, <\sigma)$. In addition, for $x$ a set,
$\ov x$ is the order type of $x$, and for $\a < \b$ ordinals,
$[\a,\b]$, $[\a,\b)$, $(\a,\b]$, and $(\a,\b)$ are as in
standard interval notation. $V_\a$ for $\a$ an ordinal will as
usual be all sets of rank $<\a$. \hb
\indent We are also assuming familiarity with the choice principles
${\rm AC}_\g$ and ${\rm DC}_\g$ where $\g$ is a (well-ordered)
cardinal. For further explanations of these principles and the
relationships amongst them, readers are urged to consult [J]. We
will just mention that ${\rm DC}_\omega$ and DC are used
synonymously.
\hfil\break
\indent We turn now to the proof of our Theorem. \hfil\break
\indent Proof of Theorem: Let $V \models$ ``ZFC + $\kappa$ is
$2^\l$ supercompact for $\l>\kappa$ a measurable cardinal''. As
in [A3], the fact that $\kappa$ is $2^\l$ supercompact for
$\l>\kappa$ a measurable cardinal implies there is a supercompact
ultrafilter $\U$ over $\pkl$ with the Menas partition property
[Me] so that $C_0 = \{p \in \pkl$ : $p \cap \kappa$ is a
measurable cardinal and $\ov p$ is the least measurable cardinal
$>$ $p \cap \kappa \} \in \U$. \hfil\break
\indent The forcing conditions $\P$ used in the proof of the
Theorem are the set of all finite sequences of the form
$\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
satisfying the following properties. \hfil\vskip .09in
\noindent
1. Each $p_i$ for $1 \le i \le n$ is an element of $C_0$, and
for $1 \le i < j \le n$, $p_i \mag p_j$, where as in [Mag],
$p_i \mag p_j$ means $p_i \subseteq p_j$ and $\ov{p_i} <
p_j \cap \kappa$. \hfil\vskip .09in
\noindent
2. $f_0 \in {\rm Col}(\omega_1, <\ov{p_1})$, for $1 \le i < n$,
$f_i \in {\rm Col}({\ov p}^+_i, < \ov{p_{i+1}})$, and
$f_n \in {\rm Col}({\ov p}^+_n, <\l)$. \hfil\vskip .09in
\noindent
3. $A \subseteq C_0$, $A \in \U$, and for every $q \in A$,
$p_n \mag q$ and the range and domain of $f_n$ are both subsets
of $q$, meaning that if $\la \la \a, \b \ra, \g \ra \in f_n$,
$\a,\b,\g \in q$.
\hfil\vskip .09in
\noindent
4. $F$ is a function defined on $A$ so that for $p \in A$,
$F(p) \in {\rm Col}({\ov p}^+, <\l) $, and if $q \in A$,
$p \mag q$, then the range and domain of $F(p)$ are both subsets
of $q$.
\hfil\break
\indent Before we can define the ordering on $\P$, we need to
define for $p,q \in A$, $p \mag q$ and $f \in
{\rm Col}({\ov p}^+, <\l)$ so that the range and domain of $f$
are subsets of $q$ the collapse of $f$ in $q$, denoted
$f^*_q$. Let $h : q \to {\ov q}$ be the unique order
isomorphism between $q$ and $\ov q$. Then $f^*_q : {\ov p}^+
\times {\ov q} \to {\ov q}$ is defined as $f^*_q(\la \a,
h^{-1}(\b) \ra) = h(f(\la \a, h^{-1}(\b) \ra))$ if
$h^{-1}(\b) \in q$. In other words, to define $f^*_q$ given
$f$, we transform using $h^{-1}$ the appropriate $\la \a,
\b \ra \in {\ov p}^+ \times {\ov q}$ into an element of
${\ov p}^+ \times \l$, apply $f$ to it, and collapse the result
using $h$. It is easily checked $f^*_q \in
{\rm Col}({\ov p}^+, < {\ov q})$. \hfil\break
\indent The ordering on $\P$ is essentially the same as given in
[Mag] modulo the above definition, i.e., if $\pi_0 =
\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra$
and $\pi_1 = \la q_1, \ldots, q_m, g_0, \ldots, g_m, B, H \ra$,
then $\pi_1$ extends $\pi_0$ iff the following conditions
hold. \hfil\vskip .09in
\noindent 1. $n \le m$, $p_i = q_i$ for $1 \le i \le n$, and
$q_i \in A$ for $n+1 \le i \le m$.
\hfil\vskip .09in\noindent
2. $f_i \subseteq g_i$ for $0 \le i < n$, and ${(f_n)}^*_{
q_{n+1}} \subseteq g_n$. If $n=m$, then $f_n \subseteq g_n$.
\hfil\vskip .09in\noindent
3. ${(H(q_i))}^*_{q_{i+1}} \subseteq g_i$ for $n+1 \le i < m$,
and $H(q_m) \subseteq g_m$.
\hfil\vskip .09in\noindent
4. $B \subseteq A$.
\hfil\vskip .09in\noindent
5. For every $p \in B$, $F(p) \subseteq H(p)$. \hfil\break
\indent Let $G$ be $V$-generic over $\P$. As in [Mag], we can define
sequences $r = \la p_i : i\in\omega-\{0\}\ra$ and $g = \la G_i : i <
\omega \ra$, where $p_i \in r$ iff $\exists \pi \in G [ p_i \in
\pi]$ and $G_i = \bigcup \{ f_i : \exists \pi \in G [\pi =
\la p_1, \ldots, p_n, f_0, \ldots, f_i, \ldots, f_n, A, F \ra
]\}$. These sequences will be well-defined by the genericity of
$G$. Further, as in [Mag], $\vrg \models ``\keqal$''. The proof
that this is so is essentially as in [Mag], pp. 11-19 and will
be expounded upon further following
the proof of Lemma 2. \hfil\break
\indent We are now in a position to describe the inner model
$N \subseteq V[G]$ which will witness the conclusions of our
Theorem. For $\d \in \kintl$, $\d$ inaccessible, let
$r \vert \d = \la p_i \cap \d : i \in \omega - \{0\} \ra$, and
let $g \vert \d = \la G^\d_i : i < \omega \ra$, where $G^\d_i
= G_i \vert {\ov {p_{i+1} \cap \d}}$. (If $f \in {\rm Col}(
\omega_1, < {\ov{p_1}})$ or $f \in {\rm Col}({\ov{p_i}}^+, <
{\ov{p_{i+1}}})$, then let $f^\d = f \vert {\ov{p_1 \cap \d}}$ or
$f^\d = f \vert {\ov{p_{i+1} \cap \d}}$.) Intuitively,
$N$ is the least model of ZF extending $V$ which contains, for
each inaccessible $\d \in \kintl$, the sequences $r \vert \d$
and $g\vert\d$. More formally, let ${\cal L}_1$ be the sublanguage of
the forcing language $\L$ with respect to $\P$ which contains
symbols $\dot v$ for each $v \in V$, a
unary predicate symbol $\dot V$ (to be interpreted ${\dot V}(
\dot v)$ iff $v \in V$), and for $\d \in \kintl$, $\d$
inaccessible, symbols ${\dot r} \vert \d$ for $r \vert \d$ and
${\dot g} \vert \d$ for $g\vert\d$. $N$ can then be defined inside $V[G]$
as follows. \hfil\break
\parindent =2.0in\indent $N_0 = \emptyset$. \hfil\vskip .09in
\parindent =2.0in\indent $N_{\a+1} = \{ x \subseteq N_\a : x$ is definable
by a term $\tau \in {\cal L}_1$ of rank $\le \a$ over the model
${\la N_\a, \in, c \ra}_{c \in N_\a}\}$. \hfil\vskip .09in
\parindent =2.0in\indent $N_\d = \bigcup_{\a < \d} N_\a$ for $\d$ a limit
ordinal. \hfil\vskip .09in
\parindent =2.0in\indent $N = \bigcup_{\a \in {\rm Ordinals}^V }N_\a$.
\hfil\vskip .09in \noindent The standard arguments show
$N \models {\rm ZF}$. \hfil\break\parindent=20pt
\noindent Lemma 1: $N \models ``\kappa = \alom$ and
$\l \le \kappa^+ = \alomp1$''. \hfil\break
\indent Proof of Lemma 1: Since by definition of $N$, $g\vert\d \in N$
for all inaccessible $\d\in\kintl$,
$N \models ``\kappa \le \alom$''. As $N \subseteq \vrg$ and
$\vrg \models ``\keqal$'', $N \models ``\keqal$''. \hfil\break
\indent As in [A3], since for $\d \in \kintl$, $\d$ inacessible,
$V[r \vert \d] \subseteq V[r \vert \d,g\vert\d] \subseteq N$ and
$V[r \vert \delta] \models ``\d$ is an
ordinal of cardinality $\kappa$'', $N \models ``\d$ is an ordinal of
cardinality $\kappa$'', i.e., $N \models ``\d$ is not a
cardinal''. As there are unboundedly many in $\l$ inaccessibles
in $\kintl$, $N \models ``\l \le \kappa^+$''. This proves
Lemma 1. \hfil\break\finpf Lemma 1\hfil\break
\indent We are now ready to state the main technical lemma used in
showing that $N \models ``\l = \alomp1$ and $\l$ is a measurable
cardinal''. \hfil\break
\noindent Lemma 2: Let $\pi =
\la p_1, \ldots, p_n, f_0, \ldots, f_n, A, F \ra \in \P$
and $\la \tau_\a : \a < \l \ra$ be a sequence of $\l$ statements
in the forcing language $\L$. There is then a condition $\pi' =
\la p_1, \ldots, p_n, f_0, \ldots, f_n, B, H \ra$ extending $\pi$
(in the terminology of [Mag], a length preserving extension of
$\pi$) so that if $\pi'' = \la p_1, \ldots, p_n, p_{n+1}, \ldots,
p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_m, C, I \ra$
extending $\pi' $ decides $\tau_\a$ and $\a \in p_{n+k}$ then
the condition $\pi''' = \la p_1, \ldots, p_n, p_{n+1}, \ldots,
p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_n, \break g_{n+1},
\ldots, g_{n+k}, H(p_{n+k+1}), \ldots, H(p_m), B',
H \vert B' \ra$ for $B' = \{ p \in B : p_m \mag p \}$ decides
$\tau_\a$ in the same way as well. \hfil\break
\indent Extending and abusing the terminology of [Mag], we will
refer to $\pi'''$ as the $k$-quasi-interpolant of $\pi'$ and
$\pi''$, written $k$-quint($\pi', \pi''$). In a further abuse of
notation, we will frequently, as was done in the statement of
Lemma 2, omit mentioning explicitly when we are taking the
collapse of a particular L\'evy collapse function $f$ in a
particular element $p \in \pkl$ when the context is clear. Lemma 2 essentially says
that any $\tau_\a$ can be decided above $\pi'$ by a finite
sequence of elements of $B$ and a finite sequence of elements taken
from the appropriate L\'evy collapses. \hfil\break
\indent Proof of Lemma 2: As in [Mag], Lemmas 2.7-2.8, it will
suffice to show that for any condition $\pi = \la p_1, \ldots,
p_n, f_0, \ldots, f_n, A, F \ra \in \P$ and any $k < \omega$
there is a condition $\pi_k = \la p_1, \ldots, p_n, f_0, \ldots,
f_n, A_k, F_k \ra$ extending $\pi$ so that for any $k$-sequence
$\la p_{n+1}, \ldots, p_{n+k} \ra$ of elements of $A_k$ (where a
0 sequence is the empty sequence), if $\pi_k' = \la p_1,
\ldots, p_n, p_{n+1}, \ldots, p_{n+k}, p_{n+k+1}, \ldots,
p_m, g_0, \ldots, g_m, C, I \ra$ extending $\pi_k$ decides
$\tau_\a$ and $\a \in p_{n+k}$ then $k$-quint($\pi_k,
\pi_k'$) decides $\tau_\a$ in the same way as well. This
suffices since we can then construct a sequence $\la \pi_k : k
< \omega \ra$ so that $\pi_0$ extends $\pi$, $\pi_{k+1}$
extends $\pi_k$, and each $\pi_k$ has the above mentioned form
and properties. The condition $\pi' = \la p_1, \ldots, p_n,
f_0, \ldots, f_n, B, H \ra$ for $B = \bigcap_{k < \omega}
A_k$ and $H(p)$ for $p \in B$ defined as $\bigcup_{k < \omega}
F_k(p)$ ($B \in \U$ by the additivity of $\U$ and $H(p)$ is
well defined by the fact each $H_k(p)$ is an element of a L\'evy
collapse which is at least ${\ov p}^+$ closed) then has the
desired properties since if $\pi_k'$ extending $\pi'$ is as
above and decides $\tau_\a$, then because $\pi'$ extends
$\pi_k$, $\pi_k'$ extends $\pi_k$, so $k$-quint($\pi_k,
\pi_k'$) decides $\tau_\a$ and is extended by $k$-quint($\pi',
\pi_k'$). We thus establish the earlier mentioned fact. As
in Lemma 2.8 of [Mag], we proceed by induction on $k$. We
consider two cases. \hfil\break
\noindent Case 1: $k=0$. In this case, we want to show that there
is a condition $\pi_0 = \la p_1, \ldots, p_n, f_0, \ldots,
f_n, A_0, F_0 \ra$ extending $\pi = \la p_1, \ldots, p_n,
f_0, \ldots, f_n, A, F \ra$ so that if $\a \in p_n$ and if
$\pi_0' = \la p_1, \ldots, p_n, p_{n+1}, \ldots, p_m,
g_0, \ldots, g_m, A_0', F_0' \ra$ extends $\pi_0$ and decides
$\tau_\a$, then the condition $\pi_0'' = \la p_1, \ldots,
p_n, p_{n+1}, \ldots, p_m, g_0, \ldots, g_n, F_0(p_{n+1}),
\ldots, F_0(p_m), A_0'', F_0' \vert A_0'' \ra$ for
$A_0'' = \{ p \in A_0 : p_m \mag p \}$ decides $\tau_\a$ as
well. Since $\pi_0''$ is an interpolant of $\pi_0$ and
$\pi_0'$ in the sense of [Mag], we can proceed as in Lemma 2.5
and Corollary 2.9 of [Mag]. Specifically, let $\la
\a_\b : \b < {\ov {p_n}} \ra$ enumerate $p_n$. Define a
sequence $\la \pi_{\a_\b} : \b < \ov{p_n} \ra$ of length
preserving extensions of $\pi$ so that $\pi_{\a_0} = \pi$ and
$\pi_{\a_{\b+1}}$ is a length preserving extension of
$\pi_{\a_\b}$ such that if $\chi$ extends $\pi_{\a_{\b+1}}$
and decides $\tau_{\a_\b}$ then the interpolant between
$\pi_{\a_{\b+1}}$ and $\chi$ does as well. Here, we are
assuming the analogue for $\P$ of Theorem 2.6 of [Mag], a fact
whose proof we will indicate at the conclusion of the proof of
this lemma. For $\b$ a limit ordinal, $\pi_{\a_\b}$
is a length preserving extension of
each $\pi_{\a_\g}$ for $\g<\b$; $\pi_{\a_\b}$ for $\b$ a
limit ordinal exists since $\ov{p_n} < \kappa$, meaning that
the arguments of Lemma 2.5 and Corollary 2.9 of [Mag] and the
first part of the proof of this lemma can be applied to show
$\pi_{\a_\b}$ exists. A length preseving extension $\pi_0$
of each $\pi_{\a_\b}$ for $\b < \ov{p_n}$ can then be
produced in the same manner. As in Corollary 2.9 of [Mag],
$\pi_0$ is the requisite condition. \hfil\break
\noindent Case 2: $k>0$. As in Lemma 2.8 of [Mag], the
inductive assumption is that the lemma holds for $k$ and any
condition $\chi$ of arbitrary length. We use this and the
methods of Lemma 2.8 to show the lemma holds for $k+1$ and
any condition $\chi$ of arbitrary length. \hfil\break
\indent Let again $\pi = \la p_1, \ldots, p_n, f_0, \ldots,
f_n, A, F \ra$. Since $\mag$ is a well founded relation on
$A$, let $\le_A$ be an extension of $\mag$ to a well ordering
of all of $A$. By induction on $\le_A$, we define a sequence
of conditions $\la \pi_p : p \in A \ra$ satisfying the following
properties. \hfil\vskip .09in
\noindent 1. $\pi_p = \la p_1, \ldots, p_n, p, f_0, \ldots,
f_{n-1}, {(f_n)}^*_p, f^p, A^p, F^p \ra$. \hfil\vskip .09in
\noindent 2. $\pi_p$ extends $\pi$. \hfil\vskip .09in
\noindent 3. If $t,q,s \in A$, $t \mag s$, $q \mag s$, and
$s \in A^t \cap A^q$, then $F^t(s)$ and $F^q(s)$ are compatible.
\hfil\break
\indent Let $\chi_p = \la p_1, \ldots, p_n, p, f_0, \ldots, f_{n-1},
{(f_n)}^*_p, f^p, B^p, H^p \ra$ where $f^p = \bigcup \{ F^q(p) : q \mag p,
q \in A \}$ (if $p \not\in A^q$, $F^q(p)=F(p)$), $B^p =
\{ t : t \in \bigcap_{q \mag t, q \le_{A} p} A^q \} \cap \{t \in
A : p \mag t \}$ (a set in $\U$ by the normality of $\U$), and
for $t \in B^p$, $H^p(t) = \bigcup \{F^q(t) : q \mag t, q \le_A
p \}$. Since $p \in A$, $\ov p$ is an inaccessible (actually
measurable) cardinal, so $\vert \{q : q \mag p \} \vert =
\ov p$, meaning, as each $F^q(p) \in {\rm Col}({\ov p}^+, <
\l )$, that $f^p \in {\rm Col}({\ov p}^+, <\l )$. Similarly,
$H^p(t)$ is well defined for $t \in B^p$, so $\chi_p$ is a
condition. Thus, by the induction hypothesis, we can let
$\pi_p = \la p_1, \ldots, p_n, p, f_0, \ldots, f_{n-1},
{(f_n)}^*_p, f^p, A^p,
F^p \ra$ extending $\chi_p$ be a condition which satisfies the
conclusions of the lemma for a condition of length $n+1$ and
$k$. This completes the inductive definition of $\la \pi_p :
p \in A \ra$ and enables us to define $\pi_{k+1} = \la p_1,
\ldots, p_n, f_0, \ldots, f_n, A_{k+1}, F_{k+1} \ra$ for
$A_{k+1} = \{ p \in A : p \in \bigcap_{q \mag p} A^q \}$ and
$F_{k+1}(p) = f^p$ for $p \in A_{k+1}$. \hfil\break
\indent To see that $\pi_{k+1}$ satisfies the lemma for $k+1$,
let $ \pi_{k+1}'= \la p_1, \ldots, p_n,\break p_{n+1}, \ldots,
p_{n+k}, p_{n+k+1}, p_{n+k+2}, \ldots, p_m, g_0, \ldots, g_m,
A_{k+1}', F_{k+1}' \ra$ extend $\pi_{k+1}$ and decide $\tau_\a$
for $\a \in p_{n+k+1}$. Since $p_{n+1}=p$ for some $p \in
A_{k+1}$, we can write $\pi_{k+1}'$ as $\la p_1, \ldots, p_n,
p, p_{n+2}, \ldots, p_{n+k+1}, p_{n+k+2}, \ldots, p_m, g_0,
\ldots, g_m, A_{k+1}', F_{k+1}' \ra$. By construction and the
arguments of [Mag], Lemma 2.8, $\pi_{k+1}'$ extends $\pi_p$,
so the induction hypothesis yields $k$-quint($\pi_p,
\pi_{k+1}'$) decides $\tau_\a$ in the same way. Again by
construction and the definition of the quasi-interpolant,
$k+1$-quint($\pi_{k+1}, \pi_{k+1}'$) extends
$k$-quint($\pi_p, \pi_{k+1}'$) and hence decides $\tau_\a$. This
proves Lemma 2. \hfil\break\finpf Lemma 2 \hfil\break
\indent Let us note that it is possible to prove a much stronger
version of Lemma 2. Specifically, it is possible to prove
that there is a length preserving extension $\pi' = \break \la p_1,
\ldots, p_n, f_0, \ldots, f_{n-1}, f'_n, B, H \ra$ of
$\pi$ so that if $\pi'' = \la p_1, \ldots, p_n, p_{n+1},
\ldots, p_{n+k}, p_{n+k+1}, \ldots, \break p_m, g_0, \ldots,
g_m, C , I \ra$ extends $\pi'$, $\a \in p_{n+k}$, and
$\pi''$ decides $\tau_\a$, then so does $\pi''' = \la
p_1, \ldots, p_{n+k}, g_0, \ldots, g_{n+k-1},
H(p_{n+k}), B', H \vert B' \ra$ for $B' = \{p \in B : p_{n+k}
\mag p\}$; when $k=0$, $\pi'''$ has the form $\la p_1, \ldots,
p_n, g_0, \ldots, g_{n-1}, f'_n, B, H \ra$. Since we do not
need this version of \break Lemma 2 to establish any of our results,
we do not include a proof here. \hb
\indent We remark that the proof of Lemma 2, with some minor
modifications, essentially gives a proof of Theorem 2.6 of
[Mag] in the context of our partial ordering $\P$. The
ingredient missing from the above proof present in the proof of
Theorem 2.6 is obtaining constant values in Lemma 2.8 for the functions
$f^p_j, \ldots, f^p_n$, something neither needed nor possible in
the case of Lemma 2. If we did want a complete proof of
Theorem 2.6 in our situation, the same proof as in [Mag] will
work to obtain the constant values for $f^p_j, \ldots,
f^p_{n-1}$. To obtain a constant value for $f^p_n$, by the
$\kappa$-additivity of $\U$, we can assume without loss of
generality that for measure 1 many $p$, ${\ov{p_n}}^+
\subseteq p$. Then, since $f^p_n \in {\rm Col}({\ov{p_n}}^+,
< {\ov p})$, we can use the order isomorphism between
$\ov p$ and $p$, the definition of $\P$, and the just mentioned
fact to code $f^p_n$ as a subset of $p$, $g^*(p)$, of
cardinality at most ${\ov{p_n}}^+ < \kappa$. By the normality
and $\kappa$-additivity of $\U$, we can obtain a constant value
$g^*(p)$ for measure 1 many $p$. Then, for measure 1 many $p$,
$g^*(p)$ can be transformed using the order isomorphism between
$p$ and $\ov p$ into a single $f^p_n \in {\rm Col}({\ov{
p_n}}^+, < {\ov p})$. This,
coupled with the methods of Lemma 2, will yield a
complete proof of the analogue of Theorem 2.6 and is the reason
we need to define $\P$ in the manner described; without this
definition, the analogue of Theorem 2.6 can't be
proved. The proof then of
Theorem 3.2 of [Mag] in our context is exactly the same, and
this shows $\vrg \models ``\kappa = \alomp1$''. \hfil\break
\indent The next two lemmas will show $N \models ``\l =
\alomp1$ is a measurable cardinal''. \hfil\break
\noindent Lemma 3: Let $x \subseteq \l$, $x \in N$. Then
$x \in V[r \vert \d, g\vert\d]$ for some $\d\in\kintl$. \hfil\break
\indent Proof of Lemma 3: Let $\pi = \la p_1, \ldots, p_n,
f_0, \ldots, f_n, A, F \ra \forces ``\tau \subseteq \l$''
where $\tau$ is a term forced to denote $x$. Since $x \in
N$, we can choose $\tau \in {\cal L}_1$, so we can assume
without loss of generality that for a single $\d$, $\tau$
mentions only one term of the form ${\dot r} \vert \d$ and one
term of the form ${\dot g} \vert \d$. Further, by Lemma 2, we can
assume that for the sequence of statements $\la \tau_\a : \a
< \l \ra$ where $\tau_\a$ is the statement $``\a\in\tau$'',
if $\pi' = \la p_1, \ldots, p_n, p_{n+1}, \ldots, p_{n+k},
p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_m, A', F' \ra$ decides
$\tau_\a$, $\pi'$ extends $\pi$, then so does $k$-quint($\pi, \pi'$). \hfil\break
\indent Working in $V[r \vert \d, g\vert\d]$ and assuming $\pi \in
G$, define a set $y$ by $\a \in y$ iff there is a condition
$\pi' \in \P$ of the form $\la p_1, \ldots, p_n, p_{n+1},
\ldots, p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_m,
A', F' \ra$ extending $\pi$ so that $\a \in p_{n+k}$,
$\pi' \forces ``\a \in \tau $'', each $g^\d_i \in G^\d_i$ for
$0 \le i < m$, $\la p_1 \cap \d, \ldots, p_m \cap \d
\ra$ is composed of the first $m$ elements of $r \vert \d$,
and for any finite $l \ge m+1$, if $r^\d_{m+1}, \ldots,
r^\d_l$ are the ${(m+1)}^{\rm st} $ through $l^{\rm th}$
elements of $r \vert \d$,
there is a sequence $\la
p_{m+1}, \ldots p_l \ra$ of elements of $A'$ so that
$p_{m+1} \mag \cdots \mag p_l$, ${({(g_m)}^*_{p_{m+1}})}^\d \in
G^\d_m$, ${({F(p_i)}^*_{p_{i+1}})}^\d \in G^\d_i$ for $m+1
\le i < l$, and $\la r^\d_{m+1}, \ldots, r^\d_l \ra =
\la p_{m+1} \cap \d, \ldots, p_l \cap \d \ra$. If $\a\in x$,
then some $\pi' \in G$ must be so that $\pi' \forces
``\a \in \tau$''. We may assume $\pi'$ extends $\pi$, so by
the genericity of $G$ and the definition of $\P$, $\pi'$ can
be assumed to be of the above form with the above
properties. Thus, $\a\in y$, so $x \subseteq y$. \hfil\break
\indent Now let $\a\in y$. Since $\pi \in G$, we can let
$\pi_0 \in G$, $\pi_0$ extending $\pi$ be so that $\pi_0$
decides ``$\a \in \tau$''. If $\pi_0 \forces ``\a \in
\tau$'', then we're done, so assume $\pi_0 \forces ``\a
\not\in\tau$''. Since the genericity of $G$ ensures $\pi_0$
can be chosen so as to have any arbitrary length, the
definitions of $\P$ and $y$ allow us, for a $\pi_0'$ witnessing
$\a \in y$, to assume $\pi_0'$ has the form $\la p_1', \ldots,
p_n', p_{n+1}', \ldots, p_{n+k}', p_{n+k+1}', \ldots, p_m',
g_0', \ldots, g_m', A_0', F_0' \ra$ and $\pi_0$ has the form
$\la p_1, \ldots, p_n, p_{n+1}, \ldots, p_{n+k}, p_{n+k+1},
\ldots, p_m, g_0, \ldots, g_m, A_0, F_0 \ra$ where $\a \in
p_{n+k}'$ and $ \a \in p_{n+k}$; if necessary, we extend
$\pi_0'$ by the definition of $y$ and $\pi_0$ by the genericity
of $G$ until $\a$ appears in the same coordinate of both
conditions and then extend either $\pi_0$ or $\pi_0'$
further until both conditions have the same length, again using
either the definition of $y$ or the genericity of $G$ if
necessary. We thus will have $\la p_1' \cap \d, \ldots,
p_m' \cap \d \ra = \la p_1 \cap \d, \ldots, p_m \cap \d \ra$
and ${(g_i')}^\d, g^\d_i \in G^\d_i$ for $0\le i \vert p_{m-1}' \vert = \vert p_{m-1} \vert > \cdots
> \vert p_1' \vert = \vert p_1 \vert$. Thus, we can let
$h_1' : p_1' - (p_1' \cap \d) \to p_1 - (p_1 \cap \d)$ be a
bijection, and for $2 \le i \le m$, let $h_i' :
(p_i' - (p_i' \cap \d)) - p_{i-1}' \to
(p_i - (p_i \cap \d)) - p_{i-1}$ be a bijection. That each
$h_i'$ can be chosen as a bijection follows from the preceding
facts. $h' = \bigcup^m_{i+1} h_i'$ will then be a bijection
between $p_m' - \d$ and $p_m - \d$, and we can let $h'' :
\l - (p_m' - \d) \to \l -(p_m - \d)$ be a bijection which is the
identity on $\d$. Hence, $h = h' \cup h''$ is a bijection of
$\l$ which is the identity on $\d$. Further, by definition,
$h(p_i') = p_i$ for $1 \le i \le m$. \hfil\break
\indent Since ${(g_i')}^\d$ and $g^\d_i$ are compatible for
$0 \le i \le n+k$, ${(F(p_i'))}^\d$ and ${(F(p_i))}^\d$ are
compatible for $n+k+1 \le i < m$, and $F(p_m') \vert \d$
and $F(p_m) \vert \d$ are compatible, we can let for $0 \le i
< m$ $\psi_i$ be an automorphism of the appropriate L\'evy
collapse generated by a function which is the identity on
$\ov{p_i' \cap \d}$ so that $\psi_i(g_i')$ is compatible with
$g_i$ for $0 \le i \le n+k$ and $\psi_i(F(p_i'))$ is compatible
with $F(p_i)$ for $n+k+1 \le i < m$. For $i=m$, we let
$\psi_m$ be an automorphism of ${\rm Col}({\ov{p_m'}}^+,
<\l)$ generated by a function which is the identity on $\d$
so that $\psi_m(F(p_m'))$ is compatible with $F(p_m)$. Also,
as in Lemma 3.4 of [Mag], $C' = \{ p \in \pkl : h''p = p \}
\in \U$. This allows us, in analogy to [Mag], to define for
$ \chi = \la q_1, \ldots, q_l, h_0,
\ldots, h_l, C, H \ra \in \P$, the function $\psi(\chi) =
\la h''q_1, \ldots, h''q_l, h_0, \ldots, h_l,
C , H \ra$ (where $\psi_i$ is the identity for
$m+1 \le i \le l$) which can be seen to generate an automorphism
of $\P$ above $\chi_0'$. (Since $C' \in \U$, we can assume
without loss of generality that $B_0' \subseteq C'$.) The
definition of $\psi$ ensures that when forcing above $\chi_0'$,
$\psi$ preserves the meaning of $r \vert \d$ and
$g \vert \d$. Thus,
$ \psi(\chi_0') = \la h''p_1',
\ldots, h''p_m', \psi_0(g_0'), \ldots, \psi_{n+k}(g_{n+k}'),\break \psi_{n+k+1}(F(p_{n+k+1}')), \ldots,
\psi_m(F(p_m')), B_0', F \vert B_0' \ra = \la p_1,
\ldots, p_m, \psi_0(g_0'), \ldots, \psi_{n+k}(g_{n+k}'),\break \psi_{n+k+1}(F(p_{n+k+1}')), \ldots,
\psi_m(F(p_m')), B_0', F \vert B_0' \ra \forces
``\a \in \tau$'' since $\tau$ can be chosen so as to be
invariant under the automorphism $\psi$ just
constructed. (Canonical terms for sets in $V$ can be chosen
so as to be invariant under any automorphism of $\P$.) Further,
$\psi(\chi_0')$ is
compatible with $\chi_0 = \la p_1, \ldots, p_m, g_0, \ldots,
g_{n+k}, F(p_{n+k}), \ldots, F(p_m), B_0, F \vert B_0 \ra$,
$\psi(\chi_0') \forces ``\a \in \tau$'', and $\chi_0 \forces
``\a \not\in \tau$''. This contradiction shows $\pi_0
\forces ``\a \in \tau$'', so $y \subseteq x$ and $x=y$. This
means $x$ is definable in $V[r \vert \d, g\vert\d]$, thus proving
Lemma 3. \hfil\break\finpf Lemma 3 \hfil\break
\indent We remark that even though the automorphism $\psi$ may
move $\a$, this has no bearing on the fact that $\psi(\chi_0')$
still decides $\tau_\a$ in the same way $\chi_0'$ does. The
ordinal $\a$ acts primarily as a coding device and, in and of
itself, doesn't necessarily have any intrinsic effect on how
$\chi_0'$ decides $``\a\in\tau$''. It is the ordinals below
$\d$ which can't be moved by $\psi$ since $\tau$ mentions
${\dot r} \vert \d$ and ${\dot g} \vert \d$. \hfil\break
\noindent Lemma 4: $N \models ``\l = \alomp1$ is a measurable
cardinal''. \hfil\break
\indent Proof of Lemma 4: Fix $\mu \in V$ a measure over
$\l$. We show that $N \models ``\mu^* = \{x \subseteq \l : x$
contains a $\mu$ measure 1 set$\}$'' is a measure over $\l$. Since
Lemma 1 shows $N \models ``\l \le \kappa^+ = \alomp1$'', this
will show $N \models `` \l=\alomp1$ is a measurable cardinal''.
\hfil\break
\indent Let $\pi = \la p_1, \ldots, p_n, f_0, \ldots, f_n,
A, F \ra \forces ``\tau \subseteq \l$ and $\tau \in \dot N$''
where $\tau$ mentions only ${\dot r} \vert \d$ and $\dot g \vert \d$. By
Lemma 2, we can assume without loss of generality that $\pi$
is so that for the sequence of statements $\la \tau_\a : \a
< \l \ra$ where $\tau_\a$ is the statement $``\a\in\tau$'', if
$\pi' = \la p_1, \ldots, p_n, p_{n+1}, \ldots, p_{n+k},
p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_m, B, H \ra$ extending
$\pi$ decides
$\tau_\a$ then so does $k$-quint($\pi,\pi'$). We show that for
some $\pi_0$ extending $\pi$, $\pi_0 \forces ``$Either $\tau$
or $\l - \tau$ contains a $\mu$ measure 1 set''. \hfil\break
\indent The key to proving the above fact and that $\mu^*$ is
$\l$-additive is to observe that for any statement $\sigma
\in {\cal L}_1$, $\sigma$ mentioning only ${\dot r} \vert\d$
and $\dot g \vert \d$, the proof of Lemma 3 actually shows that if some
$\chi_0 = \la q_1, \ldots, q_m, g_0, \ldots, g_m, B, H
\vert B \ra$ decides $\sigma$ and $\chi_0' = \la q_1', \ldots,
q_m', g_0, \ldots, g_m, B', H \vert B' \ra$ is so that
($*$) $q_i \cap \d = q_i' \cap \d$ for $1 \le i \le m$ and
($**$) $\{ \la \la r_1 \cap \d, h^\d_1 \ra, \ldots, \la r_{l-1} \cap \d,
h^\d_{l-1} \ra, \la r_l \cap \d, h_l \vert \d \ra \ra : l < \omega$, $r_1 \mag \cdots \mag r_l$, $r_1, \ldots,
r_l \in B$, $H(r_i) = h_i$ for $1 \le i \le l\} = \{ \la \la
r_1 \cap \d, h^\d_1 \ra, \ldots, \la r_{l-1} \cap \d, h^\d_{l-1} \ra, \la r_l \cap \d, h_l \vert \d \ra \ra : l
< \omega$, $r_1 \mag \cdots \mag r_l$, $r_1, \ldots, r_l \in B'$, $
H(r_i) = h_i$ for $1 \le i \le l\}$, then $\chi_0'$ decides
$\sigma$ in the same way as $\chi_0$. To see this, assume
without loss of generality that $\chi_0 \forces \sigma$. If
$\chi_0' \not\forces\sigma$, then let $\chi_1'$ extending
$\chi_0'$ $\chi_1' = \la q_1', \ldots, q_m', q_{m+1}', \ldots,
q_k', h_0', \ldots, h_m', h_{m+1}', \ldots, h_k', C', I' \ra
\forces\neg\sigma$. By ($**$), we can find $q_{m+1} \mag
\cdots \mag q_k$, $q_{m+1}, \ldots, q_k \in B$ so that
$q_i \cap \d = q_i' \cap \d$ for $m+1 \le i\le k$, ${(H(q_i))}^\d
={(H(q_i'))}^\d$ for $m+1 \le i < k$, and $H(q_k) \vert \d = H(q_k') \vert \d$. Since ${(H(q_i))}^\d = {(H(q_i'))}^\d
\subseteq {(h_i')}^\d$ for $m+1 \le i < k$, $H(q_k) \vert \d
=H(q_k')\vert\d\subseteq h_k' \vert\d$, and $H(q)\subseteq I'(q)$ for $q \in C'$, the condition $\chi_1 = \la q_1, \ldots,
q_m, q_{m+1}, \ldots, q_k, h_0,\ldots, {(h_m)}^*_{q_{m+1}},
H(q_{m+1}), \ldots, H(q_k), C, I' \vert C \ra$ for
$ C = C' \cap \{q \in B : q_k,
q_k' \mag q\}$ extends $\chi_0$, meaning $\chi_1 \forces
\sigma$. The proof of Lemma 3 then yields an automorphism $\psi$
of $\P$ on the set of conditions above $\chi_1$ so that $\psi(\chi_1)$ is compatible with $\chi_1'$
and $\psi(\chi_1) \forces \sigma$. As before, since $\chi_1'
\forces \neg\sigma$, this is again a contradiction. \hfil\break
\indent Using the above fact, define an equivalence relation
$\sim_\d$ on $S_0 = \{\chi$ extending $\pi : \chi$ is of the
form $\la p_1, \ldots, p_n, p_{n+1}, \ldots, p_m, g_0, \ldots,
g_m, B, F \vert B \ra$ by $\chi \sim_\d \chi'$ iff both ($*$)
and ($**$) are satisfied by $\chi'$ for $\chi$ and $\d$. Readers
may verify that $\sim_\d$ is indeed an equivalence relation on
$S_0$. Further, $T = \{[\chi]_{\sim_\d} : \chi \in S_0\}$ is
so that $\vert T \vert < \l$. This follows since $\l$ is
inaccessible, so for $S_1 = \{s_B : B \subseteq A$ and $B \in
\U\}$, $s_B$ for $B \subseteq A$, $B \in \U$ defined as
$\{\la q_1 \cap \d, \ldots, q_m \cap \d \ra : m < \omega$, $q_1,
\ldots, q_m \in B$, $q_1 \mag \cdots \mag q_m\}$, $\vert S_1
\vert = \vert \{ {[B]}^{<\omega} : B \subseteq P_\kappa(\d)\}
\vert = 2^\d < \l$. For $S_2 = \{\la g_0, \ldots, g_m \ra : m
< \omega$, $g_0 \in {\rm Col}(\omega_1, <\g_1)$ for some inaccessible
$\g_1 < \kappa$, for $1 \le i < m$, $g_i \in {\rm Col}(
\g^+_i, <\g_{i+1})$ where $\g_i < \g_{i+1} < \kappa$ are
inaccessible, and $g_m \in {\rm Col}( \g^+_m , <\delta)\}$,
$\vert S_2 \vert = \delta < \l$, and for $S_3 = \{t_B : B
\subseteq A$ and $B \in \U\}$, $t_B$ for $B \subseteq A, B \in
\U$ defined as $\{\la\la q_1 \cap \d, g^\d_1 \ra, \ldots,
\la q_{m-1} \cap \d, g^\d_{m-1} \ra , \la
q_m \cap \d, g_m \vert \d \ra\ra : m < \omega$, $q_1, \ldots, q_m \in
B$, $q_1 \mag \cdots \mag q_m$, and for $1 \le i \le m$, $F(q_i)
=g_i\}$, $\vert S_3 \vert \le \vert S_1 \times S_2 \vert =
2^\d \times \delta = 2^\d < \l$. Hence, $\vert T \vert \le
\vert S_1 \times S_2 \times S_3 \vert \le 2^\d < \l$.
\hfil\break
\indent Note now that for any $\a < \l$ there must be $\chi \in
S_0$ so that $\chi$ decides $\tau_\a$. This is since if
$\pi'$ as in the second paragraph of the proof of this lemma
extending $\pi$ decides $\tau_\a$, then so does
$k$-quint($\pi,\pi') \in S_0$. (Any $\pi'$ extending $\pi$
deciding $\tau_\a$ may be extended further if necessary so as to
have $\a \in p_{n+k}$.) Thus, if $\le_T$ is a well-ordering in
$V$ of $T$, we can define a function $f:\l\to T$ in $V$ by
$f(\a)$ = The $\le_T$ least element ${[\chi]}_{\sim_\d}$ of $T$
so that for some $\chi' \in \chid$, $\chi'$ decides
$\tau_\a$. By the preceding remarks, $f(\a)$ is well defined for
all $\a < \l$. Since $\mu$ is $\l$-additive and $\vert T \vert
< \l$, there must be some $y \subseteq \l$, $y \in \mu$ so that
$f$ is a constant value $\chid$ on $y$. We then know that for
any $\a\in y$ there is some $\chi' \in \chid$ so that $\chi'$
decides $\tau_\a$. By our earlier work, if one $\chi'$ in
$\chid$ decides $\tau_\a$, then any arbitrary $\chi'' \in
\chid$ decides $\tau_\a$ in the same way. Thus, if we pick
some $\pi_0 \in \chid$, $\pi_0$ extends $\pi$ and $\pi_0$
decides $\tau_\a$, i.e., $\pi_0$ decides ``$\a\in\tau$'' for
any $\a\in y$. Again by the $\l$-additivity of $\mu$, either
$y_0 = \{\a\in y : \pi_0 \forces ``\a\in\tau$''$\}$ or
$y_1 = \{\a\in y : \pi_0 \forces ``\a\not\in\tau$''$\}$ has
$\mu$ measure 1. In the first case, $\pi_0 \forces ``y_0
\subseteq \tau$'', and in the second case, $\pi_0 \forces
``y_1 \subseteq \l - \tau$''. Hence, $\pi_0 \forces ``$Either
$\tau$ or $\l - \tau$ contains a $\mu$ measure 1 set''.\hfil\break
\indent Let now $\pi = \la p_1, \ldots, p_n, f_0, \ldots, f_n,
A, F \ra \forces ``\la \sigma_\a : \a<\g<\l \ra$ is a sequence of
subsets of $\l$ in ${\dot N}$ so that each $\sigma_\a$
contains a $\mu$ measure 1 set''. If the term $\bigcap_{\a
<\g} \sigma_\a$ doesn't denote a $\mu^*$ measure 1 set, then the
preceding shows the term $\l - \bigcap_{\a<\g} \sigma_\a$, i.e.,
the term $\bigcup_{\a<\g} (\l - \sigma_\a)$ must denote a $\mu^*$
measure 1 set, so without loss of generality we can let $y
\subseteq \l$, $y \in \mu$ be so that $\pi \forces ``y
\subseteq \bigcup_{\a<\g} (\l - \sigma_\a)$''. Further, since
$\g<\l$, $\vert \{ \tau_{\a,\b}' : \b \in y$, $\a < \g$, and
$ \tau_{\a,\b}' $ is the statement $``\b \in \l -
\sigma_\a$''$\} \vert = \l$, so we can let $\la \tau_\a : \a <
\l \ra$ be an enumeration of these statements. Lemma 2 then
allows us without loss of generality to assume that $\pi$ is
so that if $\pi' = \la p_1, \ldots, p_n, p_{n+1}, \ldots,
p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0, \ldots, g_m, B, H \ra$
extends $\pi$, $\a \in p_{n+k}$, and $\pi'$ decides $\tau_\a$,
then $k$-quint($\pi,\pi'$) decides $\tau_\a$ as well. \hfil\break
\indent Since $\la \l - \sigma_\a : \a < \g < \l \ra$ denotes a
sequence of subsets of $\l$ in $N$, we can let $\d$ be so that
each $\l - \sigma_\a$ for $\a<\g$ and the sequence $\la \l -
\sigma_\a : \a<\g<\l \ra$ mentions only ${\dot r} \vert \d$
and $\dot g\vert\d$. Further, since $\pi \forces ``y \subseteq
\bigcup_{\a<\g}(\l - \sigma_\a)$'', we know that for any $\b
\in y$ there must be some $\pi'$ extending $\pi$ and some
$\a<\g$ so that $\pi' \forces ``\b \in \l - \sigma_\a$''. As
before, for $\rho < \l$ the ordinal so that $\tau_\rho$ is
the statement ``$\b \in \l - \sigma_\a$'', we can assume that
$\pi'$ has the form of the last paragraph and $\rho \in
p_{n+k}$. Thus, $k$-quint($\pi,\pi') \forces ``\b \in \l -
\sigma_\a$''. This allows us, for $S_0 = \{\chi$ extending
$\pi : \chi$ is of the form $\la p_1, \ldots, p_n, p_{n+1},
\ldots, p_m, g_0, \ldots, g_m, B, F \vert B \ra\}$, $T=
\{\chid : \chi \in S_0\}$, and $\le_T$ a well-ordering in $V$
of $T $, to define the function $f_0 : y \to T$ by
$f_0(\b)$ = The $\le_T$ least element $\chid$ of $T$ so that for
some $\a<\g$ and some $\chi' \in \chid$, $\chi' \forces
``\b \in \l - \sigma_\a$''. As earlier, since
$k$-quint($\pi,\pi') \in S_0$, $\vert T \vert < \l$, and $\mu$ is
$\l$-additive, there is a fixed $\chid \in T$ and some $y_0
\subseteq y$, $y_0 \in \mu$ so that for $\b \in y_0$,
$f_0(\b) = \chid$. Thus, as before, for any $\b \in y_0$ and
any $\chi' \in \chid$, there is some $\a<\g$ so that $\chi'
\forces ``\b \in \l - \sigma_\a$''. If we now pick some
$\pi_0 \in \chid$, we can therefore define the function
$f_1 : y_0 \to \g$ by $f_1(\b)$ = The least $\a<\g$ so that
$\pi_0 \forces ``\b \in \l - \sigma_\a$''. There must then by
the $\l$-additivity of $\mu$ be some $y_1 \subseteq y_0$,
$y_1 \in \mu$ so that for a fixed $\a < \g$, if $\b \in y_1$,
$\pi_0 \forces ``\b \in \l - \sigma_\a$'', i.e., $\pi_0
\forces ``y_1 \subseteq \l - \sigma_\a$''. We thus have
$\pi_0 \forces ``$Both $\sigma_\a$ and $\l - \sigma_\a$
contain $\mu$ measure 1 sets'', an absurdity. Hence, $N \models
``\mu^*$ is a measure over $\l$''. This proves Lemma 4.
\hfil\break\finpf Lemma 4 \hfil\break
\indent We observe that if $\mu$ were normal then we could prove
$\mu^*$ were normal as well. Since we will be showing $N \models
\fordc$, we will have $N \models {\rm DC}$, so any non-normal
measure can be normalized. We thus leave it as an exercise for
the readers that the L\'evy-Solovay [LS] arguments can be
modified in the above context to show if $\mu$ is normal then so
is $\mu^*$. \hfil\break
\indent We turn now to the proof that $N \models \fordc$. For
concreteness, we recall that for $\aleph$ a cardinal,
${\rm DC}_\aleph$ means that for $R \subseteq {[X]}^{<\aleph}
\times X$ a binary relation, if $\la x_\a : \a<\b<\aleph \ra$ a
sequence of elements of $X$ implies there is some $x \in X$ so
that $\la x_\a : \a<\b<\aleph \ra R x$, then there is a sequence
$\la x_\a : \a<\aleph \ra$ so that for any $\b<\aleph$, $\la
x_\a : \a<\b \ra R x_\b$. \hfil\break
\indent As mentioned at the beginning of this paper, the proof uses
ideas given by Kafkoulis in [Ka1]. In
analogy to [Ka1], we first observe that $N=V[\H]$ for ${\cal H} =
\bigcup_{\d<\l} \pvk$. This is true since for every $\d<\l$,
$V[r \vert \d, g\vert\d] \subseteq N$ and $\pvk$ is definable in $N$,
so ${\cal H} \in N$ and $V[{\cal H}] \subseteq N$. It is also the case that
the ordered pair $\la r \vert \d, g\vert\d \ra$ can be coded as a subset of
$\kappa$, meaning any term written using only terms of the form
$\la {\dot r} \vert \d, {\dot g}\vert\d \ra$ can be rewritten using terms
for the appropriate subset of $\kappa$ in $V[r \vert \d, g\vert\d]$. Thus,
$N \subseteq V[{\cal H}]$, so $N = V[{\cal H}]$. Further, ${\cal H} =
p^N(\kappa)$. The above makes it clear that ${\cal H} \subseteq
p^N(\kappa)$. By Lemma 3, since $\kappa < \l$, any $x \subseteq
\kappa$, $x \in N$ is an element of $V[r \vert \d, g\vert\d]$ for the
appropriate $\d<\l$. Thus, $p^N(\kappa) \subseteq {\cal H}$, so
${\cal H} = p^N(\kappa)$. \hfil\break
\indent We state now precisely our plan of attack in showing
$N \models \fordc$. We first show, in the spirit of [Ka1] but
with a somewhat different method,
that $N \models \forall n < \omega [{\rm DC}_{\aleph_n}({\cal K})]$,
where ${\cal K} \in N$ is a certain partial ordering to be described
below. Then, as in [Ka2], we force over $N$ with ${\cal K}$ to add a
well-ordering of length $\l$ of ${\cal H}$. The resulting model $M$
will be so that $M \models {\rm AC}$ and has the same $<\l$
sequences of elements of $N$ as in $N$. This will immediately
yield $N \models \fordc$. \hfil\break
\indent We begin with a precise definition of ${\cal K} \in N$ and
exactly what ${\rm DC}_\theta({\cal K})$ for $\theta < \kappa$ a
cardinal will mean. ${\cal K} = \{ f:\g \to {\cal H} : f$ is a 1-1
function with dom($f) = \g < \l \}$, ordered by $f_2$ extends
$f_1$ iff $f_2 \supseteq f_1$. We will refer to ${\cal K} \cap
V[r \vert \d, g\vert\d]$ as ${\cal K}_\d$. To state ${\rm DC}_\theta({\cal K})$,
we first define for $\g < \theta$ ${[{\cal K}]}^\g$ as $\{ \la f_\a :
\a<\g \ra$ : Each $f_\a \in {\cal K}$ and $\a<\b<\g$ implies $f_\a
\subseteq f_\b \}$, i.e., ${[{\cal K}]}^\g$ is the set of increasing
chains of elements of ${\cal K}$ of length $\g$, and we define
${[{\cal K}]}^{<\theta} = \bigcup_{\g<\theta} {[{\cal K}]}^\g$. ${\rm DC}_{
\theta}({\cal K})$ will then mean that if $R \subseteq
{[{\cal K}]}^{<\theta} \times {\cal K}$ is so that $\forall x \in
{[{\cal K}]}^{<\theta} \exists y \in {\cal K}[y$ extends each element of $x$
and $xRy]$, there is a sequence $\la x_\a : \a < \theta \ra$ of
elements of ${\cal K}$ with $\a<\b<\theta$ implying $x_\a \subseteq
x_\b$ so that for any $\b < \theta$, $\la x_\a : \a < \b \ra R
x_\b$. For a relation $R$ as just described, we will say that
$R$ has the ${\rm DC}_\theta({\cal K})$ property. \hfil\break
\indent Observe now that any element of $\K$ or for $\theta <
\l$, any $\theta$ sequence of elements of $\K$ present in $N$,
chain or otherwise, can be viewed as an element of $\H$. If
$f \in \K$, dom($f) = \g$, then we can find some $\d \in
\kintl$ so that $\g < \d$. Thus, $V[ r \vert \d, g\vert\d] \models
``\vert \g \vert \le \kappa$'', so $N \models ``\vert \g \vert
\le \kappa$'', meaning it is possible in $N$ to code $f$ as a
$\le \kappa$ sequence of subsets of $\kappa$, i.e., as an
element of $\H$. Further, if $\la f_\a : \a < \theta \ra \in
N$ is a $\theta$ sequence of elements of $\K$, dom($f_\a) =
\g_\a$, then as $\l$ is regular in $N$, $\g = \bigcup_{\a <
\theta} \g_\a < \l$. Therefore, again $N \models ``\vert \g
\vert \le \kappa$'', so it is possible in $N$ to code $f$ as a
$\le \kappa$ sequence of subsets of $\kappa$, i.e., as an
element of $\H$. This means $\K = \bigcup_{\d \in \kintl}
\K_\d$ and ${({[\K]}^\theta)}^N = \bigcup_{\d \in \kintl}
{({[\K_\d]}^\theta)}^{V[r \vert \d, g\vert\d]}$. \hfil\break
\indent Let $\theta < \kappa$ and $\dot R$ be so that $\pi_0
= \la p_1, \ldots, p_n, f_0, \ldots, f_n, A_0, F_0 \ra \forces
``{\dot R} \subseteq {[{\dot {\cal K}}]}^{<\theta} \times {\dot {\cal K}},
{\dot R} \in N$ has the ${\rm DC}_\theta({\dot {\cal K}})$
property''. We begin by showing that Lemma 2 implies a length
preserving extension $\pi_1$ of $\pi_0$ can be used so as to
define, for all $\d \in [\d_0, \l)$ where $\d_0$ is so that
$\dot R$ mentions only a term of the form $\la
{\dot r} \vert \d_0, {\dot g}\vert\d_0\ra$, $R \cap V[r \vert \d, g\vert\d]$ in
$V[r \vert \d, g\vert\d]$. To see this, we first observe that for any
$\d < \l$, $V[r \vert \d, g\vert\d] \models ``\l$ is (strongly)
inaccessible''. (This makes sense since $V[r \vert \d, g\vert\d]
\models {\rm AC}$.) Clearly, since $V[r \vert \d, g\vert\d]
\subseteq N$ and $N \models ``\l$ is regular'', $V[r \vert \d,
g\vert\d] \models ``\l$ is regular''. Further, if $V[r \vert \d,
g\vert\d] \models ``$For some $\g<\l$, $2^\g \ge \l$'', then as
$N \supseteq V[r \vert \d, g\vert\d] \models {\rm AC}$, there is a
$\l$ sequence of subsets of $\g$ in $N$, contradicting $N
\models ``\l$ is measurable''. Thus, $V[r \vert \d, g\vert\d] \models
``\l$ is inaccessible''. \hfil\break
\indent Since $V[ r \vert \d, g\vert\d] \models ``\l$ is inacessible'',
for any fixed $\d < \l$ we can find in $V$ a sequence $\la
\tau^\d_\a : \a < \g_\d < \l \ra$ of terms for all of the
elements of $\K_\d$ and all of the $\theta$ sequences of elements
of $\K_\d$ in $V[r \vert \d, g\vert\d]$. Thus, since $V \models ``\l$
is inaccessible'', our earlier observations allow us to find in
$V$ a sequence $\la \tau_\a : \a < \l \ra$ of terms for all of
the elements of $\K$ and all of the elements of
${({[\K]}^\theta)}^N$. Hence, there is in $V$ a sequence $\la
\sigma_\a : \a < \l \ra$ of terms where each $\sigma_\a$ is of
the form $\tau_{\a_0} {\dot R} \tau_{\a_1}$, $\tau_{\a_0},
\tau_{\a_1} \in \la \tau_\a : \a < \l \ra$ terms of the
appropriate form. And, since $\tau_{\a_0}$, $\dot R$, and
$\tau_{\a_1}$ all denote sets in $N$, as before we can assume
without loss of generality that each $\sigma_\a$ mentions only
one term of the form $\la {\dot r} \vert \d, {\dot g}\vert\d \ra$
for some $\d < \l$, $\d \in [\d_0, \l)$. \hfil\break
\indent Let now by Lemma 2 $\pi_1 = \la p_1, \ldots, p_n, f_0,
\ldots, f_n, A_1, F_1 \ra$ be a length preserving extension of
$\pi_0$ satisfying the conclusions of Lemma 2 for $\la \sigma_\a
: \a < \l \ra$. Using Lemma 3 and the last paragraph, we can define
in $V[r \vert \d, g\vert\d]$ for $\d \in [\d_0, \kappa)$ the relation
$R_\d$ by $x_1 R_\d x_2$ iff there are terms $\dot{x_1}$ and
$\dot {x_2}$, an $\a <\l$, and some $\pi_2 = \la p_1, \ldots,
p_n, p_{n+1}, \ldots, p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0,
\ldots, g_m, A_2, F_2 \ra$ extending $\pi_1$ so that
$\dot {x_1}$ and $\dot {x_2}$ mention only $\la {\dot r}
\vert \d, {\dot g}\vert\d \ra$, $\dot {x_1}$ and $\dot {x_2}$
denote $x_1$ and $x_2$,
$\sigma_\a$ is the term $\dot{x_1} \dot{R} \dot{x_2}$,
$\a \in p_{n+k}$, $\pi_2 \forces
``{\dot {x_1}} {\dot R} {\dot {x_2}}$'', each $g^\d_i \in G^\d_i$ for
$0 \le i < m$, $\la p_1 \cap \d, \ldots, p_m \cap \d \ra$ is
composed of the first $m$ elements of $r \vert \d$, and for any
finite $l \ge m+1$, if $r^\d_{m+1}, \ldots, r^\d_l$ are the
${(m+1)}^{\rm st}$ through $l^{\rm th}$ elements of $r \vert \d$,
there is a sequence $\la p_{m+1}, \ldots, p_l \ra$ of elements of
$A_2$ so that $p_{m+1} \mag \cdots \mag p_l$,
${({(g_m)}^*_{p_{m+1}})}^\d \in G^\d_m$,
${({(F(p_i))}^*_{p_{i+1}})}^\d \in G^\d_i$
for $m+1 \le i < l$, and $\la r^\d_{m+1}, \ldots, r^\d_l \ra =
\la p_{m+1} \cap \d, \ldots, p_l \cap \d \ra$. Since $\d_0
\le \d $, the proof of Lemma 3 and our earlier remarks show
$R_\d$ is a well-defined binary relation in $V[r \vert \d, g\vert\d]$
on ${({[\K_\d]}^{<\theta})}^{V[r \vert \d, g\vert\d]} \times \K_\d$
and for $x_1 \in {({[\K_\d]}^{<\theta})}^{V[r \vert \d, g\vert\d]}$,
$x_2 \in \K_\d$, $N \models ``x_1 R x_2$'' iff $V[r \vert \d,
g\vert\d] \models ``x_1 R_\d x_2$''. Thus, $R \cap V[r \vert \d,
g\vert\d]$ is definable in $V[r \vert \d, g\vert\d]$ as $R_\d$, and for
$\d_0<\d_1<\l$, $R_{\d_0} \subseteq R_{\d_1} \subseteq
R$. \hfil\break
\noindent Lemma 5: $N \models \forall n < \omega
[{\rm DC}_{\aleph_n}(\K)]$. \hfil\break
\indent Proof of Lemma 5: We
work in $N$. Fix $\theta < \kappa$ a cardinal in
$N$. We show $N \models {\rm DC}_\theta(\K)$. Since $N \models
``\keqal$'', this will prove Lemma 5. Accordingly, let
$\pi_0 = \la p_1, \ldots, p_n, f_0, \ldots, f_n, A_0, F_0 \ra
\forces ``{\dot R} \subseteq {[\dot{\cal K}]}^{<\theta} \times
{\dot {\cal K}}, {\dot R} \in {\dot N}$ has the ${\rm DC}_{
\theta}({\dot {\cal K}})$ property''. We can assume that
$\pi_0$ satisfies the conclusions of Lemma 2 for $\la
\sigma_\a : \a < \lambda \ra$, where $\la \sigma_\a : \a <
\lambda \ra$ is as in the two paragraphs immediately
preceding the statement of Lemma 5. Thus, $\pi_0$ can be used
in the definition of $R_\d$ for appropriate values of
$\d$. Further, for $\la \tau_\a : \a < \lambda \ra$ as in
the two paragraphs immediately preceding the statement of Lemma
5, since there are just $\lambda$ statements of the form
$``\exists \tau'[\tau \dot{R} \tau'$ and $\tau'$ has support
$\b$, i.e., $\tau'$ mentions only $\la \dot{r} \vert \b,
\dot{g} \vert \b \ra]$'' where $\tau \in \la \tau_\a : \a
< \lambda \ra$ is a term of the appropriate form and
$\b < \lambda$, we can assume without loss of generality that
$\la \sigma_\a : \a < \lambda \ra$ contains all such
statements. \hb
\indent Work inductively, assuming $\eta = \la \eta_\a : \a
< \d < \theta \ra$, a term having support $< \lambda$, has
been defined so that for any $\a < \d$, $\pi_0 \forces ``
\la \eta_\b : \b < \a \ra \dot{R} \eta_\a$''. Let
$\eta$ and $\dot R$ be supported by $\g \in (\kappa, \l)$,
and let $M \prec V_\rho$, where $\rho > \l$ is sufficiently
large, be so that $\g, \pi_0, \eta, \dot{R} \in M$ and
$M \cap V_\l = V_{\b_0}$ for some $\b_0 < \l$. We claim that
$\pi_0 \forces ``\exists \tau'[\tau'$ has support $\b_0$
and $\eta \dot{R} \tau']$''. \hb
\indent If the claim fails, let $\pi_1$ extending $\pi_0$
be so that $\pi_1 \forces ``\neg \exists \tau'[\tau'$ has
support $\b_0$ and $\eta \dot{R} \tau']$ and $\eta \dot{R}
\ov{\tau}$ for $\ov\tau$ of support $> \b_0$''. By a
slight weakening of Lemma 2, $\pi_1$ can be assumed to be of
the form $\la p_1, \ldots, p_n, p_{n+1}, \ldots,
p_{n+k}, p_{n+k+1}, \ldots, p_m, g_0, \ldots,
g_m, B, F \vert B \ra$, where the indices for
the statements $``\exists \tau'[\tau'$ has support
$\b_0$ and $\eta \dot{R} \tau']$'' and $``\eta \dot{R}
\ov{\tau}$'' are elements of $p_{n+k}$ and $B = \{p
\in A_0 : p_m \mag p \}$. \hb
\indent Let $f = g_m \vert \b_0$. We have $V_\rho \models
``$There exists an extension $\pi^*_1$ of $\pi_0$ of the
form $\la p_1, \ldots, p_n, p^*_{n+1}, \ldots,
p^*_{n+k}, p^*_{n+k+1}, \ldots, p^*_m,
g_0, \ldots, g_{m-1}, g^*_m, C, H \ra$
so that: \hfil\vskip .09in\no
1. $p_{ i} \cap \g = p^*_{ i} \cap \g$ for
$n+1 \le i \le m$. \hfil\vskip .09in\no
2. $g^*_m$ extends $f$. \hfil\vskip .09in\no
3. For some $\ov{\ov{\tau}}$, $\pi^*_1 \forces `` \eta
\dot{R} \ov{\ov{\tau}}$''. \hfil\vskip .09in\no
4. The index $\a_0$ for $``\eta \dot{R} \ov{\ov{\tau}}$'' is an
element of $p^*_{n+k}$''. \hfil\vskip .09in\no
Since $\pi_0 \in M$, $\la p^*_{n+1} \cap \g, \ldots,
p^*_m \cap \g \ra \in M$, $\la g_0, \ldots, g_{m-1}, f \ra
\in M$, and $M \prec V_\rho$, we can assume without loss of
generality that $\a_0, \pi^*_1, \ov{\ov{\tau}} \in
M$. Thus, since $M \cap V_\l = V_{\b_0}$, $\ov{\ov{\tau}}$
has support $\b_0$, and $\pi^*_1 \forces ``\eta \dot{R}
\ov{\ov{\tau}}$''. By again a slight weakening of Lemma 2,
$\pi^{**}_1 = \la p_1, \ldots, p_n,
p^*_{n+1}, \ldots, p^*_{n+k}, p^*_{n+k+1},
\ldots, p^*_m, g_0, \ldots, g_{m-1}, F(p^*_m),
C^*, F \vert C^* \ra \forces ``\eta \dot{R}
\ov{\ov{\tau}}$'' for $C^* = \{ p \in C :
p^*_m \mag p \}$. \hb
\indent It is the case that $g_m$ and $F(p^*_m)$ are
compatible, since $\pi^*_1 \in M$, so $F(p^*_m)$, which
extends $f = g_m \vert \b_0$, is bounded below $\b_0$. This
allows us, without loss of generality, to assume that $g_m$
in $\pi_1$ and $F(p^*_m)$ in $\pi^{**}_1$ have been replaced
by $g_m \cup F(p^*_m)$. Thus, as in Lemma 3, there is an
automorphism $\psi$ of $\P$ generated by functions which are
the identity on $\g$ so that $\psi(\pi^{**}_1)$ is
compatible with $\pi_1$; further, since the collapse maps in
$\pi_1$ and $\pi^{**}_1$ are the same, the methods of Lemma 3
tell us $\psi$ can be assumed to preserve the meaning of
the generic sequence of collapse maps $g$ and
of each element $p_i$ of the generic sequence $r$ for
$i>m$. Therefore, since $\g < \b_0$ and $\pi^{**}_1
\forces ``\exists \tau'[\tau'$ has support $\b_0$ and
$\eta \dot{R} \tau']$'', i.e., $\pi^{**}_1 \forces
``\exists \tau'[\tau'$ is $V$-definable from $\la \dot{r}
\vert \b_0, \dot{g} \vert \b_0 \ra$ and $\eta \dot{R}
\tau']$ ($\tau'$ is $\ov{\ov{\tau}}$), $\psi(\pi^{**}_1)
\forces ``\exists \tau'[\tau'$ is $V$-definable from
$\psi(\la \dot{r} \vert \b_0, \dot{g} \vert \b_0 \ra)$ and
$\eta \dot{R} \tau']$''. Since $\psi(\la \dot{r} \vert
\b_0, \dot{g} \vert \b_0 \ra)$ is $V$-definable from
$\la \dot{r} \vert \b_0, \dot{g} \vert \b_0 \ra$ via the
definition ``Take $\la \dot{r} \vert \b_0, \dot{g} \vert \b_0 \ra$,
keep the term $\dot{g} \vert \b_0$ the same, and replace
the first $m$ elements $\la p_1 \cap \b_0, \ldots,
p_n \cap \b_0, p^*_{n+1} \cap \b_0, \ldots, p^*_m \cap
\b_0 \ra$ of $\dot{r} \vert \b_0$ by $\la p_1 \cap \b_0,
\ldots, p_n \cap \b_0, \psi(p^*_{n+1}) \cap \b_0, \ldots,
\psi(p^*_m) \cap \b_0 \ra$'', $\psi(\pi^{**}_1) \forces
``\exists \tau'[\tau'$ is $V$-definable from $\la \dot{r}
\vert \b_0, \dot{g} \vert \b_0 \ra$ and $\eta \dot{R}
\tau']$'', i.e., $\psi(\pi^{**}_1) \forces ``\exists
\tau'[\tau'$ has support $\b_0$ and $\eta \dot{R}
\tau']$''. But $\psi(\pi^{**}_1)$ is compatible with
$\pi_1$ and $\pi_1 \forces ``\neg \exists \tau'[\tau'$ has
support $\b_0$ and $\eta \dot{R} \tau']$'', a
contradiction. This means there must be some term
$\eta_\d$ of support $\b_0<\l$ so that $\pi_0 \forces ``\la
\eta_\a : \a < \d \ra \dot{R} \eta_\d$''. Since $\d <
\theta$ was arbitrary, $\l$ is regular, and $\theta
< \l$, there must be a term $\eta = \la \eta_\a : \a <
\theta \ra$ of some support $\g_0 < \l$ so that for any
$\a < \theta$, $\pi_0 \forces `` \la \eta_\b : \b < \a \ra
\dot{R} \eta_\a$''. By the remarks immediately preceding the
statement of this lemma, this term can be evaluated in
$V[r \vert \g_0, g \vert \g_0] \subseteq N$, thus yielding a
${\rm DC}_\theta({\cal K})$ sequence for $R$ in $N$. This proves
Lemma 5. \hfil\break\finpf Lemma 5 \hfil\break
\indent We remark that in the preceding proof we could have had
$\theta < \l$. However, since no ordinal $\theta \in (\kappa,
\l)$ is a cardinal in $N$, and since $\forall n < \omega
[{\rm DC}_{\aleph_n}(\K)]$ and ${\rm DC}_{\aleph_\omega}(\K)$
are equivalent by the singularity of $\alom$, we have proven
Lemma 5 in the form stated. \hfil\break
\indent We examine some further properties of $\K$ prior to
showing that forcing over $N$ with $\K$ adds no new $\theta$
sequences of elements of $N$ for $\theta < \kappa$. We note that
for $G_{\cal K}$ $N$-generic over $\K$, $\bigcup G_{\cal K} =
F_{\cal K}$ is a 1-1 onto function from $\l$ to $\H$. The
definition of $\K$ clearly ensures $F_{\cal K}$ is 1-1. If $f
\in \K$, dom($f)=\g<\l$, then as we have already observed, $f \in
V[r \vert \d_0,g\vert\d_0]$ for some $\d_0 \in \kintl$. Since
$p^{V[r \vert \d_0,g\vert\d_0]}(\kappa) \ne p^N(\kappa)$ ($r \vert \d_1
\not\in V[r \vert \d_0,g\vert\d_0]$ for $\d_1 \in (\d_0, \l)$), there must
be some $x \in \H$, $x \not\in{\rm rng}(f)$. Then $h = f \cup
\la \g, x \ra$ extends $f$. Density now yields $F_{\cal K}$ is
onto $\H$ with domain $\l$. And, as $F_{\cal K}$ well-orders
$\H$ and $N=V[\H]$, $M=N[G_{\cal K}]=V[{\cal H}][G_{\cal K}] \models
{\rm AC}$. \hfil\break
\noindent Lemma 6: Suppose $p \in \K$, $\theta < \kappa$,
$p \forces ``{\dot f}:\theta \to {\dot N}$ is a function''. Then
for some $q$ extending $p$, $q$ decides a value for ${\dot f}(\a)$
for every $\a<\theta$. \hfil\break
\indent Proof of Lemma 6: Lemma 6 will be implied by Lemma
5. Specifically, we proceed by induction, taking as our inductive
hypothesis that for any $p \in \K$ and any $\theta < \kappa$,
there is a $q$ extending $p$ so that $q$ decides a value for
${\dot f}(\a)$ for every $\a < \theta$. If $\theta = \b+1$ and
the hypothesis is true for $\b$, let $p \in K$ and $q'$
extending $p$ be so that $q'$ decides a value for ${\dot f}(\a)$
for every $\a < \b$. We can then pick $q$ extending $q'$ so that
$q$ decides a value for ${\dot f}(\b)$. This $q$ is the
required condition. \hfil\break
\indent Assume now that $\theta < \kappa$ is a limit ordinal and
the hypothesis is true for every $\b < \theta$. Fix $p \in
\K$. Define now a relationship $R \in N$ on ${[\K]}^{<\theta}
\times \K$ by $\la p_\a : \a<\b<\theta \ra R q$ iff $q$
extends $p_\a$ for $\a<\b$, $\la p_\a : \a<\b \ra$ is so that
$\a<\g<\b$ implies $p_\a \subseteq p_\g$, and $q$ decides a value
for ${\dot f}(\a)$ for $\a<\b$. It is easy to verify that $\K$
is $\g$-additive for any $\g<\l$, meaning $\bigcup_{\a<\b} p_\a =
q' \in \K$ is so that $q'$ extends $p_\a$ for $\a<\b$. By
induction, we can let $q''$ extending $q'$ be so that $q''$
decides a value for ${\dot f}(\a)$ for every $\b < \a$. Thus,
$R$ has the ${\rm DC}_\theta(\K)$ property, so by Lemma 5, let
$\la p_\a: \a<\theta \ra$ be a ${\rm DC}_\theta(\K)$ sequence for
$R$. $q = \bigcup_{\a<\theta} p_\a$ will then decide a value for
${\dot f}(\a)$ for every $\a<\theta$. This proves Lemma 6.
\hfil\break\finpf Lemma 6 \hfil\break
\noindent Lemma 7: $N \models \fordc$. \hfil\break
\indent Proof of Lemma 7: Let $\theta < \kappa$ be
arbitrary. Since $N \models ``\keqal$'', Lemma 7 will be proven
if we can show that $N \models {\rm DC}_\theta$. Thus, let $R
\subseteq {[X]}^{<\theta} \times X$, $X,R \in N$ be a relation
so that $N \models ``\forall x \in {[X]}^{<\theta} \exists y \in X[x
R y]$''. (In this case, ${[X]}^{<\theta}$ is just $<\theta$
sequences of elements of $X$. We presume no other relationship on
elements of $X$.) \hfil\break
\indent By Lemma 6, $M$ and $N$ contain the same $\theta$ sequences
of elements of $N$, since if $p_0 \forces ``{\dot f} : \theta \to
\dot{N}$ is a function'' and $p_1$ extending $p_0$ is so that $p_1$
decides a value for ${\dot f}(\a)$ for every $\a<\theta$, then
$\dot f$ is completely determined in $N$ by $h(\a)$ = The value
$p_1$ decides ${\dot f}(\a)$ has. As $N \subseteq M$ and
$R,X \in N$, $R,X \in M$, and since $M$ and $N$ contain the same
$\theta$ sequences of elements of $N$, $R$ is so that
$M \models ``\forall x \in {[X]}^{<\theta} \exists y
[xRy]$''. Thus, since $M \models {\rm AC}$, let $\la x_\a : \a
<\theta \ra \in M$ be a ${\rm DC}_\theta$ sequence for $R$. Since
$M$ contains no new $\theta$ sequences of elements of $N$,
$\la x_\a : \a < \theta \ra \in N$. Hence, $N \models
{\rm DC}_\theta$. This proves Lemma 7. \hfil\break
\finpf Lemma 7 \hfil\break
\indent We again remark that as before, in Lemmas 6 and 7, $\theta$
could be any ordinal $<\l$. \hfil\break
\indent Lemmas 1-7 complete the proof of our Theorem. \hfil\break
\finpf Theorem \hfil\break
\indent Let us observe that since $N \models \fordc$, there can be
none of the usual large cardinals (weakly compact, Ramsey,
Jonsson, Rowbottom, measurable, etc.) below $\alom$ in $N$. In
fact, if GCH below $\kappa$ holds in $V$, we can show $N
\models ``\forall n < \omega[2^{\aleph_n} = \aleph_{n+1}]$'' as
well. This leaves open, of course, the question of a large
cardinal property for $\alom$ in $N$. The most natural question
to ask is if $\alom$ is a Rowbottom cardinal in $N$. Indeed,
under AD + DC (see [Kl]) and in the models constructed via
forcing [A3], [A4], [AH] in which $\alomp1$ is a measurable
cardinal, $\alom$ is a Rowbottom cardinal. Unfortunately, it
seems highly unlikely that $\alom$ is a Rowbottom cardinal in
$N$. In fact, we conjecture that producing a model for ``ZF +
$\fordc$ + $\alom$ is a Rowbottom cardinal'' is probably as
difficult as producing a model for ``ZFC + $\alom$ is a
Rowbottom cardinal'', the last remaining open problem from
Silver's thesis [Si]. \hfil\break
\indent One may then wonder if it is possible to weaken our
requirements a little and produce a model for ``ZF +
${\rm DC}_{\aleph_n}$ + $\alom$ is a Rowbottom cardinal +
$\alomp1$ is a measurable cardinal'' for some arbitrary $n <
\omega$. In fact, since the methods of [A3], [A4], and [AH] will
yield a model for ``ZF + $\neg{{\rm AC}_\omega}$ +
$\aleph_{2n+1}$ is a measurable cardinal for $n<\omega$ +
$\alom$ is a Rowbottom cardinal + $\alomp1$ is a measurable
cardinal'', and since a model for ``ZF + ${\rm DC}_{\aleph_n}$ +
$\alom$ is a Rowbottom cardinal'' for arbitrary $n < \omega$ is
constructed in [A2], we can ask if it is possible for arbitrary
$n < \omega$, using some variant of the methods of this paper, to
produce a model for ``ZF + ${\rm DC}_{\aleph_n}$ + For
$k < \omega$, $\aleph_{n+2k+1}$ is a measurable cardinal +
$\alom$ is a Rowbottom cardinal + $\alomp1$ is a measurable
cardinal''. Although we haven't yet done this, we feel the
answer to the preceding is yes. \hfil\break
As promised at the beginning, we outline how the results of this paper
can be obtained from the assumption of the existence of cardinals
$\kappa < \l$ so that $\kappa$ is $\l$ supercompact and $\l$ is
measurable. Note that we used the fact that $\kappa$ is
$2^\l$ supercompact for $\l>\kappa$ measurable to obtain a
supercompact ultrafilter $\U$ over $\pkl$ so that $\U$ satisfies
the Menas partition property [Me] and so that $C_0 = \{p \in
\pkl : p \cap \kappa$ is a measurable cardinal and $\ov p$ is
the least measurable cardinal $> p \cap \kappa \} \in
\U$. The only place we need to use the Menas partition property
is in the proof that $V[r,g] \models ``\keqal$'' (Theorem 3.2 of
[Mag]), and we need $C_0 \in \U$ in order to construct the
automorphism $\psi$ of Lemmas 3 and 5 and in order to know that each
$p \cap \kappa$ for $p \in A \subseteq C_0$, $A \in \U$ is
inaccessible so the proof of the analogue of Lemma 2.8 of [Mag]
and the proof of Lemma 2 will go through. However, as in [A1],
the weaker property that there is some normal measure
${\cal U}^*$ over $\pkl$ so that (+) for some $C^*_0 \in
{\cal U}^*$, if
$p \in C^*_0$, $q \in C^*_0$, $p \cap \kappa = q \cap \kappa$, then
$\vert p \vert = \vert q \vert$ will suffice in the construction
of the automorphism $\psi$, and the fact $V^{\pkl} /
{\cal U}^* \models ``\l$ is inaccessible and $\kappa$ is
inaccessible'' will let us assume that for $p \in C^*_0$, $p \cap
\kappa$ is inaccessible and $\ov p$ is inaccessible, thus
allowing the proofs of the analogues of Lemma 2.8 of [Mag] and
Lemma 2 to be carried out. Lemma 3 of [AH] then shows that the
proof that $V[r,g] \models ``\keqal$'' can be done without using
the Menas partition property. Thus, it will
suffice to show that if
$\kappa$ is $\l$ supercompact for $\l > \kappa$ a measurable
cardinal, then there is some normal measure ${\cal U}^*$ over
$\pkl$ satisfying property (+). \hfil\break
\indent To see this, as in [A1], the fact that $\l$ is
inaccessible allows us to conclude that for every inaccessible
$\d\in\kintl$, there is a supercompact ultrafilter ${\cal U}_\d$
over $P_\kappa(\d)$ satisfying property (+). Therefore, we can
define a tree ${\cal T}$ of ultrafilters by ${\cal U}'
\le_{\cal T} {\cal U}''$ iff ${\cal U}'$ is a supercompact
ultrafilter over $P_\kappa(\d')$, ${\cal U}''$ is a supercompact
ultrafilter over $P_\kappa(\d'')$, $\d' \le \d''$, $\d'$ and
$\d''$ are inaccessible, ${\cal U}'$ and ${\cal U}''$ both
satisfy property (+), and ${\cal U}' = {\cal U}'' \vert
\d'$, where ${\cal U}'' \vert \d' = \{A \vert \d': A \in
{\cal U}'' \}$ for $A \vert \d' = \{ p \cap \d' : p \in
A \}$. Since
there are unboundedly many in $\l$ inaccessibles in
$\kintl$, and since if $\d' \le \d''$, ${\cal U}''$ a supercompact
ultrafilter over $P_\kappa(\d'')$ satisfying property (+) means
${\cal U}'' \vert \d'$ is a supercompact ultrafilter over
$P_\kappa(\d')$ satisfying property (+), for any $\d\in\kintl$,
there is a branch in ${\cal T}$ of size at least $\d$. Further,
for any $\d\in\kintl$, there are $<\l$ supercompact ultrafilters
over $P_\kappa(\d)$. Thus, since $\l$ is weakly compact, we can
apply the tree property to obtain a size $\l$ branch of
${\cal T}$. Write this branch as $\la {\cal U}_{\d_\a} : \d_\a
< \l \ra$ where $\la \d_\a : \a < \l \ra$ enumerates the
inaccessibles in $\kintl$. But it can then be verified that for
$\mu$ any normal measure over $\l$, the ultrafilter
${\cal U}^*$ over $\pkl$ given by $A \in {\cal U}^*$ iff
$\{\d<\l : (A \cap P_\kappa(\d)) \in {\cal U}_\d \} \in \mu$ is a
normal measure over $\pkl$ satisfying property (+).
\hfil\break
In conclusion, we mention that Shelah has shown [Sh] that the
result of this paper is the best possible. Specifically, he
has shown that ZF + DC $\vdash$ If $\vert H(\kappa) \vert =
\kappa$ and $\aleph_0 < {\rm cof}(\kappa) < \kappa$, then $\kappa^+$ is
regular but non-measurable; in fact, under these circumstances,
the partition relation $\kappa^+ \to
{(\kappa^+)}^2_2$ fails, i.e., $\kappa^+$ isn't even
weakly compact. Thus, a model
for ``ZF + ${\rm DC}_\kappa$ + $\kappa^+$ is measurable + $
\aleph_0 < {\rm cof}(\kappa) < \kappa$'' is impossible. \hb
\noindent Acknowledgement: We are grateful to George Kafkoulis for
several discussions, by e-mail and in person, concerning his
proof of [Ka1].
\hfil\break
\vfill\eject
\frenchspacing\vskip 1in\centerline{References}\vskip .50in
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\hfil\break
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\hfil\break
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\hfil\break
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\hfil\break
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\hfil\break\vfill\eject\end