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\def\forces{\hbox{$\|\hskip-2pt\hbox{--}$\hskip2pt}}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
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\def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3}
\def\finpf{\hfill$\square$}
\def\a{ \alpha }
\def\b{ \beta }
\def\g{ \gamma }
\def\f{ \varphi }
\def\d{ \delta }
\def\l{ \lambda }
\def\k0{\kappa_0}
\def\k00{\kappa^*_0}
\def\la{\langle}
\def\ra{\rangle}
\def\an{\a_\nu}
\def\anp{\a_{\nu + 1}}
\def\prt{R_{< \kappa^+_0}}
\def\un{\underline}
\def\ov{\overline}
\def\mag{ \subset_{{}_{{}_{\!\!\!\!\!\sim}}} }
$\ $ \vskip 1.5in
\centerline{``On the Class of Measurable Cardinals Without the
Axiom of Choice''}
\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: Using techniques of Gitik in conjunction with
a large cardinal hypothesis whose
consistency strength is strictly in between that of a supercompact
and an almost huge cardinal, we obtain the relative consistency of
the theory ``$ZF + \neg{AC_\omega} + \kappa > \omega$ is measurable iff
$\kappa$ is the successor of a singular cardinal''.
\hfil\break\vfill\eject
\vskip 1in
\centerline{``On the Class of Measurable Cardinals Without the
Axiom of Choice''}
\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 It is well known that when the Axiom of Choice (AC)
becomes false, the structure of the set theoretic universe can be
radically altered. As an example, $\aleph_1$ can be singular
[L]. Indeed, all uncountable successor and limit cardinals can be
singular [G1], or in fact, any desired uncountable successor
cardinal, along with all limit cardinals, can be singular [G2].
\hfil\break
\indent The bizarre behavior of the set theoretic universe in the
absence of AC extends to large cardinals as well. Large cardinals
such as Ramsey and measurable cardinals can be successor instead of
limit cardinals [J1], [T], [A3], [A4], [A5], [AH]. It is even
possible to have non-vacuously the consistency of the theory
``$ZF + \neg{AC_\omega} + \kappa > \omega$ is regular iff $\kappa$ is
measurable'' [A1]. \hfil\break
\indent The purpose of this paper is to show that yet more unusual
possibilities for the structure of the class of measurable cardinals
without the Axiom of Choice can occur. It was shown in [A4],
[A5], and [AH] that it is possible to force and obtain a model in
which the successor of a singular cardinal is measurable. (See
[K] for a discussion of this result in the context of the Axiom of
Determinacy (AD).) We generalize this result in the spirit of
[A1] to show the consistency of the theory ``$ZF + \neg {AC_\omega}
+ \kappa > \omega$ is measurable iff $\kappa$ is the successor of a
singular cardinal'' relative to a certain large cardinal
assumption. Specifically, we prove the following \hfil\vskip .09in \noindent
Theorem: Let $V \models ``ZFC +$ There is a cardinal $\kappa_0$ so
that: \hfil\vskip .09in \noindent
1. $\kappa_0$ is superstrong via $j$, i.e., there is a $j:V \to M$
with crit($j) = \kappa_0$ so that $V_{j(\kappa_0)} \subseteq M$.
\hfil\vskip .09in \noindent
2. $\kappa_0$ is $< j(\kappa_0)$ supercompact, i.e., for all $\a < \kappa_0$,
$\kappa_0$ is $\a$ supercompact. \hfil\vskip .09in \noindent
3. The inner model $M$ is so that $V_{j(\kappa^*_0 + 1)} \subseteq M$,
where $\kappa^*_0$ is the least cardinal $> \kappa_0$ which is $<
j(\kappa_0)$ supercompact''. \hfil\vskip .09in \noindent
Let, in addition, $A, B \subseteq \kappa_0$, $A, B \in V$ satisfy the
following properties: \hfil\vskip .09in \noindent
a. $A \cap B = \emptyset$ and $A \cup B = \kappa_0$. \hfil\vskip .09in \noindent
b. If $\l < \kappa_0$ is a limit ordinal, then $\l \in A$. \hfil\vskip .09in \noindent
c. If $\nu \in A$, then $\nu + 1, \nu + 2 \in B$. \hfil\vskip .09in \noindent
There is then a sequence $\la \a_\nu : \nu < \kappa_0 \ra$ and a model
$N_A$ of height $\kappa_0$ for the theory ``$ZF + \neg{AC_\omega} +$ The
cardinal $\g$ is singular iff $\g = \a_\nu$ for some $\nu \in
A$ + For all $\nu \in A$, $\g^+ = \a^+_\nu$ is measurable and
carries a normal measure + If $\g$ is not the successor of a
singular cardinal, then $\g$ isn't measurable''. \hfil\break
\indent Note that the conditions on $A$ and $B$ imply that $B$ is
composed entirely of successor ordinals $< \kappa_0$ and that the final
model $N_A$ will satisfy ``$\kappa$ is measurable iff $\kappa$ is the
successor of a singular cardinal''. Since $N_A \models
\neg{AC_\omega}$, i.e., since $N_A \not \models DC$, the fact that
each measurable cardinal carries a normal measure is
significant. Note further that if $\kappa$ is an almost huge cardinal,
i.e., if there is a $j:V \to M$ with crit$(j) = \kappa$ so that
$M^{< j(\kappa)} \subseteq M$, then $\kappa$ possesses properties
(1)-(3) above, and as shown in [A2], $\kappa$ has a normal measure
concentrating on cardinals satisfying properties (1)-(3)
above. Thus, the $\kappa_0$ used in the construction of the model
$N_A$ is strictly weaker in consistency strength than an almost
huge cardinal (and since $\kappa^*_0 > \kappa_0$ is inaccessible, is
strictly stronger in consistency strength than a supercompact
cardinal). \hfil\break
\indent Turning now to our proof, the proof of the Theorem uses
Gitik's techniques of [G2] (see also [A1], [A2], and [A3]) to
construct $N_A$. To begin, if $V$ satisfies the hypotheses of
the Theorem, let $A$, $B$, $\kappa_0$, and $j:V \to M$ be as in
these hypotheses. The first step in the proof is to define a Radin
sequence of measures $\mu_{< \kappa_0^+} = \la \mu_\a : \a <
\kappa_0^+ \ra$ of length $\kappa_0^+$ over $P_{\kappa_0}(\kappa_0^*)$. Specifically, if
$\a = 0$, $\mu_\a$ is defined by $X \in \mu_\a$ iff
$\la j(\b) : \b < \kappa^*_0 \ra \in j(X)$, and if $\a > 0$, $\mu_\a$ is
defined by $X \in \mu_\a$ iff $\la \mu_\b : \b < \a \ra
=_{\rm df} \mu_{< \a} \in j(X)$. As in [A2], properties (1) and (3)
above ensure that this definition makes sense. \hfil\break
\indent Next, using $\mu_{< \kappa_0^+}$, we let $R_{< \kappa_0^+}$ be
supercompact Radin forcing defined over $V_{\kappa_0} \times
P_{\kappa_0}(\kappa^*_0)$. The particulars of the definition can be found in
[G2] and [A3]; however, in the interest of completeness and clarity,
we repeat the definition here. $R_{< \kappa_0^+}$ is composed of all
finite sequences of the form $\la \la p_0, u_0, C_0, \ra, \ldots,
\la p_n, u_n, C_n \ra , \la \mu_{<\kappa_0^+}, C \ra \ra$ satisfying
the following properties: \hfil \vskip .09in
\noindent 1. For $0 \le i < j \le n$, $p_i \mag p_j$, where for
$p, q \in P_{\kappa_0}(\kappa^*_0)$,
$\ p \,\mag\, q$ means $p \subseteq q$ and
$\overline{p} < q \cap \kappa$. ($\overline{p}$ is the order
type of p.) \hfil\vskip .09in
\noindent 2. For $0 \le i \le n$, $p_i \cap \kappa_0$ is a
$<\kappa_0$ supercompact cardinal. \hfil
\vskip .09in \noindent 3. $\overline{p_i}$ is the least cardinal
$> p_i \cap \kappa_0$ which is a $< \kappa_0$ supercompact
cardinal. In analogy to our earlier notation and the notation of
[G2], we write $\overline{p_i} = {(p_i \cap \kappa_0)}^*$.
\hfil\vskip .09in\noindent 4. For $0 \le i \le n$, $u_i$ is a
Radin sequence of measures over $V_{p_i \cap \kappa_0} \times
P_{p_i \cap \kappa_0}(\overline{p_i})$ with ${(u_i)}_0$, the
$0^{\rm th}$ coordinate of $u_i$, a supercompact measure over
$P_{p_i \cap \kappa_0}(\overline{p_i})$. \hfil\vskip .09in
\noindent 5. $C_i$ is a sequence of measure 1 sets for $u_i$.
\hfil\vskip .09in\noindent 6. $C$ is a sequence of measure 1
sets for $\mu_{<\kappa^+_0}$. \hfil\vskip .09in
\noindent 7. For each $p \in {(C)}_0$, where ${(C)}_0$ is the
coordinate of $C$ so that ${(C)}_0 \in \mu_0$,
$\cup^n_{i = 0} p_i \mag p$. \hfil\vskip .09in
\noindent 8. For each $p \in {(C)}_0$, $\overline{p} =
{(p \cap \kappa_0)}^*$ and $p \cap \kappa_0$ is a $< \kappa_0$
supercompact cardinal. \hfil\break
\indent Properties (4), (5), and (6) are all standard properties
of Radin forcing. Properties (1), (2), (3), (7), and (8) all
follow since properties (1)-(3) of $\kappa_0$ of the hypotheses
of the Theorem imply $M \models ``\kappa_0$ is $< j(\kappa_0)$
supercompact and $\kappa^*_0$ is the least cardinal $> \kappa_0$
which is $< j(\kappa_0)$ supercompact'', so by reflection,
$\{ p \in P_{\kappa_0}(\kappa^*_0) : p \cap \kappa_0$ is a
$< \kappa_0$ supercompact cardinal and $\overline{p}$ is the
least $< \kappa_0$ supercompact cardinal $> p \cap \kappa_0
\} \in \mu_0$. \hfil\break
\indent We recall now the definition of the ordering on
$R_{< \kappa^+_0}$. If $\pi_0 = \la \la p_0, u_0, C_0 \ra ,
\ldots , \la p_n, u_n, C_n \ra ,\hfil\break \la \mu_{< \kappa^+_0} ,
C \ra \ra$ and $\pi_1 = \la \la q_0, v_0, D_0 \ra , \ldots ,
\la q_m, v_m, D_m \ra , \la \mu_{\kappa^+_0} , D \ra \ra$, then
$\pi_1$ extends $\pi_0$ iff the following conditions hold:
\hfil\vskip .09in \noindent 1. For each $\la p_j, u_j, C_j \ra$
which appears in $\pi_0$ there is a $\la q_i, v_i, D_i \ra$ which
appears in $\pi_1$ so that $\la q_i, v_i \ra = \la p_j, u_j \ra$
and $D_i \subseteq C_j$, i.e., for each coordinate ${(D_i)}_\a$
and ${(C_j)}_\a$, ${(D_i)}_\a \subseteq {(C_j)}_\a$. \hfil
\vskip .09in \noindent 2. $D \subseteq C$. \hfil\vskip .09in
\noindent 3. $n \le m$. \hfil\vskip .09in \noindent
4. If $\la q_i, v_i, D_i \ra$ does not appear in $\pi_0$, let
$\la p_j, u_j, C_j \ra$ (or $\la \mu_{< \kappa^+_0}, C \ra$) be
the first element of $\pi_0$ so that $p_j \cap \kappa_0 >
q_i \cap \kappa_0$. Then \hfil\vskip .06in \indent a) $q_i$ is
order isomorphic to some $q \in {(C_j)}_0$. \hfil\vskip .06in
\indent b) There exits an $\a < \a_0$, where $\a_0$ is the length
of $u_j$, so that $v_i$ is isomorphic ``in a natural way'' to an
ultrafilter sequence $v \in {(C_j)}_\a$. \hfil\vskip .06in
\indent c) For $\b_0$ the length of $v_i$, there is a function
$f: \b_0 \to \a_0$ so that for $\b < \b_0$, ${(D_i)}_\b$ is a
set of ultrafilter sequences so that for some subset
${{(D_i)}_\b}'$ of ${(C_j)}_{f(\b)}$, each ultrafilter sequence
in ${(D_i)}_\b$ is isomorphic ``in a natural way'' to an ultrafilter
sequence in ${{(D_i)}_\b}'$. \hfil\vskip .09in
\noindent For further information on the definition of the ordering
on $R_{< \kappa^+_0}$ (including the meaning of ``in a natural
way''), readers are referred to [A3] and [FW]. \hfil\break
\indent Before giving the definition of the partial ordering used in
the construction of the model for our Theorem, we recall the
definition of two key partial orderings. If $\a < \b$ are regular
cardinals, then $Col( \a, < \b)$ is just the usual L\'evy collapse
of all cardinals in the interval $(\a, \b)$ to $\a$, i.e.,
$Col(\a, < \b) = \{ f: \a \times \b \to \a : f$ is a function
so that $\vert {\rm dom}(f) \vert < \a$ and $f(\la \g, \d \ra) < \d
\}$ ordered by inclusion. For $\sigma \in (\a, \b)$ a regular
cardinal, $f \in Col(\a,<\b)$, $f \vert \sigma = \{ \la \la
\g,\d \ra,\rho \ra \in f : \d<\sigma \}$. If $G$ is $V$-generic
over $Col(\a,<\b)$, then $G \vert \sigma = \{ f \vert \sigma :
f \in G \}$ is $V$-generic over $\{ f \vert \sigma : f \in
Col(\a,<\b) \} = Col(\a,<\b) \vert \sigma = Col(\a,<\sigma)$.
\hfil\break
\indent If $\a$ is $\b$ supercompact, then let ${\cal U}$ be a
normal measure over $P_\a(\b)$ satisfying the Menas partition
property [M]. (Such an ultrafilter will always exist if $\a$
is $2^\b$ supercompact, a restriction which will cause no problems
since we will be working with cardinals $\a < \b < \kappa_0$, so
$\a$ can be chosen to be $< \kappa_0$ supercompact.) Supercompact
Prikry forcing $SC(\a, \b)$ is then defined as all sequences of
the form $\la p_0, \ldots, p_n, C \ra$ so that: \hfil\vskip .09in
\noindent 1. $n \in \omega$ and $C \in {\cal U}$. \hfil\vskip .09in
\noindent 2. For $0 \le i \le n$, $p_i \in P_\a(\b)$.
\hfil\vskip .09in\noindent 3. For $0 \le i < j \le n$,
$p_i \mag p_j$. \hfil\vskip .09in\noindent 4. For each $q \in C$,
$p_n \mag q$. \hfil\break
\indent For $\pi_1 = \la p_0, \ldots, p_n, C \ra$ and $\pi_2 =
\la q_0, \ldots, q_m, D \ra$, $\pi_2$ extends $\pi_1$ iff:
\hfil\vskip .09in\noindent 1. $n \le m$. \hfil\vskip .09in
\noindent 2. For $0 \le i \le n$, $p_i = q_i$. \hfil\vskip .09in
\noindent 3. For $n+1 \le i \le m$, $q_i \in C$. \hfil\vskip .09in
\noindent 4. $D \subseteq C$. \hfil\break
\indent We now define a partial ordering $P'$ by $$P' =
R_{<\kappa^+_0} \times \prod_{\{ \la \a, \b \ra : \a < \b <
\kappa_0 {\rm \ are \ regular \ cardinals}\!\}}Col(\a, <\b)
$$ \break $$\times
\prod_{\{ \la \a, \b \ra : \a < \b < \kappa_0 {\rm \ is \ so \ that \ }
\b {\rm \ is \ a \ regular \ cardinal \ and \ } \a {\rm \ is} \
< \kappa_0 {\ \rm supercompact}\!\}} SC(\a, \b)$$ ordered
componentwise, and let $P$ be the subordering of $P'$ consisting of
all conditions of finite support, also ordered componentwise. Let
$G$ be $V$-generic over $P$. The model $N_A$ for $A$ as in the
statement of the Theorem will be a submodel of $V[G]$ similar to
the models $N_A$ of [G2] and [A3]. We describe this model in more
detail below. \hfil\break
\indent Let $G_0$ be the projection of $G$ onto
$R_{< \kappa^+_0}$. For any condition $\pi = \la \la p_0,
u_0, C_0 \ra, \ldots, \la p_n, u_n, C_n \ra, \hfil\break\la
\mu_{< \kappa^+_0}, C \ra \ra \in R_{< \kappa^+_0}$ or any
condition $\pi = \la p_0, \ldots, p_n, C \ra \in
SC(\a, \b)$ call $\la p_0, \ldots, p_n \ra$ the p-part of
$\pi$. Let $R = \{p : \exists \pi \in G_0[p \in$ p-part($
\pi)]\}$ and let $R_l = \{p : p \in R$ and $p$ is a limit
point of $R\}$. We define three sets $E_0$, $E_1$, and $E_2$ by
$E_0 = \{\a :$ For some $\pi \in G_0$ and some $p \in $p-part$(
\pi)$, $p \cap \kappa_0 = \a\}$, $E_1 = \{\a : \a$ is a limit
point of $E_0\}$, and $E_2 = E_1 \cup \{ \omega \} \cup
\{\b : \exists \a \in E_1[\b = \a^*]\}$. Let $\la \a_\nu :
\nu < \kappa_0 \ra$ be the continuous increasing enumeration of
$E_2$, and let $\nu = \nu' + n$ for some $n \in \omega$. For
$\b$ where $\b \in [\a_\nu, \a_{\nu + 1})$ in the first four
cases, $\b \in [\a^+_\nu, \a_{\nu + 1})$ in the fifth case, and
$\b = \a_{\nu + 1}$ in the last two cases, sets $C_i(\a_\nu,
\b)$ and $C_i(\a^+_\nu, \b)$ are defined according to specific
conditions on $\nu$ and $\nu'$ in the following manner:
\hfil\vskip .09in\noindent 1. $\nu' = \nu \ne 0$ and
$n=0$. Let then $p(\a_\nu)$ be the element $p$ of $R$ so that
$p \cap \kappa_0 = \a_\nu$, and let $h_{p(\a_\nu)} : p(\a_\nu)
\to \overline{p(\a_\nu)}$ be the order isomorphism between
$p(\a_\nu)$ and $\overline{p(\a_\nu)}$. $C_1(\a_\nu, \b) =
\{{h_{p(\a_\nu)}}''p \cap \b : p \in R_l$, $ p \subseteq p(\a_\nu),$
and $h^{-1}_{p(\a_\nu)}(\b) \in p\}$. \hfil\vskip .09in
\noindent 2. $\nu' \ne \nu$ and $n = 2k$. Let $C_2(\a_\nu,
\b) = \{{h_{p(\a_\nu)}}''p \cap \b : p \in R$, and if $(\nu' \ne
0)$ or $(\nu' = 0$ and $k \ge 1)$, then $p(\a_{\nu' + 2(k-1)})
\subset p \subseteq p(\a_\nu)\}$. \hfil\vskip .09in
\noindent 3. $\nu' \ne \nu$ and $n=2k+1$. Let $G(\a_\nu,
\a_{\nu + 1})$ be the projection of $G$ onto $SC(\a_\nu, \a_{
\nu + 1})$. $C_3(\a_\nu, \b) = \{p \cap \b : \exists \pi \in
G(\a_\nu, \a_{\nu + 1})[p \in $p-part($\pi)]\}$. \hfil\vskip .09in
\noindent 4. $n \ne 0$ or $\nu' = n = 0$. Let $H(\a_\nu, \a_{\nu
+1})$ be the projection of $G$ onto $Col(\a_\nu, <\a_{\nu+1})
$. $C_4(\a_\nu, \b) = H(\a_\nu. \a_{\nu+1}) \vert \b$. \hfil
\vskip .09in\noindent 5. $n \ne 0$. Let $H(\a^+_\nu, \anp)$ be the
projection of $G$ onto $Col(\a^+_\nu, <\anp)$. $C_5(\a^+_\nu, \b) =
H(\a^+_\nu, \anp) \vert \b$. \hfil\vskip .09in
\noindent 6. $n \ne 0$ or $\nu' = n = 0$. With $H(\an, \anp)$
having the same meaning as in (4) above, $C_6(\an, \anp) =
H(\an, \anp)$. \hfil\vskip .09in \noindent 7. $n \ne 0$. With
$H(\a^+_\nu, \anp)$ having the same meaning as in (5) above,
$C_7(\a^+_\nu, \anp) = H(\a^+_\nu, \anp)$. \hfil\break
\indent We can now give a description of the model $N_A$
witnessing the conclusions of our Theorem. Intuitively, $N_A$
is $V_{\kappa_0}$ of the least model of $ZF$ extending $V$
which contains, for $\b$ as above, $C_1(\an, \b)$ if $\nu$ is a
limit ordinal, $C_2(\an, \b)$ if $\nu = \nu' + 2k$ and $\nu
\in A$, $C_3(\an, \b)$ if $\nu = \nu' + 2k + 1$ and $\nu \in A$,
$C_4(\an, \b)$ if $\nu \in B$, $\nu + 1 \in A$, and for $\nu - 1$
the immediate predecessor of $\nu$ (which exists since $B$ is
composed entirely of successor ordinals), $\nu - 1 \in B \cup
\{0\}$, $C_5(\a^+_\nu, \b)$ if $\nu \in B$, $\nu + 1 \in A$, and
$\nu - 1 \in A$, $C_6(\an, \anp)$ if $\nu - 1, \nu, \nu + 1 \in
B \cup \{0\}$, and $C_7(\a^+_\nu, \anp)$ if $\nu, \nu + 1 \in B$
and $\nu - 1 \in A$. The $C_i$ have been chosen so as to ensure
that successors of singular cardinals are measurable and
successors of regular cardinals are non-measurable. \hfil\break
\indent To define $N_A$ more precisely, it is necessary to define
canonical names $\underline{\an}$ for the $\an$'s and canonical
names $\underline{C_i(\nu, \b)}$ and $\underline{C_i(\nu,
\nu + 1)}$ for the seven sets just described. Recall that it is
possible to decide $p(\an)$ (and hence $\overline{p(\an)}$) by
writing $\omega \cdot \nu = \omega^{\sigma_0} \cdot n_0 +
\omega^{\sigma_1} \cdot n_1 + \cdots + \omega^{\sigma_m} \cdot
n_m$ (where $\sigma_0 > \sigma_1 > \cdots > \sigma_m$ are
ordinals, $n_0, \ldots, n_m > 0$ are integers, and $+$,
$\cdot$, and exponentiation are as in ordinal arithmetic), letting
$\pi = \la{\la p_{ij_i}, u_{ij_i}, C_{ij_i} \ra}_{i \le m,
1 \le j_i\le n_i}, \la \mu_{<\kappa^+_0}, C \ra \ra$ be so that
min($p_{i1} \cap \kappa_0, \omega^{{\rm length}(u_{i1})}) =
\sigma_i$ and length($u_{ij_1}) = {\rm min}(p_{i1} \cap \kappa_0,
{\rm length}(u_{i1}))$ for $1 \le j_i \le n_i$, and letting
$p(\an)$ be $p_{mn_m}$. Further, $D_\nu = \{ r \in P : r \vert
R_{<\kappa^+_0}$ extends a condition $\pi$ of the above form$\}$
is a dense open subset of $P$. $\underline{\an}$ is the name of
the $\an$ determined by any element of $D_\nu \cap G$; in the
notation of [G2], $\underline{\an} = \{ \la r, \check \a_\nu(r)
\ra : r \in D_\nu \}$, where $\an(r)$ is the $\an$ determined
by the condition $r$. \hfil\break
\indent The canonical names $\underline{C_i(\nu, \b)}$ and
$\underline{C_i(\nu, \nu + 1)}$ are defined in a manner so as to
be invariant under the appropriate group of
automorphisms. Specifically, there are seven cases to
consider. We again write $\nu = \nu' + n$ and let $\b$ be as
before. We also assume without loss of generality that as in
[G2], $\anp$ is determined by $D_\nu$. Further, we adopt
throughout each of the seven cases the notation of [G2].
\hfil\vskip .09in\noindent
1. $\nu' \ne \nu \ne 0$ and $n=0$. $\un{C_1(\nu, \b)} = \{
\la r, (\check r \vert \prt) \vert (\an(r), \b) \ra : r \in
D_\nu \}$, where for $r \in P$, $\pi =\hfil\break r \vert \prt$, $\pi \vert
(\an(r), \b) = \{ {{h_{p(\an)(r)}}}''p\cap \b : p \in$p-part$
(\pi)$, $ p \subseteq p(\an)(r)$, $ p \in R_l \vert \pi$, and
$h^{-1}_{p(\an)(r)}(\b) \in p $. \hfil\vskip .09in\noindent
2. $\nu \in A$, $\nu' \ne \nu$, and $n=2k$. Note that as in
[G2] we can assume without loss of generality that\hfil\break for any
$r \in D_\nu$, $r$ determines $\a_{\nu' + 2(k-1)}$. $\un{C_2(\nu,
\b)} = \{ \la r, (\check r \vert \prt) \vert (\an(r), \b) \ra : r
\in D_\nu \}$, where this time for $r \in P$, $\pi = r \vert
\prt$, $\pi \vert (\an(r), \b) = \{ {{h_{p(\an)(r)}}}''p \cap \b
: p \in$p-part$(\pi)$, $ p \in R \vert \pi$, $ p(\a_{\nu' +
2(k-1)})(r) \subseteq p \subseteq p(\an)(r)$, and $h^{-1}_{
p(\an)(r)}(\b) \in p \}$. \hfil\vskip .09in\noindent
3. $\nu \in A$, $\nu' \ne \nu$, and $n=2k+1$. $\un{C_3(\nu,
\b)} = \{ \la r, (\check r \vert SC(\an(r), \a_{\nu+1}(r)))\vert
(\an(r), \b) \ra : r \in D_\nu\}$, where for $r \in P$, $\pi =
r \vert SC(\an(r), \a_{\nu+1}(r))$, $\pi \vert (\an(r), \b) =
\{p \cap \b : p \in$p-part$(\pi)\}$. \hfil\vskip .09in
\noindent 4. $\nu-1 \in B \cup \{0\}$, $\nu \in B$, and $\nu+1
\in A$. $\un{C_4(\nu, \b)} = \{ \la r, (\check r \vert
Col(\an(r), \a_{\nu+1}(r))) \vert \b \ra : r \in D_\nu \}$.
\hfil\vskip .09in\noindent 5. $\nu \in B$, $\nu-1,\nu+1 \in
A$. $\un{C_5(\nu, \b)} = \{ \la r, (\check r \vert
Col(\a^+_\nu(r), \a_{\nu+1}(r))) \vert \b \ra : r \in
D_\nu \}$. \hfil\vskip .09in\noindent 6. $\nu-1, \nu, \nu+1
\in B \cup \{0\}$. $\un{C_6(\nu, \nu+1)} = \{ \la r,
(\check r \vert Col(\an(r), \a_{\nu+1}(r))) \ra : r \in
D_\nu \}$. \hfil\vskip .09in\noindent 7. $\nu, \nu+1 \in
B$ and $\nu-1 \in A$. $\un{C_7(\nu, \nu+1)} = \{ \la
r, (\check r \vert Col(\a^+_\nu(r), \a_{\nu+1}(r))) \ra :
r \in D_\nu \}$. \hfil\vskip.09in\noindent
As in [G2], since for any $r, r' \in D_\nu \cap G$,
$p(\an)(r) = p(\an)(r')$, each of the definitions above is
unambiguous. \hfil\break
\indent Let ${\cal G}$ be the group of automorphisms of [G2],
and let $\un{C(G)} = \bigcup^5_{i=1} \{ \pi(\un{C_i(\nu,\b)}) : \pi
\in {\cal G}$, $ 0<\nu<\kappa_0$, and $\b \in [\nu,\kappa_0)$ is a
cardinal$\} \cup \bigcup^7_{i=6} \{ \pi(\un{C_i(\nu,\nu+1)}) : \pi
\in {\cal G}$ and $0<\nu<\kappa_0\}$. $C(G) = \bigcup^5_{i=1}
\{i_G(\pi(\un{C_i(\nu,\b)})) : \pi \in {\cal G}$, $ 0<\nu<\kappa_0$,
and $\b \in [\nu,\kappa_0)$ is a cardinal$\} \cup \bigcup^7_{i=6}\{
i_G(\pi(\un{C_i(\nu,\nu+1)})) : \pi \in {\cal G}$ and $0
<\nu<\kappa_0\} = i_G(\un{C(G)})$. $N_A$ is then the set of all
sets of rank $<\kappa_0$ of the model consisting of all sets
which are hereditarily $V$ definable from $C(G)$, i.e.,
$N_A = V^{HVD(C(G))}_{\kappa_0}$. \hfil\break
\indent The arguments of [G2] show that $N_A \models ZF+
\neg{AC_{\omega}}$. In addition, we know that for any ordinal
$\g$ and any set $x \subseteq \g$, $x \in N_A$, $x = \{\a<\g :
V[G] \models \f(\a, i_G(\pi_1(\un{C_{i_1}(\nu_1,\b_1)})),\ldots,
i_G(\pi_n(\un{C_{i_n}(\nu_n,\b_n)})), C(G))\}$, where $i_j$ is
an integer, $1 \le j \le n$, $1 \le i_j \le 7$, each $\pi_i
\in {\cal G}$, each $\b_i$ is an appropriate ordinal for $i_j$,
and $\f(x_0, \ldots, x_{n+1})$ is a formula which may also
contain some parameters from $V$ which we shall suppress.
\hfil\break
\indent Let $$\ov{P} = \prod_{\{i_j : i_j \in \{4,6\}, j \le n
\}} Col(\a_{\nu_j}, \b_j) \times \prod_{\{i_j : i_j \in
\{5,7\}, j \le n\}} Col(\a^+_{\nu_j}, \b_j)$$ \hfil\break
$$\times \prod_{\{i_j : i_j = 3, j \le n\}} SC(\a_{\nu_j},
\b_j)\times \prt .$$ \hfil\break\noindent For $\pi \in \prt$
and $\g$ an arbitrary ordinal, let $\pi \vert \g = \{\la q,
u, C \ra \in \pi : q \cap \kappa_0 \le \g \}$, and for
$p \in \ov{P}$, $p = \la p_1, \ldots, p_m, \pi \ra$, $m \le n$,
$\pi \in \prt$, let $p \vert \g = \la q_1, \ldots, q_m, \pi \vert
\g \ra$, where $q_j = p_j$ if either $\a_{\nu_j}$ or
$\a^+_{\nu_j}$ is $\le \g$ and $q_j = \emptyset$ otherwise. In
other words, $p \vert \g$ is the part of p below or at
$\g$. Without loss of generality, we ignore the empty coordinates
and let $\ov{P} \vert \g = \{p \vert \g : p \in \ov{P} \}$. Let
$G \vert \g$ be the projection of $G$ onto $\ov{P} \vert \g$. An
analogous fact to Theorem 3.2.11 of [G2] holds, using the same
proof as in [G2], namely for any $x \subseteq \g$, $x \in
V[G \vert \g]$. In addition, the elements of $\ov{P} \vert \g$
can be partitioned into equivalence classes (the ``almost
similar'' equivalence classes of [G2]) with respect to
$\un{C_{i_1}(\nu_1,\b_1)}, \ldots, \un{C_{i_n}(\nu_n,\b_n)}$
so that if $\sigma < \g$, $\tau$ is a term for $x$, and $p
\,\forces\, \sigma \in \tau$, for any $q$ in the same equivalence class
as $p$, $q \,\forces\, \sigma \in \tau$. Further, if $\nu \in A$, then
the arguments of [G2] show that for $\g = \anp$ there are
$<\anp$ such equivalence classes. It is this last fact, in
tandem with the way in which $N_A$ was defined, that allows us
to show that $N_A$ is our desired model. \hfil\break
\noindent Lemma 1: $N_A \models ``\g$ is a singular cardinal''
iff $\g=\an$ for some $\nu \in A$. \hfil\break
\indent Proof of Lemma 1: In order to prove this lemma, we must
first ascertain the nature of the cardinal structure of
$N_A$. Specifically, we show that all (well-ordered) cardinals of
$N_A$ are either an $\an$ or an ${(\a^+_\nu)}^V$ if $\nu = \sigma
+1$ and $\sigma \in A$. Thus, we begin by showing that any $\g$
for $\g=\an$ or $\g={(\a^+_\nu)}^V$ if $\nu=\sigma+1$ and $\sigma
\in A$ remains a cardinal in $N_A$. \hfil\break
\indent Let $\g$ be as just stated. If $x \subseteq \g$, $x \in
N_A$, then as mentioned before, $x \in V[G \vert \g]$ where
$G \vert \g$ is $V$-generic over $\ov{P} \vert \g$ for $\ov{P}
\vert \g$, $G \vert \g$ as previously described. Thus, it suffices
to show that $\g$ remains a cardinal in $V[G \vert \g]$. To see
this, observe that we can write $\ov{P} \vert \g$ as $Q_0 \times
Q_1$, where $Q_0$ is a partial ordering (possibly trivial)
defined over $\g$ and some ordinal $\b > \g$, and $Q_1$ is the
rest of $\ov{P} \vert \g$. Since by the definition of $N_A$,
$Q_0$ will be either trivial (if $\g=\an$ and $\nu-1 \in A$), a
partial ordering of the form $Col(\g, <\b)$, a partial ordering of
the form $SC(\g,\b)$, or a supercompact Radin forcing defined over
$P_\g(\b)$ isomorphic to a partial ordering of the form
$SC(\g,\b)$, forcing with $Q_0$ will preserve the fact that $\g$
is a cardinal and preserve the same bounded subsets of $\g$ as
in $V$. Working now in $V^{Q_0}$, we can factor $Q_1$ as
$Q_2 \times Q_3$, where $Q_2$ is a partial ordering defined over
ordinals $\g'<\b' \le \g$ and $Q_3$ is the rest of $Q_1$. The
fact that $\ov{P} \vert \g$ is a finite product allows us to assume that
$\g'$ and $\b'$ are the maximum such ordinals. Further, the fact
that $V$ and $V^{Q_0}$ have the same bounded subsets of $\g$
ensures that $Q_2$ can be partitioned into $<\g$ almost similar
equivalence classes in both $V$ and $V^{Q_0}$ unless $Q_2$ is of
the form $Col(\g',<\g)$. In this case, the definition of $N_A$
ensures that $V^{Q_0} \models ``\g$ is a regular cardinal'', so
forcing over $V^{Q_0}$ with $Q_2$ preserves the fact that $\g$
is a regular cardinal; further, $V^{Q_0 \times Q_2} \models
``Q_3$ can be partitioned into $<\g$ almost similar equivalence
classes''. Thus, in either case, since $G \vert \g$ is
$V$-generic over $\ov{P} \vert \g = Q_0 \times Q_1 = Q_0 \times
Q_2 \times Q_3$, $V[G \vert \g] \models ``\g$ is a
cardinal''. \hfil\break
\indent Let now $\la \b_\nu : \nu<\kappa_0 \ra$ be the continuous
increasing enumeration of $\{\an : \nu < \kappa_0\} \cup
\{{(\a^+_\nu)}^V : \nu=\sigma+1$ and $\sigma \in A\}$. As in
[G2], by the fact that the definition of $N_A$ ensures that
$N_A$ contains collapse maps for each $V$ cardinal in the interval
$(\b_\nu, \b_{\nu+1})$ where $\nu < \kappa_0$ is arbitrary, it is
inductively the case that $N_A \models ``\forall \nu [\b_\nu
\le \aleph_\nu]$''. Since each $\b_\nu$ is a cardinal in $N_A$,
$N_A \models ``\forall \nu[\b_\nu = \aleph_\nu]$''. Thus, the
$\b_\nu$'s and the cardinals of $N_A$ coincide. \hfil\break
\indent If $\g=\an$ for some $\nu \in A$, then since the definition
of $N_A$ ensures that $N_A$ contains a cofinal $\omega$ sequence
for $\an$, $N_A \models ``\an$ is a singular cardinal''. If
$\g \ne \an$ for some $\nu \in A$ and $N_A \models ``\g$ is a
cardinal'', then by the preceding paragraph, either $\g=\an$ for
$\nu \in B$ or $\g={(\a^+_\nu)}^V$ for $\nu=\sigma+1$ and
$\sigma \in A$. No matter which of these were true, if $x \in
N_A$ coded a sequence witnessing the singularity of $\g$, then
$x \in V[G \vert \g]$ for $G \vert \g$ as earlier. When
factoring $\ov{P} \vert \g$ into $Q_0 \times Q_1$, since this
case ensures $Q_0$ must be a L\'evy collapse, $V^{Q_0} \models
``\g$ is regular''. Further, when factoring $Q_1$ into $Q_2
\times Q_3$, since our earlier discussion shows either $Q_2$ is
of the form $Col(\g', <\g)$ or is so that $V^{Q_1} \models
``Q_2$ can be partitioned into $<\g$ almost similar equivalence
classes'', $V^{Q_0 \times Q_2} \models ``\g$ is
regular''. Therefore, since $V^{Q_0 \times Q_2} \models ``Q_3$
can be partitioned into $<\g$ almost similar equivalence classes''
and $G \vert \g$ is $V$-generic over $Q_0 \times Q_2
\times Q_3$, $V[G \vert \g] \models ``\g$ is regular''. Thus,
$x$ cannot code a sequence witnessing the singularity of
$\g$. This proves Lemma 1. \hfil\break
\finpf Lemma 1 \hfil\break
\noindent Lemma 2: If $N_A \models ``\g$ is the successor of a
singular cardinal'', then $N_A \models ``\g$ is measurable via
some normal measure''. \hfil\break
\indent Proof of Lemma 2: By Lemma 1, for any $\g$ as in the
hypotheses, $N_A \models ``\g=\a^+_\nu$'' where $\nu \in
A$. Further, the definition of $N_A$ ensures that $\g
=\anp$. \hfil\break
\indent Fix $\mu \in V$ a normal measure over $\g$. In
$N_A$, define
$\mu^* = \{y \subseteq \g : y$ contains a $\mu$ measure 1
set$\}$. We show $N_A \models ``\mu^*$ is a normal measure
over $\g$''. If $x \subseteq \g$, $x \in N_A$, then $x \in
V[G \vert \g]$ for $G \vert \g$ $V$-generic over $\ov{P} \vert
\g$, $\ov{P} \vert \g$, $G \vert \g$ as before. Further, as
mentioned in the sentences immediately preceding the statement
of Lemma 1, the elements of $\ov{P} \vert \g$ can be partitioned
into $<\anp$ many almost similar equivalence classes so that if
$p$ and $q$ are in the same equivalence class, $\tau$ is a
term for $x$, and $p$ decides ``$\sigma \in \tau$'', then $q$
decides ``$\sigma \in \tau$'' in the same way. Thus, as in the
proof of Lemma 1.3 of [A3], in $V[G \vert \g]$, the
L\'evy-Solovay arguments [LS] show $\mu' =\{ y \subseteq \g :
y$ contains a $\mu$ measure 1 set$\}$ is a normal measure
over $\g$. In particular, since $x \in V[G \vert \g]$, either
$x$ or $\g-x$ will contain a $\mu$ measure 1 set. Further, if
$N_A \models ``\la x_\b : \b<\d<\g \ra$ is a sequence of $\mu^*$
measure 1 sets'', then since $\la x_\b : \b<\d<\g \ra$ can be
coded by a single $x \subseteq \g$, for the appropriate
$\ov{P} \vert \g$ and $G \vert \g$, both $x$ and $\la x_\b
: \b<\d<\g \ra$ are elements of $V[G \vert \g]$. Thus,
$V[G \vert \g] \models ``\cap_{\b<\d} x_\b \in \mu'$'', so
$N_A \models ``\cap_{\b<\d} x_\b \in \mu^*$''. Finally, if
$N_A \models ``f:\g \to \g$ is a regressive function'', then
since $f$ can be coded by a set of ordinals, $f \in V[G \vert
\g]$ for the appropriate $\ov{P} \vert \g$ and $G \vert
\g$. Thus, $V[G \vert \g] \models ``f$ is constant on a $\mu'$
measure 1 set'', so $N_A \models ``f$ is constant on a $\mu^*$
measure 1 set''. This proves Lemma 2. \hfil\break
\finpf Lemma 2 \hfil\break
\noindent Lemma 3: If $N_A \models ``\g$ is not the successor of
a singular cardinal'', then $N_A \models``\g$ is not
measurable''. \hfil\break
\indent Proof of Lemma 3: By Lemma 1, for any $\g$ as in the
hypotheses, either $\g={(\a^+_\nu)}^V$ for $\nu=\sigma+1$ and
$\sigma \in A$ or
$\g = \an$ for some $\nu \in B$ so that $\nu-1\in B$. If
$\g={(\a^+_\nu)}^V$, then since $V \models AC$, $V$ contains a
sequence of length ${(\a^+_\nu)}^V$ of subsets of $\an$. Since
$V \subseteq N_A$, this sequence is present in $N_A$ also. It is
well known (see [J2], Lemma 27.2, p. 298) that if such a sequence
exists, regardless of whether AC is true, $\g$ can't be
measurable. \hfil\break
\indent If $\g=\an$ for some $\nu\in B$ so that $\nu-1\in B$, then
by the construction of $N_A$, $N_A$ contains a set of the form
$C_6(\a_{\nu-1}, \an)$ or a set of the form $C_7({\a^+_{\nu-1}},
\an)$. Since $V[C_6(\a_{\nu-1}, \an)] \subseteq N_A$ or
$V[C_7(\a^+_{\nu-1}, \an)] \subseteq N_A$, in either case, there
will be present in $N_A$ a sequence of subsets of some smaller
cardinal of length $\an$. As in the last paragraph, the presence
of such a sequence contradicts the measurability of $\g$. This
proves Lemma 3. \hfil\break
\finpf Lemma 3 \hfil\break
\indent The above three lemmas complete the proof of our
Theorem. \hfil\break
\finpf Theorem \hfil\break
\indent Let us observe that the current state of forcing technology
requires that if $N_A \models ``\an$ is measurable'', then
$N_A \models ``\a^+_\nu$ is regular'' (and of course, by our
requirements, is non-measurable). If we wanted to have $N_A
\models ``\an$ is measurable and $\a^+_\nu$ is singular'', then
there would have to be some way to collapse a singular cardinal
to be the successor of a measurable cardinal $\kappa$ while
preserving the measurability of $\kappa$. Unless $\kappa$ is
to become $\aleph_1$ (see [A1]), it is unknown how to do
this. It is for this reason we require that $N_A$ contains
sets of the form $C_5(\a^+_\nu, \b)$ and $C_7(\a^+_\nu,
\anp)$, since their presence provides enough of a ``gap'' to
ensure that $\an$ remains measurable in $N_A$. More
specifically, their presence ensures that in the analogue to
Theorem 3.2.11 of [G2], since the $Q_0$ of Lemma 1 is trivial,
the partial ordering $\ov{P} \vert \an$ can be partitioned into
$<\an$ almost similar equivalence classes, the $V$-measurable
cardinal which becomes $\a^+_{\nu-1}$ in $N_A$. This, as just
shown, allows $\an$ to remain measurable in $N_A$ while
preserving ${(\a^+_\nu)}^V$ as a regular cardinal. \hfil\break
\indent In conclusion, we remark that from a weaker hypothesis
than that assumed for our Theorem, i.e., from a cardinal
$\kappa_0$ so that $\kappa_0$ is $2^\lambda$ supercompact
for $\l>\kappa_0$ measurable, it is possible to construct a
model for the theory ``$ZF + \neg{AC_\omega} + \kappa >
\omega$ is measurable iff $\kappa$ is the successor of a
singular cardinal''. In this model, all limit cardinals will be
singular and all successor cardinals will be regular, so the
only measurable cardinals will be successors of limit
cardinals. Thus, there is somewhat less flexibility as to what
cardinals can be singular. An outline of the proof is as
follows: If $j:V \to M$ witnesses that $\kappa_0$ is
$2^{\l}$ supercompact for $\l>\kappa_0$ measurable, let
$\kappa^*_0$ in this case be the least measurable $>\kappa_0$,
and let $\prt$ as before be supercompact Radin forcing over
$P_{\kappa_0}(\kappa^*_0)$ defined using $j$. (The fact that
$\kappa_0$ is $2^{\l}$ supercompact ensures this definition
can be given.) Let $$P' = \prt \times
\prod_{\{\la \a, \b \ra : \a<\b<\kappa_0 \ {\rm are \ regular
\ cardinals}\}} Col(\a, <\b),$$ and let $P$ be the subordering
consisting of all conditions of finite support. For $\la \an
: \nu<\kappa_0 \ra$ as before, let $N_A$ be $V_{\kappa_0}$
of the least model of $ZF$ extending $V$ which contains the
appropriate analogues of the sets $C_1(\an, \b)$ if $\nu<
\kappa_0$ is a limit ordinal and $\b\in [\an, \anp)$,
$C_7(\a^+_\nu, \anp)$ if $\nu$ is the successor of a limit
ordinal, and $C_6(\an,\anp)$ if $\nu$ is neither a limit ordinal
nor the successor of a limit ordinal. $N_A$ can then be shown
to be our desired model. Further, as the referee has pointed out,
the methods of this paper can be used to construct, from the
hypotheses of this paragraph, a model for the theory ``$ZF +
\neg{AC_\omega}$ + For every ordinal $\a$, either $\aleph_\a$
or $\aleph_{\a + 1}$ is measurable'', i.e., a model in which
every second cardinal is measurable.
\hfil\break\vfill\eject
\frenchspacing\vskip 1in
\centerline{References}
\vskip .75in
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\bye