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\title{Vaught's conjecture on analytic sets\footnote{1991 {\it Mathematics
Subject Classification.} Primary 03E15. {\it Key words and phrases.} Polish
group. Group actions. Topological Vaught conjecture.}}
\author{Greg Hjorth \footnote{Research partially supported by NSF grant DMS 96-22977}}
\date{\today}
\maketitle
\abstract{Let $G$ be a Polish group. We characterize when there is a
Polish space $X$ with a continuous $G$-action and an analytic set
(that is to say, the Borel image of some Borel set in some Polish
space) $A\subset X$ having uncountably many orbits but no perfect
set of orbit inequivalent points.
Such a Polish $G$-space $X$ and analytic $A$ exist exactly when there
is a continuous, surjective homomorphism from
a closed subgroup of $G$ onto the infinite
symmetric group, $S_\infty$, consisting of all permutations of $\N$
equipped with the topology of pointwise convergence.}
\bigskip
\bigskip
{\bf $\S$0 Prehistory}
The Vaught conjecture
stands as a major problem in mathematical
logic and may be stated as follows:
\medskip
{\bf 0.1 Conjecture} (Original Vaught conjecture) Let $T$ be
a first order theory in a countable language. Then either
$T$ has $\leq \aleph_0$ many non-isomorphic countable models or it
has $2^{\aleph_0}$ many non-isomorphic countable models.
\medskip
The study of this conjecture
and the proof or refutation of its specializations and generalizations --
found in papers such as
\cite{becker2}, \cite{buechler},
\cite{hart}, \cite{hjorthsolecki}, \cite{sami},
\cite{shelah}, \cite{steel} --
has almost become a research area in its own right.
One natural extension of the Vaught conjecture is to classes
of models defined by countably infinitary sentences
in the sense of \cite{keisler}.
\medskip
{\bf 0.2 Conjecture} (Vaught conjecture for infinitary logic)
Let $\sigma\in {\cal L}_{\omega_1, \omega}$.
Then either
$\sigma$ has $\leq \aleph_0$ many non-isomorphic
countable models or it
has $2^{\aleph_0}$ many non-isomorphic countable models.
\medskip
0.2, like 0.1, remains open. Under the assumption of the continuum
hypothesis both these conjectures are trivially true, and for this
reason it is often customary to consider a strengthened version
which has the same spirit as the original. To phrase this
stronger version we need to equip the collection of all models whose
underlying set is $\N$ with a topology, whose basic open sets consist
of all ${\cal M}$ for which
\[{\cal M}\models \varphi(a_1, a_2, ..., a_n),\]
for some first order formula $\varphi$ and $\vec a=a_1,..., a_n\in \N$.
It is known from \cite{grmory} that the space of all structures or models
on $\N$ in that given topology forms a Polish space.
The collection of models satisfying a given theory $T$ is then closed
inside this space, and thus again a Polish space.
\medskip
{\bf 0.3 Conjecture} Let $T$ be
a first order theory in a countable language ${\cal L}$. Then either
$T$ has $\leq \aleph_0$ many non-isomorphic countable models or
there is a perfect set $P$ included in the above described
space of ${\cal L}$-structures
modeling $T$ for which any two models in $P$ are non-isomorphic.
\medskip
{\bf 0.4 Conjecture} Let $\sigma\in {\cal L}_{\omega_1, \omega}$.
Then either
$\sigma$ has $\leq \aleph_0$ many non-isomorphic countable models or
there is a perfect set $P$ included in the space of ${\cal L}$-structures
which model $\sigma$
for which any two models in $P$ are non-isomorphic.
\medskip
Since any perfect set has size $2^{\aleph_0}$, 0.3 and 0.4 clearly
imply 0.1 and 0.2 respectively. More subtle is the fact that
0.3 and 0.4 are {\it equiprovable} over ZFC with their counterparts
0.1 and 0.2. A proof of this further fact can be found in
\cite{steel}.
It was demonstrated in \cite{vaught} that the isomorphism invariant
Borel subsets
of the space of ${\cal L}$-structures on $\N$ are exactly those
defined by some infinitary sentence
$\sigma\in {\cal L}_{\omega_1, \omega}$. Thus 0.4 asserts that
any invariant Borel set in our space of countable structures
has either only countably many orbits or it has
{\it perfectly many} orbits, in the sense of there being a perfect
set of pairwise non-isomorphic models.
Thus we might be led to asking whether the Vaught
conjecture can hold even on {\it analytic sets}.
\medskip
{\bf 0.5 Definition} Let $X$ be a Polish space. $A\subset X$ is
{\it analytic}, or $\Ubf{\Sigma}^1_1$, if there is some other Polish
space $Y$ and some Borel function
\[f:Y\rightarrow X\]
and some Borel $B\subset Y$ with
\[f[B]=A.\]
\medskip
{\bf 0.6 Fact} There are countable languages ${\cal L}\subset
{\cal L^*}$ and a first order ${\cal L}^*$ theory $S$ such that the
class
\[A_S\]
of ${\cal L}$-structures on $\N$ admitting an expansion to a model
of $S$ has uncountably many non-isomorphic countable models
but not perfectly many non-isomorphic models. (See \cite{becker}
for a discussion of this and related results.)
\medskip
Since the process of assigning to each ${\cal L}^*$ structure on
$\N$ its reduction to an ${\cal L}$ structure is Borel, we obtain
the failure of Vaught's conjecture on $\Ubf{\Sigma}^1_1$ sets: We {\it
can} have an analytic subset of the space of countable ${\cal L}$ structures
having uncountably many
but not perfectly many non-isomorphic models.
A second direction in which we might try to extend the Vaught conjecture
is to consider general actions of Polish groups. Here it is important
to note that the Polish group $S_\infty$, consisting of all permutations
of the natural numbers in the topology of pointwise convergence, acts
continuously on any such space of ${\cal L}$ structures with underlying
set $\N$, and moreover its orbit equivalence relation is exactly the
isomorphism relation on these countable structures.
\medskip
{\bf 0.7 Conjecture} (Topological Vaught conjecture) Let $G$ be a Polish
group acting continuously on a Polish space $X$. Then either $X$ has only
countably many orbits under this action or it has perfectly many orbits.
\medskip
This conjecture just for the group $S_\infty$ implies 0.3, and hence 0.1,
since the
subspace of models for $T$ is a closed subset of the space of structures on
$\N$. By a clever choice of the topology on our space of countable
structures it can also be shown to imply 0.4. More recently Becker
and Kechris in \cite{beckerkechris} have shown that 0.4 is in fact
{\it equivalent} to the topological Vaught conjecture for the
Polish group $S_\infty$.
0.7, like 0.1-0.4, remains open. It has however been proved for
various classes of Polish groups:-
\medskip
{\bf 0.8 Theorem} (Folklore) All locally compact Polish groups satisfy Vaught's
conjecture, in the sense that if $G$ is a locally compact Polish group acting
continuously on a Polish space $X$ then either $|X/G|\leq\aleph_0$ or there is a
perfect set of points with different orbits (and hence $|X/G|\geq 2^{\aleph_0}$).
\medskip
{\bf 0.9 Theorem} (Sami; see \cite{sami})
Abelian Polish groups satisfy Vaught's conjecture.
\medskip
{\bf 0.10 Theorem} (Hjorth, Solecki; \cite{hjorthsolecki})
Invariantly metrizable and nilpotent Polish
groups
satisfy Vaught's conjecture.
\medskip
{\bf 0.11 Theorem} (Becker; \cite{becker}) Complete left invariant metric
groups satisfy Vaught's conjecture.
\medskip
As discussed in \cite{becker}, this implies the Vaught conjecture for
solvable groups, as well as the version obtained at 0.10.
In each of these cases the result was shortly or immediately after extended to
analytic sets.
\medskip
{\bf 0.12 Definition} For $G$ a Polish group let
TVC$(G,\Ubf{\Sigma}^1_1)$ be the assertion that whenever
$G$ acts continuously on a Polish space $X$ and $A\subset X$ is
analytic then either $|A/G|\leq\aleph_0$ or there is a
perfect set of orbit inequivalent points in $A$.
\medskip
Thus we have TVC$(G,\Ubf{\Sigma}^1_1)$
for $G$ in
each of the collections groups mentioned in 0.8-0.11 above.
But it is known that
TVC$(S_{\infty},\Ubf{\Sigma}^1_1)$ {\it fails};
$A_S$ from 0.6 provides a counterexample.
In this paper we show that the presence of $S_{\infty}$ is a necessary condition for
TVC$(G,\Ubf{\Sigma}^1_1)$ to fail:
\medskip
{\bf 0.13 Theorem} If $G$ is a Polish group for which the Vaught conjecture fails on
analytic sets then there is a closed subgroup
of $G$ that has $S_{\infty}$ as a continuous homomorphic image.
\medskip
From the failure of
TVC$(G,\Ubf{\Sigma}^1_1)$ we may obtain the homomorphism onto $S_{\infty}$.
The converse of 0.13 is known and follows by 2.3.5 of
\cite{beckerkechris}. Thus we have an exact characterization of
TVC$(G,\Ubf{\Sigma}^1_1)$. If as widely suspected the Vaught conjecture
should fail for $S_{\infty}$ then this would as well characterize the groups for
which the topological Vaught conjecture holds.
0.13 implies the earlier results at 0.8-0.11. For instance one
can use \cite{gao} to show that no group having $S_\infty$
as its continuous homomorphic image can be given a compatible
complete left invariant metric.
\newpage
{\bf $\S$1 Preliminaries}
\medskip
This paper uses much the same techniques as \cite{hjorthorbit}.
The notation is also similar to \cite{hjorthorbit}, though in this
regard
\cite{beckerkechris} probably
provides a better reference to the notation relating to
Polish group actions.
In other respects we follow the usual set theoretic conventions,
as can be found in \cite{jech}. Perhaps the most alarming of these
is to use $\omega$ for
\[\N=\{0, 1, 2, 3, ...\}.\]
A basic tool in the descriptive set theory of Polish group actions is
provided by the Effros lemma.
\medskip
{\bf 1.1 Lemma} (Effros; see \cite{effros} or 2.2.2 \cite{beckerkechris})
Let $G$ be a Polish group acting continuously on
a Polish space $X$ (in other words, let $X$ be a {\it Polish $G$-space}).
For $x\in X$ we have $[x]_G\in \Ubf{\Pi}^0_2$ if and only if
\[G\rightarrow [x]_G,\]
\[g\mapsto g\cdot x\]
is open.
\medskip
{\bf 1.2 Corollary} Let $G$ be a Polish group and
$X$ a Polish $G$-space.
Suppose that $[x]_G$ is $\Ubf{\Pi}^0_2$.
Then for all $V$ containing the identity we may find open
$U$ containing $x$
such that for all $ x'\in U\cap[x]_G$ and $U'\subset X$ open
\[[x]_G\cap U'\cap U\neq\emptyset\]
implies that there exists $g\in V$
such that
\[g\cdot x'\in U'.\]
Proof. Choose $W$ an open neighborhood of the identity with $W^{-1}=W$ and
$W^2\subset V$. Then by 1.1 let $U$ be an open set with $U\cap [x]_G=W\cdot x$.
For all $x'\in [x]_G\cap U$ and open $U'$ with
$[x]_G\cap U'\cap U\neq\emptyset$ we choose some $\hat{x}\in
[x]_G\cap U'\cap U$. Since $x',\hat{x}\in U\cap [x]_G$, the assumption on
$U$ provides $g_0, g_1\in W$ with
\[g_0\cdot x=x',\]
\[g_1\cdot x=\hat{x}.\]
Then $g_1g_0^{-1}$ is as required. \hfill $\Box$
\medskip
{\bf 1.3 Definition} Let $X$ be a Polish space and ${\cal B}$ a basis. Let ${\cal L}({\cal B})$ be the
propositional language formed from the atomic propositions $\dot{x}\in U$, for $U\in{\cal B}$.
Let ${\cal L}_{\infty,0}({\cal B})$ be the infinitary version, obtained by closing under negation and
arbitrary disjunction and conjunction.
$F\subset$ ${\cal L}_{\infty, 0}({\cal B})$ is a {\it fragment} if it is closed under subformulas and
the finitary Boolean operations of negation and finite disjunction and finite conjunction
and it includes all atomic propositions.
For a point $x\in X$ and $\varphi\in$${\cal L}_{\infty 0}({\cal B})$,
we can then define
$x\models \varphi$ by induction in the usual fashion, starting with
\[x\models \dot{x}\in U\]
if in fact $x\in U$, then
$x\models \bigvee_{i\in\Lambda}(\varphi_i)$ if there
is some $i\in \Lambda$ with
$x\models \varphi_i$, and $x\models \neg \varphi$ if $x$ does not
model $\varphi$. For $\V[H]$ a generic extension of $\V$ and $x, \varphi\in {\Bbb V}$ we
have $\V\models (x\models \varphi)$ if and only if
$\V[H]\models(x\models\varphi)$.
In the case that $X$ is a
Polish $G$-space and $V\subset G$ open we may also define the Vaught
transform
$\varphi^{\Delta V}$ by induction on the logical complexity of $\varphi$,
mimicking the definition of the usual Vaught transforms for Borel sets.
For this purpose let $(V_i)_{i\in\N}$ be an enumeration of the
non-empty basic open subsets of $G$.
$(\dot{x}\in U)^{\Delta V}$ is just
\[\bigvee_{U'\in {\cal B}'} \dot{x}\in U',\]
where ${\cal B}'$ is the set of
basic open sets $U'$ for which there is some $i$ with
$V_i\cdot U'\subset U$ and $V_i\subset V$.
\[(\bigvee_{i\in\Lambda}\varphi_i)^{\Delta V}\]
is
\[\bigvee_{i\in\Lambda}(\varphi_i^{\Delta V}).\]
\[(\neg \varphi)^{\Delta V}\]
is
\[\bigvee\{\neg(\varphi^{\Delta V_n}):
V_n\subset V\}.\]
In any generic extension in which $\varphi$ is hereditarily countable
\[x\models \varphi^{\Delta V}\]
if and only if
\[\exists ^* g\in V (g\cdot x\models\varphi)\]
(where $\exists^*$ is the categoricity quantifier
``there exists non-meagerly many'').
In general $\varphi^{\Delta V}$ depends on the choice of ${\cal B}_0$
and ${\cal B}$. Our notation suppresses this dependence.
Note that if $\varphi\in {\cal L}_{\infty, 0}({\cal B})$ and $g\in G$ is
such that ${\cal B}$ is fixed set wise under $g$ translation, then we
can canonically define $g\varphi$ with the property that
\[x\models g\varphi\]
if and only if $g^{-1}\cdot x\models \varphi$. We define $g\varphi$ by
induction the complexity of $\varphi$, the base case given by
$g(\dot{x}\in U)$ being the sentence $\dot{x}\in g\cdot U$, and the inductive
steps through the infinitary boolean operations carried out in
the usual way.
\medskip
In what follows there is a frequent peril that we may confuse the usual name for the
generic object -- $\dot{G}$ -- with the customary name for the group -- $G$.
For this reason I will generally resist explicit mention of the generic. If $\sigma$ is
a term, existing in our ground model, then $\dot{\sigma}$ will denote the effect of
applying the term to the generic object in the generic extension. At times we will wish
to consider product forcing, and then I will use $\dot{G}_l\times \dot{G}_r$ to name the
generic object on a product forcing ${\Bbb P}\times {\Bbb Q}$.
Similarly to avoid confusion with an open set, we use the embellished font $\V$, rather than
the plain $V$, to indicate the universe of all sets.
$d_X$ will always denote a complete metric on $X$ and $d_G$ a complete metric on
the group $G$.
$G_x$ indicates the stabilizer of $x$, that is to say
\[G_x=\{g\in G: g\cdot x=x\}.\]
\medskip
{\bf 1.4 Lemma} Let $X$ be a Polish $G$-space. ${\Bbb P}$ a forcing notion, $p\in {\Bbb P}$
a condition, $\sigma$ a ${\Bbb P}$-term.
Suppose that ${\cal B}$ is a countable basis for $X$ and ${\cal B}_0$ a countable
basis for $G$. Suppose that $G_0$ is a countable dense subgroup of $G$ and ${\cal B}$ is
closed under $G_0$ translation and that ${\cal B}_0$ is closed under left and right
$G_0$ translation.
%\[p\Vdash_{\Bbb P}\dot{\sigma}\in X\]
%and that $p$ decides the equivalence class of $\sigma$ in the sense that
%\[(p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l]E_G\sigma[\dot{G}_r].\]
Then there is a formula $\varphi_0$ and a fragment $F_0$ containing $\varphi_0$
so that:
\leftskip 0.5in
\noindent (i)
\[\{\{x\in X: x\models \psi^{\Delta V}\}:
\psi\in F_0, V\in {\cal B}_0\}\]
provides a basis for a topology $\tau_0(F_0)$,
and in any generic extension in which $F_0$ becomes countable $(X,\tau_0(F_0))$
is a Polish $G$-space;
\noindent (ii) $\varphi_0$ absolutely
describes the equivalence class (if any) indicated
by the triple
$({\Bbb P},p,\sigma)$, in the sense that in any generic extension $\V[H]$ we find
\[\V[H]\models\forall x\in X(p\Vdash_{\Bbb P}x E_G\dot{\sigma}\Leftrightarrow x\models
\varphi_0).\]
\leftskip 0in
Proof. Note that if $F$ is any fragment and $\tau(F)$ is the topology
with basis $\{\{x\in X: x\models \psi\}:
\psi\in F\}$, then $\tau(F)$ includes the original topology in virtue of
$F$ including all the atomic propositions.
Claim(1): If $F$ is a fragment, then in any generic extension in which $F$ becomes countable,
$\tau(F)$ generates a Polish topology.
Proof of claim. Recall from 13.2 \cite{kechris1} that whenever we have a
Polish topology $\tau_Y$ on a Polish space $Y$ and $A\subset Y$ a
$\tau_Y$-open set, then the minimal topology obtained from
$\tau_Y$ by adding $A$ as a clopen set
gives a topology on $Y$ equivalent to taking
the disjoint union of $(A,\tau_Y)$ and $(Y\setminus A,\tau_Y)$,
and is therefore again Polish. Recall also that if $\tau_i$ is an
increasing sequence of Polish topologies on $Y$, then the minimal
topology including the union is again Polish, since it corresponds
to
the diagonal in
the resulting space
\[\prod_{i\in\omega}(Y,\tau_i).\]
Iterating these two operations one shows by induction on the logical
complexity of the infinitary sentences in $F$ that $\tau(F)$ is Polish
in any generic extension in which $F$ becomes countable.
\hfill (Claim(1)$\Box$)
Let $F_0$ be a fragment closed under the
Vaught transforms ($\psi\mapsto \psi^{\Delta V}$ for $V\subset G$
basic open) and such that for each $\psi\in F_0$ and $g\in G_0$
we have a corresponding $g\psi\in F_0$ with the property that
through all generic extensions
\[(x\models g\psi)\Leftrightarrow (g^{-1}\cdot x\models \psi).\]
By the Becker-Kechris theorem on changing topologies, as found at
\cite{beckerkechris2} or as
in
$\S$5.2 and the proof of 7.1.3 of \cite{beckerkechris}, we have that
\[\{\{x\in X: x\models \psi^{\Delta V}\}:
\psi\in F_0, V\in {\cal B}_0\}\]
generates a topology $\tau_0(F_0)$, with $(X, \tau_0(F_0))$
a Polish $G$-space whenever $F_0$ becomes countable. Since the
sets of the form
\[\{x\in X: x\models \psi \},\] $\psi\in F_0$, form an
algebra closed under $G_0$ translation, it is easily seen that
the collection $\{\{x\in X: x\models \psi^{\Delta V}\}:
\psi\in F_0, V\in {\cal B}_0\}$ provides a basis for the topology
$\tau_0(F_0)$.
In what follows it is crucial that
the properties of most interest are absolute,
and thus we can be ambiguous about the generic extension in which we
make an evaluation. In particular,
for any $x$ appearing in any generic extension,
the assertion $p\Vdash_{\Bbb P} x E_G\dot{\sigma}$ depends solely on $x$, and
not the generic extension in which we consider this question:-
Claim(2): If $\V[H_1]\subset {\Bbb V}[H_1][H_2]$ are two generic extensions of
$\V$ and $x\in {\Bbb V}[H_1]$, then
\[\V[H_1]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}\]
if and only if
\[\V[H_1][H_2]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}.\]
Proof of claim. Otherwise we may find $H\subset {\Bbb P}$ that is
generic for both models, and such that the generic extensions
$\V[H_1][H]$ and
$\V[H_1][H_2][H]$ disagree on $x E_G\sigma[H]$, with a violation of
absoluteness of $\Ubf{\Sigma}^1_1$. \hfill (Claim(2)$\Box$)
Claim(3): There is a generic extension of $\V[H]$ and $x\in {\Bbb V}[H]$ with
\[\V[H]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}\]
if and only if
\[\V\models (p, p)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l] E_G
\sigma[\dot{G}_r].\]
Proof of claim. Clearly if $ (p, p)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l] E_G
\sigma[\dot{G}_r]$ then we can let $H\subset {\Bbb P}$ be generic below
$p$, and obtain some $x=\sigma[H]$ in $\V[H]$ with
\[\V[H]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}.\]
Conversely, if we have $\V[H]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}$ in some
generic extension $\V[H]$ of $\V$ and $q_1, q_2\leq p$ with
\[\V\models (q_1, q_2)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l]\neg E_G
\sigma[\dot{G}_r],\]
then we may choose $H_1\times H_2$ a $\V[H]$-generic filter on ${\Bbb P}\times
{\Bbb P}$ below $(q_1, q_2)$. Then
\[\sigma[H_1]\neg E_G\sigma[H_2],\]
by assumption on $(q_1, q_2)$, and hence for some $i\in \{1, 2\}$ we have
\[\sigma[H_i]\neg E_G x,\]
contradicting assumption on $p$.
\hfill (Claim(3)$\Box$)
Claim(4): The set of $x$ for which
\[p\Vdash_{\Bbb P}x E_G \dot{\sigma}\]
is $\Ubf{\Sigma}^1_1$ in any generic extension in which
$({\cal P}({\Bbb P}))^\V$ (=$_{df}$ Power set of ${\Bbb P}$ as calculated in $\V$)
becomes countable.
Proof of claim: It suffices by claim(3) to show that if
\[\V\models (p, p)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l] E_G
\sigma[\dot{G}_r]\]
then for any two
$H_1, H_2\subset {\Bbb P}$ meeting all the dense open sets in $\V$
we have
\[\sigma[H_1] E_G\sigma[H_2].\]
But given $H_1, H_2$ we can
go to a third generic extension $\V[H_3]$ where
$H_3\subset {\Bbb P}$ is $\V[H_1]$ and $\V[H_2]$ generic below $p$,
and thus $\sigma[H_3] E_G\sigma[H_i]$ for {\it both} $i=1$ and
$i=2$.
\hfill (Claim(4)$\Box$)
Thus we may find a transfinite sequence $(\psi_{\alpha})_{\alpha
\in\:{\rm Ord}}$ of infinitary propositions in ${\cal L}_{\infty 0}({\cal B})$ such that
in any extension $\V[H]$ in which $|{\cal P}({\Bbb P})^\V|\leq \aleph_0$
and for any $x\in {\Bbb V}[H]$ we have
\[(p\Vdash_{\Bbb P}x E_G \dot{\sigma})\Leftrightarrow \exists \alpha<\omega_1^{\V[H]}
(x\models \psi_{\alpha}).\]
This is obtained either by appeal to the usual decomposition of $\Ubf{\Sigma}^1_1$
into Borel sets or by letting $\psi_{\alpha}$ be chosen canonically so that for all $x$ in any
generic extension
\[(x\models\psi_{\alpha})\Leftrightarrow (L_{\alpha}(G, X, \sigma, ({\cal P}({\Bbb P}))^\V, x)\models
p\Vdash_{\Bbb P}x E_G \dot{\sigma}).\]
Claim(5):
In any generic extension $\V[H]$ in which $({\cal P}({\Bbb P}))^\V$
becomes countable and for any $x\in {\Bbb V}[H]$ with
$p\Vdash_{\Bbb P}x E_G \dot{\sigma}$ we have some $\alpha<\omega_1^{\V[H]}$
with
\[\exists^*g\in G(g\cdot x\models \psi_{\alpha}).\]
Proof of claim. Since for any particular $\alpha<\omega_1^{\V[H]}$ the
displayed sentence is absolute, we may as well assume MA$_{\aleph_1}$.
By invariance of the set in question we have that for all $g\in G$ there
will be some $\alpha\in \omega_1^{\V[H]}$ with
\[g\cdot x\models \psi_{\alpha}.\]
By MA$_{\aleph_1}$, for some $\alpha$ the set of $g$ for which
$g\cdot x\models \psi_{\alpha}$ must be non-meager. \hfill (Claim(5)$\Box$)
So far there is no guarantee that {\it any}
$\psi_{\alpha}$ is non-trivial;
nothing yet has ruled against us {\it never} having the situation that
$p\Vdash_{\Bbb P}x E_G \dot{\sigma}$.
Claim(6): If there is some generic extension
in which
\[\V[H]\models p\Vdash_{\Bbb P} x E_G\dot{\sigma}\]
then there is some $\alpha$ for which
$\varphi_0=(\psi_{\alpha})^{\Delta G}$ satisfies (ii) from the
statement of the lemma.
Proof of claim. Following claim(4) and applying the absoluteness
of $\Ubf{\Sigma}^1_1$ we
can assume that $\V[H]$ arises by simply collapsing
$ ({\cal P}({\Bbb P}))^\V$ to $\aleph_0$.
Following claim(5) let us choose some $q\in$ Coll$(\omega, ({\cal P}({\Bbb P}))^\V)$
and $\alpha< (|{\cal P}({\Bbb P})|^+)^\V$ such that
\[q\Vdash_{{\rm Coll}(\omega, ({\cal P}({\Bbb P}))^\V)}\exists x \exists^*g
(g\cdot x\models \psi_\alpha).\]
Now suppose that $\V[H_1]$ is any generic extension with some
$x_1\in {\Bbb V}[H_1]$ such that
\[\V[H_1]\models (p\Vdash_{\Bbb P} x_1 E_G\dot{\sigma}).\]
Then we may choose $H_2\subset {\rm Coll}(\omega, ({\cal P}({\Bbb P}))^\V)$
to be $\V[H_1]$-generic below $q$. Thus by assumption on $q$ we may
find some $x_2\in {\Bbb V}[H_2]$ with
\[\V[H_2]\models \exists^* g\in G(x_2\models \psi_\alpha),\]
and thus in particular
\[\V[H_2]\models (p\Vdash_{\Bbb P} x_2 E_G\dot{\sigma}).\]
And then by claim(2)
\[\V[H_1][H_2]\models (p\Vdash_{\Bbb P} x_2 E_G\dot{\sigma}),\]
\[\V[H_1][H_2]\models (p\Vdash_{\Bbb P} x_1 E_G\dot{\sigma}).\]
So if we let $H\subset
{\Bbb P}$
be $\V[H_1][H_2]$-generic below
$p$ we obtain
\[\V[H_1][H_2][H]\models x_2 E_G\sigma[H],\]
\[\V[H_1][H_2][H]\models x_1 E_G\sigma[H].\]
Thus $x_1 E_G x_2$.
But $(\psi_\alpha)^{\Delta G}$ is absolute between forcing
extensions, and
thus
\[\V[H_1][H_2][H]\models\exists^* g\in G(g\cdot x_2\models \psi_\alpha),\]
and then by invariance
\[\V[H_1][H_2][H]\models\exists^* g\in G(g\cdot x_1\models \psi_\alpha).\]
One final application of absoluteness gives
\[\V[H_1]\models (x_1\models \psi_\alpha^{\Delta G}).\]
\hfill (Claim(6)$\Box$)
\hfill $\Box$
\medskip
Nothing in the statement of the lemma rules out (ii) holding
trivially. It could just be the case that in every
extension $\V[H_0]$ we have $\V[H_0]\models \forall x(p\Vdash_{\Bbb P}
x\neg E_G \dot{\sigma})$. However in the non-trivial case where there
is a generic extension $\V[H_0]$ satisfying
$\exists x(p\Vdash_{\Bbb P}
x E_G \dot{\sigma})$ it will follow by absoluteness for
$\Ubf{\Sigma}^1_1$ that for any generic extension $\V[H_1]$ in which
$F_0$ becomes countable we must have $\V[H_1]\models \exists x (x\models
\varphi_0)$, and therefore
$\V[H_1]\models \exists x(p\Vdash_{\Bbb P}
x E_G \dot{\sigma})$.
\medskip
{\bf 1.5 Lemma} Let $G$ be a Polish group, $X$ a Polish $G$-space, $A\subset X$ a
$\Ubf{\Sigma}^1_1$ set displaying a counterexample to TVC$(G,\Ubf{\Sigma}^1_1)$ --
so that $A/G$ has uncountably many orbits, but no perfect set of $E_G$-inequivalent
points.
Then for each ordinal $\delta$ there is a sequence $({\Bbb P}_{\alpha},
p_{\alpha},\sigma_{\alpha})_{\alpha <\delta}$ so that:
\leftskip 0.5in
\noindent (0) for each $\alpha<\delta$
\[p_{\alpha}\Vdash_{{\Bbb P}_{\alpha}}\dot{\sigma}_{\alpha}\in A;\]
\noindent (i) for each $\alpha<\delta$
\[(p_{\alpha},p_{\alpha})\Vdash_{{\Bbb P}_{\alpha}\times{\Bbb P}_{\alpha}}
\sigma_{\alpha}[\dot{G}_l]E_G\sigma_{\alpha} [\dot{G}_r];\]
\noindent (ii) for each $\alpha<\beta< \delta$
\[(p_{\alpha},p_{\beta})\Vdash_{{\Bbb P}_{\alpha}\times{\Bbb P}_{\beta}}
\neg(\sigma_{\alpha}[\dot{G}_l] E_G\sigma_{\beta} [\dot{G}_r]).\]
\leftskip 0in
Proof.
Claim(1): If ${\Bbb P}$ is any forcing notion, $\sigma$ a ${\Bbb P}$-term for
an element of $A$, then there is $p\in{\Bbb P}$ such that
$(p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}
\sigma[\dot{G}_l]E_G\sigma [\dot{G}_r].$
Proof of claim. It is known from \cite{burgess}, \cite{stern}, and recorded in many
places, that whenever we have a $\Ubf{\Sigma}^1_1$ equivalence relation $F$ on
a Polish space $Y$ for which there is no perfect set of inequivalent
reals, then for all ${\Bbb P}\Vdash \dot{\tau}\in Y$ we may find some $p$ with
\[(p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}
\tau[\dot{G}_l]F\tau [\dot{G}_r].\]
Taking $\pi:Y\twoheadrightarrow A$ to be the continuous map witnessing
$A\in \Ubf{\Sigma}^1_1$ we may let $F$ be the pullback of $E_G$. Since $A$ has no
perfect set of $E_G$-inequivalent reals, $Y$ has no perfect set of
$F$-inequivalent reals. Thus if $\sigma$ is a term for
an element of $A$ then we let $\tau$ be a term for an element of $Y$ with
\[{\Bbb P}\Vdash \pi(\dot{\tau})=\dot{\sigma},\]
and we obtain as above some $p\in{\Bbb P}$ with
\[(p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}
\tau[\dot{G}_l]F\tau [\dot{G}_r],\]
\[\therefore (p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}
\sigma[\dot{G}_l]E_G\sigma [\dot{G}_r].\]
\hfill (Claim(1)$\Box$)
Claim(2): In every generic extension of $\V$ we have $|A/G|\geq \aleph_1$.
Proof of claim (compare also \cite{sami}):
We use the fact that every orbit in $X$ is Borel in
any code for its stabilizer. (See 7.1.2 \cite{beckerkechris}.)
Let ${\cal F}(G)$ be the standard Borel space of closed subsets of
$G$ in the Effros Borel structure. Let $S\subset X\times {\cal F}(G)$
be $\Ubf{\Pi}^1_1$ with $S=\{(x, G_x):x\in X\}$. Let
\[E\subset
X\times X\times{\cal F}(G)\]
be $\Ubf{\Pi}^1_1$ such that \[(x, y, G_x)\in E\]
if and only if $xE_G y$.
Then the uncountability of $A/G$ becomes the $\Ubf{\Pi}^1_2$ statement
\[\forall (x_i, F_i)_{i\in\omega}\in (X\times{\cal F}(X))^{\omega} \exists y \in A
(\exists i(\neg S(x_i, F_i))\vee \forall i (\neg E(x_i, y, F_i))),\]
and is therefore absolute. \hfill (Claim(2)$\Box$)
Thus we may successively choose ${\Bbb P}_{\alpha}$ and $\sigma_{\alpha}$ such that
\[{\Bbb P}_{\alpha}\Vdash \dot{\sigma}_{\alpha}\in A\]
and for all $\beta<\alpha$ and $q\in {\Bbb P}_{\alpha}$
\[(q, p_{\beta})\Vdash _{{\Bbb P}_{\alpha}\times{\Bbb P}_{\beta}}
\neg(\sigma_{\alpha}[\dot{G}_l] E_G\sigma_{\beta} [\dot{G}_r]).\]
Refining to some $p_{\alpha}\in {\Bbb P}_{\alpha}$ with
\[(p_{\alpha},p_{\alpha})\Vdash_{{\Bbb P}_{\alpha}\times{\Bbb P}_{\alpha}}
\sigma_{\alpha}[\dot{G}_l]E_G\sigma_{\alpha} [\dot{G}_r]\]
we finish. \hfill $\Box$
\medskip
Note that in the situation of
1.5 we may find for each $\alpha<\delta$ some corresponding $\varphi_{0,\alpha}$
for the forcing notion ${\Bbb P}_{\alpha}$ below $p_{\alpha}$ as
in 1.4. We are guaranteed that this be non-trivial, in the sense that if
$H\subset {\Bbb P}_{\alpha}$ is $\V$-generic below $p_{\alpha}$,
and $x=\sigma_{\alpha}[H]$,
then
\[p_{\alpha}\Vdash_{\Bbb P}x E_G \dot{\sigma}_{\alpha},\]
\[\therefore x\models \varphi_{0,\alpha}.\]
On the other hand,
if $x_0\in {\Bbb V}[H_0]$ is an element of some generic extension
$\V[H_0]$ and $x\models\varphi_{0,\alpha}$ and $H_1\subset {\Bbb P}_{\alpha}$
is any $\V$-generic with $x_1=\sigma_{\alpha}[H_1]$,
then we may choose
$H_2\subset{\Bbb P}_{\alpha}$ that is generic for
both $\V[H_1]$ and $\V[H_2]$. Thus
\[x_0E_G \sigma_{\alpha}[H_2]\]
by assumption on $\varphi_{0,\alpha}$.
Moreover
\[x_1 E_G \sigma_{\alpha}[H_2]\]
by the assumption
${\Bbb P}_{\alpha},
p_{\alpha},\sigma_{\alpha}$.
Hence $x_0 E_G x_1$.
In this sense we can associate to $({\Bbb P}_{\alpha},
p_{\alpha},\sigma_{\alpha})_{\alpha <\delta}$ a corresponding
sequence
$(\varphi_{0,\alpha})_{\alpha\in\delta}$, each $\varphi_{0, \alpha}$
providing an
$\infty$-Borel code
for the indicated equivalence class
$[\sigma_{\alpha}[H]]_G$ whenever $H\subset{\Bbb P}_{\alpha}$ is $\V$-generic
below $p_{\alpha}$.
\newpage
{\bf $\S$2 Proof}
\medskip
{\bf 2.1 Definition} $U$ is a {\it regular open} set if
$U$ equals the interior of its closure:
\[(\overline{U})^o=U\]
For $A$ a set let $RO(A)=(\overline{A})^o$.
\medskip
Note that $RO(A)$ is always regular open.
\medskip
{\bf 2.2 Lemma} Let $G$ be a Polish group. For $V_0, V_1\subset G$
regular open sets,
\[\{g\in G: V_0\cdot g= V_1\}\]
is a closed subset of $G$.
Proof. The set in question arises as the intersection of the closed sets
\[\{g\in G: V_0\cdot g\subset\overline{V_1}\}\]
and
\[\{g\in G: \overline{V_0}\cdot g\supset V_1\}.\]
\hfill $\Box$
\medskip
I need that the reader is willing to allow that we may speak of an
$\omega$-model of set theory containing a Polish space, group, action,
Borel set, and so on, provided suitable codes exist in the well founded
part. I will lean on the usual set theoretical identifications, and speak
of open sets and Borel sets being ``in" an $\omega$-model, when really it is
only the case that the model has some suitable subset of the natural numbers
which codes them.
Illfounded $\omega$-models are essential to the arguments below.
The structure of the argument is to use the large sequence of forcing notions
layed out in 1.5 to obtain a suitable $\omega$-model with generating
indiscernibles of order type $({\Bbb Q},<)$.
Thus we can inject Aut$({\Bbb Q},<)$ into the automorphisms of the $\omega$-model,
and then using a kind of ``back-and-forth" argument at 2.3 obtain for each
automorphism some corresponding
element of the group
$G$. Since wellfounded models are rigid and since we want automorphisms in great number,
the passage to an $\omega$-model
occupies a necessary step in the proofs.
Let ZFC$^*$ be some large fragment of ZFC; at the
very least
strong enough to prove all the lemmas of $\S$1, but weak enough to admit a finite
axiomatization.
The next lemma gives the connection between automorphisms of an
$\omega$-model and the corresponding elements of $G$. Here it should
be understood that we do everything honestly: We are supposing that the
$M$-generic $H$ exists in $\V$, our ground model, and conclude with
$\hat{g}$ again in the ground model. We are working inside a single
class model of ZFC which sees not only $M$ and $\pi$ but also
$X$, $G$, $G_0$,
${\Bbb P}$,
$F_0$, ${\cal B}$, ${\cal B}_0$, $\varphi_0$, and $H$.
\medskip
{\bf 2.3 Lemma} Let $M$ be an $\omega$-model of ZFC$^*$. Let $X$, $G$, $G_0$,
${\Bbb P}$,
$F_0$, ${\cal B}$, ${\cal B}_0$, $\varphi_0$
satisfy 1.4 inside $M$. Suppose
\[\pi:M\cong M\]
is an automorphism of $M$ fixing $X$, $G$, $G_0$, ${\Bbb P}$, $F_0$, $\varphi_0$, and
all elements of ${\cal B}$ and ${\cal B}_0$.
Suppose $H\subset$ Coll($\omega, F_0$) is $M$-generic and
$x\in X^{M[H]}$ with
\[M[H]\models(x\models \varphi_0).\]
Then there exists $\hat{g}\in G$ so that for all
$\psi \in F_0$ and $V\in{\cal B}_0$
\[RO(\{g\in G_0: M[H]\models(g\cdot x\models \psi^{\Delta V})\})\hat{g}^{-1}=
RO(\{g\in G_0: M[H]\models(g\cdot x\models \pi(\psi)^{\Delta V})\}).\]
\noindent Proof. It suffices to find $g_0, g_1\in G$ so that
\[RO(\{g\in G_0: M[H]\models(g\cdot x\models \psi^{\Delta V})\}){g}_0^{-1}=
RO(\{g\in G_0: M[H]\models(g\cdot x\models \pi(\psi)^{\Delta V})\})g^{-1}_1\]
for all $\psi$ and $V$.
Let ${{\Bbb P}_0} $ be the forcing notion Coll$(\omega, F_0)$ in $M$.
Fixing $d_G$ a complete metric on $G$ we build $h_i, h_i'\in G_0$,
$\psi_i, \psi_i'\in F_0$, $W_i, W_i', V_i\in {\cal B}_0$ so that
\leftskip 0.5in
\noindent (i) $W_i$, $W'_i$ are open neighborhoods of the
identity, with
\[W_{i+1}\subset W_i,\]
\[W_{i+1}'\subset W_i',\]
\[d_G(W_i)<2^{-i},\]
\[d_G(W_i')<2^{-i};\]
\noindent (ii) $\pi(\psi_i)=\psi_i'$;
\noindent (iii) $h_{2i}=h_{2i+1}$; for all $g\in
W_{2i+1}h_{2i}$
\[d_G(g, h_{2i})<2^{-i};\]
\noindent (iv) $h_{2i+1}'=h_{2i+2}'$; for all $g\in
W_{2i+2}h_{2i+1}'$
\[d_G(g, h_{2i+1}')<2^{-i};\]
\noindent (v) $h_{i+1}\in W_ih_i$ and $h_{i+1}'\in W_ih_i'$;
\noindent (vi) $M[H]\models (h_i\cdot x\models (\psi_i)^{\Delta V_i})$;
\noindent (vii) $M[H]\models (h_i'\cdot x\models (\psi_i')^{\Delta V_i})$;
\noindent (viii) $M^{{\Bbb P}_0} $ satisfies that for all $y_0, y_1\in X$
all $\psi\in F_0$, and all $V\in {\cal B}_0$,
if
\[y_0\models \varphi_0\wedge (\psi_i)^{\Delta V_i}\]
\[y_1\models \varphi_0\wedge (\psi_i)^{\Delta V_i}\wedge \psi^{\Delta V}\]
then
\[y_0\models ((\psi_i)^{\Delta V_i}\wedge \psi^{\Delta V})^{\Delta W_i};\]
\noindent (ix) conversely $M^{{\Bbb P}_0} $ satisfies that for all $y_0, y_1\in X$,
$\psi\in F_0$, $V\in {\cal B}_0$,
if
\[y_0\models \varphi_0\wedge (\psi_i')^{\Delta V_i}\]
\[y_1\models \varphi_0\wedge (\psi_i')^{\Delta V_i}\wedge \psi^{\Delta V}\]
then
\[y_0\models ((\psi_i')^{\Delta V_i}\wedge \psi^{\Delta V})^{\Delta W_i}.\]
\leftskip 0in
\noindent Actually (ix) follows from (viii), (ii), and the elementarity of
$\pi$.
Before verifying that we may produce $h_i, h_i'\in G_0$,
$\psi_i, \psi_i'\in F_0$, $W_i, V_i\in {\cal B}_0$ as above, let us imagine that
it is already completed and see how to finish. Using (iii), (iv), and (v) we may obtain
\[g_0= {\rm lim}_{i\rightarrow \infty} h_i\]
and
\[g_1={\rm lim}_{i\rightarrow \infty} h_i'.\]
It suffices to check that for all $\psi\in F_0$, $V\in {\cal B}_0$, and
\[ g\in RO(\{h\in G_0: M[H]\models(h\cdot x\models \psi^{\Delta V})\}){g}_0^{-1}\]
we have
\[g\in \overline{(\{h\in G_0:
M[H]\models(h\cdot x\models \pi(\psi)^{\Delta V})\})} g_1^{-1}\]
(the converse implication will be exactly symmetric).
Fix $g$ in
\[RO(\{h\in G_0: M[H]\models(h\cdot x\models \psi^{\Delta V})\}){g}_0^{-1}.\]
By assumption on $g$ there are arbitrarily large $i$ and $h\in G_0$ with
$h(h_i)^{-1}$ arbitrarily close to $g$ and
$M[H]\models(h\cdot x\models \psi^{\Delta V})$.
Thus replacing $h(h_i)^{-1}$ with $\hat{g}$
for sufficiently large $i$ we may choose a sufficiently small
open neighbourhood
$W\in{\cal B}_0$ of the identity and $\hat{g}\in G_0$ sufficiently close to $g$
so that $W\hat{g} W_i$ is an arbitrarily
small neighbourhood of $g$ and
\[M[H]\models (\hat{g} h_i\cdot x\models \psi^{\Delta V})\]
\[\therefore M[H]\models ( h_i\cdot x\models
(\psi^{\Delta V})^{\Delta W\hat{g}})\]
hence, as witnessed by $y=h_i\cdot x$,
\[M^{{\Bbb P}_0}\models \exists y( y\models \varphi_0\wedge (\psi_i)^{\Delta V_i}
\wedge (\psi^{\Delta V})^{\Delta W\hat{g}}),\]
\[\therefore M^{{\Bbb P}_0}\models \exists y( y\models \varphi_0\wedge (\psi_i')^{\Delta V_i}
\wedge (\pi(\psi)^{\Delta V})^{\Delta W\hat{g}})\]
by (ii) and elementarity of $\pi$,
\[\therefore M[H]\models (h_i'\cdot x\models
((\pi(\psi)^{\Delta V})^{\Delta W\hat{g}})
^{\Delta W_i})\]
by (ix). So there exists some $\bar{g}\in W\hat{g} W_i$ such that
\[M[H]\models (\bar{g}h_i'\cdot x\models \pi(\psi)^{\Delta V}).\]
By letting $d_G(W\hat{g} W_i)\rightarrow 0$ and $h_i'\rightarrow g_1$ we get
\[g\in \overline{\{h\in G_0:
M[H]\models(h\cdot x\models \pi(\psi)^{\Delta V})\}} g_1^{-1},\]
as required.
We are left to hammer out the sequences.
Suppose that we have $\psi_j, \psi'_j, W_j,V_j, h_j, h_j'$ for $j\leq 2i$. Immediately
we may set $h_{2i+1}=h_{2i}$ and
find $W_{2i+1}\subset W_{2i}$ giving (iii), and then by 1.2 and 1.4(i) we
can produce $\psi_{2i+1}$ and $V_{2i+1}$ satisfying (viii) and such that
\[M[H]\models h_{2i}\cdot x=_{df} h_{2i+1}\cdot x\models (\psi_{2i+1})^{V_{2i+1}}.\]
Then by considering that $\pi$ is elementary
\[M^{{\Bbb P}_0} \models \exists y(y\models \varphi_0\wedge \pi(\psi_{2i})^{\Delta V_{2i}}
\wedge \pi(\psi_{2i+1})^{\Delta V_{2i+1}}).\]
Thus by (ix) we may find $h'\in G_0\cap W_{2i}$ so that
\[M[H]\models (h'h'_{2i}\cdot x\models \pi(\psi_{2i})^{\Delta V_{2i}}
\wedge \pi(\psi_{2i+1})^{\Delta V_{2i+1}}).\]
In other words, by (ii), if we let $\psi_{2i+1}'=\pi(\psi_{2i+1})$ then
\[M[H]\models (h'h'_{2i}\cdot x\models (\psi_{2i}')^{\Delta V_{2i}}
\wedge (\psi_{2i+1}')^{\Delta V_{2i+1}}).\]
Taking $h_{2i+1}'=h'h_{2i}'$ we complete the transition from $2i$ to $2i+1$.
The further step of producing $\psi_{2i+2}, \psi_{2i+2}',
W_{2i+2}, h_{2i+2}$, $V_{2i+2}$ and $h_{2i+2}'$ is completely symmetrical. \hfill $\Box$
\medskip
{\bf 2.4 Definition} $S_{\infty}$ {\it divides} a Polish
group $G$ if there is a closed subgroup $H\kappa$ so that $\V_{\theta}\models$
ZFC$^*$ and choose an elementary substructure
\[A {\prec} \V_{\theta}\]
so that
\[|A|=\delta,\]
\[\delta+1\subset A,\]
and $X$, $G$, $F_0$, $\varphi_0$, and so on, in $A$. Let $N$ be the transitive
collapse of $A$ and
\[\pi: N\rightarrow {\Bbb V}_{\theta}\]
the inverse of the collapsing map. Set $\hat{{\Bbb P}}=\pi^{-1}({\Bbb P}_0)$
(where ${{\Bbb P}_0}$= Coll$(\omega, F_0)$), $ \hat{\varphi}_0=\pi^{-1}(\varphi_0)$,
$\hat{F}_0=\pi^{-1}(F_0)$,
choose
\[\hat{H}\subset \hat{\Bbb P},\]
\[H\subset {\Bbb P}_0\]
to be $\V$-generic, and choose $\hat{x}\in N[\hat{H}]$ and $x\in {\Bbb V}[H]$ so that
\[N[\hat{H}]\models (\hat{x}\models \hat{\varphi}_0),\]
\[\V[H]\models (x\models\varphi_0).\]
($x$ and $\hat{x}$ exist for the reasons mentioned following 1.4.)
Note that for each $V\in {\cal B}_0$ there will be a maximal $R$-discrete subset of
${\cal B}(V)$ included in $N$.
It suffices to show
\[\hat{x}E_Gx.\]
As in the proof of 2.3 find $h_i, h_i'\in G_0$, $\psi_i\in F_0$,
$\psi_i'\in {\hat{F}}_0$, $V_i, V'_i \in{\cal B}_0$, $W_i\in {\cal B}_0$
and $U_i\subset X$ basic open so that :
\leftskip 0.5in
\noindent (i) $W_{i+1}\subset W_i$, $W_i=(W_i)^{-1}$, $d_G(W_i)<2^{-i},$ $1_G\in W_i$;
$U_{i+1}\subset U_i$,
$d_X(U_i)<2^{-i}$;
\noindent (ii) $h_{2i+1}=h_{2i}$; for all $g\in (W_{2i+1})^3 h_{2i}$
\[d_G(g, h_{2i})<2^{-i};\]
\noindent (iii) $h_{2i+2}'=h_{2i+1}'$; for all $g\in (W_{2i+2})^3 h_{2i+1}'$
\[d_G(g, h_{2i+1}')<2^{-i};\]
\noindent (iv) $h_{i+1}\in (W_i)^3 h_i$ and $h_{i+1}'\in (W_i)^3 h_i'$;
\noindent (v) $\V[H]\models (h_i\cdot {x}\models (\psi_i)^{\Delta V_i})$;
\noindent (vi) $N[\hat{H}]\models (h_i'\cdot \hat{x}\models (\psi_i')^{\Delta V_i'})$;
\noindent (vii) $\V\models (\psi_i, V_i)\in {\cal B}(W_i)$;
\noindent (viii) $N \models (\psi_i', V_i')\in {\cal B}(W_i)$;
\noindent (ix) $(\pi(\psi_i'), V_i') R(\psi_i, V_i)$;
\noindent (x) $h_i\cdot x, h_i'\cdot \hat{x} \in U_i$.
\leftskip 0in
Granting all this may be found we finish quickly. By (ii), (iii), and (iv) we
get $g_0$= lim $h_i$ and $g_1$= lim $h_i'$, whence
\[g_0\cdot x=g_1\cdot \hat{x}\]
by (x) and (i).
This would contradict ${\hat{\Bbb P}}$ being too small to introduce
a representative of $[x]_G$.
So instead suppose we have built $V_j, V_j', \psi_j$ and so on for $j\leq 2i$ and
concentrate on trying to show that we may continue the construction up to
$2i+2$.
First choose $W_{2i+1}\subset W_{2i}$ in accordance with (i) and (ii) and then for
(x) and (i) choose $U_{2i+1}\subset U_{2i}$ containing $h_{2i}\cdot x (=_{df}
h_{2i+1}\cdot x)$ with $d_X(U_{2i+1})<2^{-2i-1}$. Then by 1.2 we may choose
$(V_{2i+1}, \psi_{2i+1})\in {\cal B}(W_{2i+1})$ with
\[h_{2i}\cdot x\models (\psi_{2i+1})^{\Delta V_{2i+1}}.\]
On the $N$ side we use the assumption on $R$ to find $V_{2i+1}'$ and $\psi_{2i+1}'$ in
$N$ so that
\[N^{\hat{\Bbb P}}\models (\psi_{2i+1}', V_{2i+1}')\in {\cal B}(W_{2i+1})\]
and
\[(\pi(\psi_{2i+1}'), V_{2i+1}') R (\psi_{2i+1}, V_{2i+1}).\]
Unwinding the definitions of ${\cal B}(W_{2i})$ and ${\cal B}(W_{2i+1})$ gives
\[\V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge \pi(\psi_{2i}')^{\Delta V_{2i}'})
\Rightarrow y\models ((\psi_{2i})^{\Delta V_{2i}}
\wedge \pi(\psi_{2i}')^{\Delta V_{2i}'})^{\Delta W_{2i}},\]
\[\V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge (\psi_{2i})^{\Delta V_{2i}})
\Rightarrow y\models ((\psi_{2i})^{\Delta V_{2i}}
\wedge (\psi_{2i+1})^{\Delta V_{2i+1}})^{\Delta W_{2i}},\]
\[\V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge (\psi_{2i+1})^{\Delta V_{2i+1}})
\Rightarrow y\models ((\psi_{2i+1})^{\Delta V_{2i+1}}
\wedge \pi(\psi_{2i+1}')^{\Delta V_{2i+1}'})^{\Delta W_{2i+1}}.\]
After possibly tightening the $\tau_0(F_0)$-open set corresponding to
$(\psi_{2i+1})^{\Delta V_{2i+1}}$
we can assume
\[(\psi_{2i+1})^{\Delta V_{2i+1}}\Rightarrow \dot{x}\in U_{2i+1},\]
and thus we have
\[\V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge \pi(\psi_{2i}')^{\Delta V_{2i}'})
\Rightarrow y\models ((\pi(\psi_{2i+1}'))^{\Delta V_{2i+1}'}
\wedge \dot{x}\in U_{2i+1})^{\Delta (W_{2i})^3}.\]
Thus by elementarity of $\pi$ we may find
$h'\in (W_{2i})^3\cap G_0$ so that $h'h_{2i}'\cdot \hat{x}\in
U_{2i+1}$ and
\[N[\hat{H}]\models (h'h_{2i}'\cdot \hat{x}\models (\psi_{2i+1}')^{\Delta V_{2i+1}'}).\]
Then setting $h_{2i+1}'=h'h_{2i}'$ completes the transition from $2i$ to $2i+1$.
The step from $2i+1$ to $2i+2$ is similar, though easier since it will be a trivial
task to find for $\V$ some $(\psi_{2i+2}, V_{2i+2})\in
{\cal B}(W_{2i+2})$ with
$(\pi(\psi_{2i+2}'), V_{2i+2}') R (\psi_{2i+2}, V_{2i+2})$. \hfill $\Box$
\medskip
We need a fact from infinitary model theory.
\medskip
{\bf 2.8 Theorem} Let $\varphi\in {\cal L}_{\omega_1,\omega}$ and suppose
\[N\models \varphi\]
and $P$ is a predicate in the language of $N$ with
\[|(P)^N|\geq \beth_{\aleph_1}.\]
Then $\varphi$ has a model with generating indiscernibles in $P$.
More precisely there is a model $M$ with language ${\cal L}^*\supset
{\cal L}$, ${\cal L}^*$ having a new symbol $<$, along with new
function symbols of the form $f_{\hat{\varphi}}$ for $\hat{\varphi}$ in the
fragment of ${\cal L}_{\omega_1,\omega}$ generated by $\varphi$,
and distinguished elements $(c_i)_{i\in \N}$,
so that:
\leftskip 0.5in
\noindent (i) $(<)^M$ linearly orders $(P)^M$;
\noindent (ii) each $f_{\hat{\varphi}}$ is a Skolem function for
$\hat{\varphi}$;
\noindent (iii) $M$ is the Skolem hull of $\{c_i:i\in\N\}$ (under the
functions of the form $f_{\hat{\varphi}}$);
\noindent (iv) each $c_i\in (P)^M$;
\noindent (v) for all $\psi$ in the fragment of ${\cal L}^*_{\omega_1,\omega}$
generated by $\varphi$, for all $i_1