\documentclass{article}
%\input{amssymb.def}
%\input{amssymb.tex}
\usepackage{graphicx}
\usepackage{amssymb}
% Set left margin - The default is 1 inch, so the following
% command sets a 1.25-inch left margin.
% Set width of the text - What is left will be the right margin.
% In this case, right margin is 8.5in - 1.25in - 6in = 1.25in.
%\setlength{\textwidth}{6.5in}
% Set top margin - The default is 1 inch, so the following
% command sets a 0.75-inch top margin.
\setlength{\topmargin}{-0.25in}
% Set height of the header
\setlength{\headheight}{0.3in}
% Set height of the text
\setlength{\textheight}{8.5in}
% Set vertical distance between the text and the
% bottom of footer
\setlength{\footskip}{0.3in}
% Set the beginning of a LaTeX document
\def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}}
\def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}}
\def\R{{\Bbb R}}
\def\V{{\Bbb V}}
\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\o{{\omega}}
\def\oo{{\omega^{\omega}}}
\def\lo{{\omega^{<\omega}}}
\def\d{{\dot{\bigcup}}}
\def\h{{\cal H}}
\def\no{\noindent}
\def\m{{\cal M}}
\def\n{{\cal N}}
\def\t{{\cal T}}
\def\s{{\cal S}}
\def\B{{\Bbb B}}
\def\Box{\square}
\newsymbol\game 2061
\begin{document}
% Redefine "plain" pagestyle
\makeatletter % `@' is now a normal "letter' for LaTeX
\title{\LARGE {\bf A boundedness lemma for iterations}\footnote{Partially
supported by NSF grant DMS 96-22977 and a fellowship from
the Sloan foundation}}
\author{Greg Hjorth}
\date{\today}
\maketitle
% Set to use the "plain" pagestyle
\pagestyle{plain}
\noindent{\large{\bf 0. Introduction}}
The purpose of this paper is to present a kind of boundedness lemma for
direct limits of coarse structural mice, and to indicate some applications
to descriptive set theory. For instance, this allows us to show that under large cardinal
or determinacy assumptions there is no prewellorder $\leq$ of length $\ubf{\delta}^1_2$
such that for some formula $\psi$ and parameter $z$
$$x\leq y$$
if and only if
$$L[x, y, z]\models \psi(x, y, z).$$
It is a peculiar experience to write up a result in this area.
Following the work of Martin, Steel, Woodin, and other
inner model theory experts,
there is an enormous overhang of
theorems and ideas, and
it only takes one wandering pebble to restart
the avalanche.
For this reason I have chosen to center the exposition around the
one pebble at 1.7 which I believe to be new. The applications discussed in
section 2 involve routine modifications of known methods.
A detailed
introduction to many of the techniques related to using the Martin-Steel inner model theory
and Woodin's free extender algebra is given in the course of \cite{hj1}.
Certainly a familiarity with the Martin-Steel papers, \cite{ms0} and \cite{ms}, is
a prerequisite, as is some knowledge of the free extender algebra.
Probably anyone interested in this paper will already know the
necessary descriptive set theory, most of which can be found in
\cite{kms}.
Discussion of earlier results in this direction can be found in
\cite{jackson} or \cite{hj2}.
\bigskip
\no{\bf Acknowledgments } I wish to thank Itay Neeman for a number of helpful conversations,
most of which took place by e-mail.
\bigskip
\noindent{\large {\bf 1. The lemma}}
In what follows, all iteration trees are $+2$ and normal.
We will frequently have call to consider illfounded models, and
I will without comment then
identify the wellfounded part of such a model with
its transitive collapse.
Throughout this section assume there exists a Woodin cardinal
with a measurable above; by the determinacy proofs of \cite{ms0} this
is certainly enough to ensure $\Ubf{\Pi}^1_4$ absoluteness between
$V$ and generic extensions obtained by forcing with posets of size less than
the Woodin. Some version of 1.7 makes sense and
can be proved
only assuming determinacy assumptions, but the
statement becomes more intricate without requiring anything
genuinely new for the proof.
\bigskip
\noindent{\bf 1.1 Definition} An expanded premouse $\m=(M, \in, \delta,
{\cal F}, \vec \nu)$ is said to be an $M$-{\it model} if:
(i) $(M, \in)$ models ZF+DC$_{\delta}$ and the ${\cal F}$ sequence is an element
of $M$;
(ii) $M=L(M_{\delta})$ where $M_{\delta}=(V_{\delta})^M$, and
$M_{\delta}$ is countable in $V$;
(iii) $M_{\delta}$ satisfies ZFC; $\delta$ is Woodin in both $\m$ and
$L(\vec E|_{{\delta}})$,
where $\vec E$ is the extender sequence derived from the Doddage ${\cal F}$;
(iv) no $\bar{\delta}$ less than $\delta$ is Woodin in
%$L(\vec E|_{\bar{\delta}})$ or
$L(M_{\bar{\delta}})$;
(v) the Doddage ${\cal F}$ witnesses that $\delta$ is Woodin in $M$:
for all $A\subset \delta$ in $M$ there is some $\kappa<\delta$ which is witnessed
by ${\cal F}$ to be $<\delta$ $A$-strong, in the sense that for all $\lambda$
strictly between $\kappa$ and $\delta$ there is some $E$ on the ${\cal F}$ sequence
such that $(V_{\lambda})^M\in$ Ult$(M, E)$ and for $j_E:M\rightarrow$ Ult$(M, E)$ the
ultrapower map we have $j_E(A)\cap \lambda =A\cap \lambda$;
(vi) $\m$ is fully iterable for countable length iteration trees,
both in the sense that II has a winning strategy for
the iteration game where I presents the iteration trees using extenders on
the ${\cal F}$ sequence of limit length and II gets
to choose the cofinal branches, and further that:
\leftskip 0.4in
\no given a sequence of countable length iteration trees $\t_0$, $\t_1$...
$\t_n$...where the cofinal branches are chosen according to the winning strategy\footnote
{Though by 1.5 below there can be only one choice of a fully wellfounded
cofinal branch,
and therefore at this level the winning strategy for II plays a trivial part.}, with
$\t_0$ an iteration tree on $\m$ with final model $\m_1$, and embedding
$\rho_{\t_0}: \m\rightarrow \m_1$ given by the direct limit along the main branch,
$\t_{i}$ an iteration tree on $\m_i$ with final model
$\m_{i+1}$, and embedding
$\rho_{\t_{i}}: \m_i\rightarrow \m_{i+1}$ given by the direct limit along the main branch,
then the direct limit model
$${\rm DirLim}(\m_i, \rho_{\t_i})$$ is wellfounded.
\leftskip 0in
For $z\in \m\cap \omega^{\omega}$ we then
let $v_l(\m, z)$ be the sup of the ordinals less than $\delta$ definable in
$\m$ from $z$, and ${\cal F}$, and the first
$l$ many uniform indiscernibles, $u_1, u_2, ..., u_l$; thus $v_0(\m, z)$ is the sup of the ordinals
less than $\delta$ that are definable from just $z$ and ${\cal F}$. Note that we could
have
equivalently defined $v_l(\m, z)$ to be the sup of the ordinals less than $\delta$ definable in
$(L_{u_{l+1}}(M_{\delta}), \in, \delta,
{\cal F}, \vec \nu)$ from $z$, ${\cal F}$, and the first
$l$ many uniform indiscernibles.
Thus the ordinal $v_i(\m, z)$ will be definable from $u_1, u_2, ..., u_i, u_{i+1}$ and $z$.
%Write $v_l(\m)$ in place of $v_l(\m, z)$
%in the special case that $z$ is recursive.
Note that (iv) above entails the sequence $(v_i(\m), z)_{i\in\o}$ being cofinal in $\delta$.
Given $\t_0$, $\t_1$...
$\t_n$... and $\rho_{\t_0}, \rho_{\t_i}, ...\rho_{\t_n}$... as in (vi) above, and
$k\in\omega$, we let
$$\rho_{\t_0 \t_{1}...\t_k}: \m\rightarrow \m_{k+1}$$
be the composition $\rho_{\t_k}\circ\rho_{\t_{k-1}}\circ...\rho_{\t_0}$.
In general I will use the phrase ``iterate" of ${\cal M}$ to include
not just models arising from a single iteration tree applied to ${\cal M}$,
but by also the models arising from finitely many compositions of iteration trees,
such as the $\m_i$'s appearing in (vi).
As a helpful but not quite correct abuse of notation, let us agree to
write $\m=L(M_{\delta})$, suppressing mention of the extra structure provided by
the Doddage.
%I will also only be interested in trees using extenders on the
%${\cal F}$ sequence, so let us further agree that in all that follows the iteration
%trees are formed solely by use of extenders on ${\cal F}$ -- in other words, where
%below is written ``iteration tree" the reader should in fact read ``iteration tree
%formed using only extenders on the ${\cal F}$ sequence;"
Below we will be working entirely with coarse structural premice from
\cite{ms}, which admittedly gives the exposition a kind of antiquated
style.
Certainly
many of the ideas will
work in a wider context.
On the other hand I do not want to spell out what kinds of mice
can be used and how the various arguments can be trivially modified,
but rather choose to give the lemma 1.7 in the simplest form.
\bigskip
Until the end of this section, fix $\m=L(M_{\delta})$, some $M$-model, and
$z\in \m\cap \o^\o$.
\bigskip
\no{\bf 1.2 Definition} Let $X_n$ be the space of all finite sequences
\[(\t_0, \m_1, \t_1, \m_2, ...\m_k, \alpha)\]
where if we define $\m_0=\m$, each
$\t_i$ is a countable iteration tree on $\m_i$ having final model $\m_{i+1}$,
with cofinal branches chosen according to the winning strategy for II,
and associated embedding
$\rho_{\t_i}: \m_i\rightarrow \m_{i+1}$ along the main branch,
and
$$\alpha \beta,$$
where $\rho_{\s_{k}\s_{k+1}...\s_{l-1}}: \n_k\rightarrow \n_l$ is the embedding indicated
by the iteration trees as in 1.1.
\bigskip
In 1.2 the notation could be made more explicit by
writing $X_n(\m, z)$ and $R_n(\m, z)$ to indicate the
dependence on $\m$ and $z$, but since these are
fixed throughout this section there should be
no confusion.
\bigskip
\no{\bf 1.3 Lemma} $R_n$ is wellfounded, in the sense of there being no infinite
sequence $(x_i)_{i\in\omega}$ of elements in $X_n$ with $x_i R_n x_{i+1}$ at each
$i$.
Proof. Otherwise let $\t_i$ be the iteration tree occuring in $x_j$ all sufficiently
large $j$ and let $\m_i$ be the $i$th model in $x_j$ for all large enough $j$. We then
obtain that $(\t_i)_{i\in \omega}$ is a sequence of iteration trees, with
$\t_{i}$ an iteration tree on $\m_i$ with final model
$\m_{i+1}$, with embedding
$$\rho_{\t_{i}}: \m_i\rightarrow \m_{i+1},$$ all arranged so that the
direct limit model DirLim$(\m_i, \rho_{\t_i})$ is illfounded.
This is in direct contradiction to our iterability assumption on the $M$-model
$\m$. \hfill $\Box$
\bigskip
The next lemma is well known, and in some form dates back to the original papers by
Martin and Steel.
\bigskip
\no{\bf 1.4 Lemma} Let $\n=L(N_{\bar{\delta}})$
where $N_{\bar{\delta}}=(V_{\bar{\delta}})^{\n}$ and
no $\gamma<\bar{\delta}$ is Woodin in $L((V_{\gamma})^{\n})$.
Let $\t$ be an iteration tree of
limit length on $\n$, and let $b$ and $c$ be cofinal
branches through $\t$. Let
$$i_b:\n\rightarrow \n_b$$
$$i_c:\n\rightarrow \n_c$$
be the direct limit maps along these cofinal branches.
Suppose $\alpha<\bar{\delta}$ is such that $i_b(\alpha)$ and $i_c(\alpha)$ are in the
wellfounded parts of $\n_b$ and $\n_c$ and moreover
$$i_b(\alpha)=i_c(\alpha)$$
and
$$i_b(\bar{\delta})=i_c(\bar{\delta}).$$
Then $i_b$ and $i_c$ agree up to $\alpha$ -- i.e.
$$i_b|_{\alpha}=i_c|_{\alpha}.$$
Proof. Let $\gamma$ be the greatest ordinal in the intersection of
$b$ and $c$; this ordinal exists since both branches are closed subsets
of the ordinals. Without loss we may assume that the least ordinal after
$\gamma$ in $b\cup c$ is in $b$ rather than $c$. Call it $\gamma_0+1$
(we are justified in the assumption it be a successor, since the limit
ordinals are all at limit levels of the tree).
Let $\n_\beta$ represent the $\beta$th model on $\t$ for $\beta$ less than
the length of $\t$. Let $E_{\beta}$ be the extender used in definition of
$\n_{\beta+1}$ from its preceding model -- so that $\n_{\beta+1}=$
Ult$(\n_{\beta^*}, E_{\beta})$, where $\beta^*$ is the predecessor
ordinal to $\beta+1$ in the tree ordering provided by $\t$.
Following the proof of the uniqueness theorem from \cite{ms}
we obtain a sequence $\gamma_0<\gamma_1<\gamma_2<...$ and
$\lambda_0<\lambda_1<\lambda_2...$ such that $\lambda_i+1$ is the
least place on $c$ after $\gamma_i+1$ and $\gamma_{i+1}+1$ is the
least place on $b$ after $\lambda_i+1$. Note then that the length of the
tree is equal to both $\bigcup_{i\in\o}\gamma_i$
and $\bigcup_{i\in\o}\lambda_i$.
%We then let
%$\hat{\lambda}_i$ be the greatest ordinal on $c$ before $\lambda_i+1$ and
%$\hat{\gamma}_{i}$ be the least ordinal on $b$ before
%$\gamma_i+1$ -- so $\lambda_i^*=\hat{\lambda}_i$ and
%$\gamma_i^*=\hat{\gamma}_i$.
The normality of the iteration tree entails
\[{\rm cp}(E_{\lambda_i})< {\rm lh}(E_{\gamma_{i}}),\]
\[{\rm cp}(E_{\gamma_{i+1}})< {\rm lh}(E_{\lambda_i}),\]
\[{\rm cp}(E_{\lambda_i})< {\rm cp}(E_{\lambda_{i+1}}),\]
\[{\rm cp}(E_{\gamma_{i}})< {\rm cp}(E_{\gamma_{i+1}}).\]
By 2.2 \cite{ms} we have that
$$\delta(\t)=\bigcup_{i\in\o}{\rm cp}(E_{\lambda_i}) =\bigcup_{i\in\o}{\rm cp}(E_{\gamma_i}) $$
is Woodin in
$$L_{i_b({\bar{\delta}})}(\m(\t)),$$
where
$$\m(\t)=\bigcup_{\alpha\in i_b({\bar{\delta}})}{\rm Lim}_{i\rightarrow \o}
(V_{\alpha})^{\n_{\gamma_i+1}}= \bigcup_{\alpha\in i_b({\bar{\delta}})}{\rm Lim}_{i\rightarrow \o}
(V_{\alpha})^{\n_{\lambda_i+1}}.$$
Thus
$$\delta(\t)=i_b({\bar{\delta}})=i_c({\bar{\delta}})$$
by minimality of $\bar{\delta}$ in $\n$.
%and hence $\deltaequal to
%$$i_b(\bar{\delta})=i_c(\bar{\delta}).$$
%\begin{figure}[ht]
%{\psfig{figure=tree.eps}}
%\caption{Overlap among the extenders on the two branches}
%\end{figure}
\leftskip 5in
\begin{figure}
\begin{picture}(0,0)%
{\includegraphics{tree.pstex}}%
%\epsfig{file=tree.pstex}%
\end{picture}%
\setlength{\unitlength}{0.00050000in}%
\begin{picture}(4951,4599)(1801,-4199)
\put(1876,-1261){\makebox(0,0)[lb]{lh$(E_{\gamma_i})$}}
\put(1801,-2236){\makebox(0,0)[lb]{cp$(E_{\gamma_i})$}}
\put(3676,-136){\makebox(0,0)[lb]{$E_{\gamma_{i+1}}$}}
\put(3676,-1561){\makebox(0,0)[lb]{$E_{\gamma_i}$}}
\put(6526,-736){\makebox(0,0)[lb]{$E_{\lambda_i}$}}
\put(4501,-586){\makebox(0,0)[lb]{b}}
\put(6526, 89){\makebox(0,0)[lb]{c}}
\end{picture}
\leftskip 0in
\caption{Overlap among the extenders on the two branches}
\end{figure}
\leftskip 0.4in
\no Thus we obtain:
\no(1) for all $\beta\geq$ inf(cp$(E_{\gamma_0})$, cp$(E_{\lambda_0})$)
with $\beta<\bar{\delta}$, we have that
$$i_{\gamma, b}(\beta)\neq i_{\gamma, c}(\beta), $$
where
$$i_{\gamma, b}:\n_{\gamma}\rightarrow \n_b$$
and
$$i_{\gamma, c}:\n_{\gamma}\rightarrow \n_c$$ are the
respective direct limits along these cofinal branches.
\no(2) Conversely for all $\beta<$inf(cp$(E_{\gamma_0})$, cp$(E_{\lambda_0})$)
we have that
$$i_{\gamma, b}(\beta)= i_{\gamma, c}(\beta). $$
\leftskip 0in
\no So by (1) we have $i_{0, \gamma}(\alpha)< $inf(cp$(E_{\gamma_0})$, cp$(E_{\lambda_0})$), where
$$i_{0, \gamma}:\n\rightarrow \n_{\gamma}$$
is the canonical embedding along the iteration tree.
Thus for all $\beta<\alpha$ we have $$i_b(\beta)=
i_{\gamma, b}(i_{0, \gamma}(\beta))=i_{\gamma, c}(i_{0, \gamma}(\beta))=
i_c(\beta)$$ by (2), just as required. \hfill $\Box$
\bigskip
\no{\bf 1.5 Theorem}(Woodin) Let $\t$ be a countable iteration tree on an $M$-model $\n$ of limit
length. Then $\t$ has exactly one wellfounded cofinal branch.
Proof. See 5.5 of \cite{ms}. \hfill $\Box$
%Let $b$ and $c$ be wellfounded cofinal branches and let
%$$i_b:\n\rightarrow \n_b$$
%$$i_c:\n\rightarrow \n_c$$
%be the associated elementary embeddings.
%Note then that $i_b(\bar{\delta})=i_c(\bar{\delta})$ since both are equal to
%the Woodin cardinal of the common model.
%Note then as in the
%proof of 1.4 that $\n_b=\n_d$ and that the
%Woodin cardinal of this common model is equal to the sup of the lengths of
%the extenders used in $\t$. In particular $i_b(\bar{\delta})=
%i_c(\bar{\delta})$, where $\bar{\delta}$ is the Woodin cardinal in $\n$.
%Now choose $D$ to be a club class of ordinals that are fixed under $i_b$ and
%$i_c$. Since this club class is definable from countable objects, it includes
%all the uniform indiscernibles. As remarked in the course of 1.1, the sequence
%$(v_i(\n))_{i\in\o}$ will be cofinal in the Woodin cardinal $\bar{\delta}$.
%Since the uniform indiscernibles
%are fixed by the two embeddings,
%$$i_b(v_i(\n))=i_c(v_i(\n))$$
%at each $i$. Since $i_b(\bar{\delta})=i_c(\bar{\delta})$ it follows that
%$i_b|_{\bar{\delta}}=i_c|_{\bar{\delta}}$, and so $b=c$. \hfill $\Box$
\bigskip
\no{\bf 1.6 Corollary} The property of being an $M$-model is $\Ubf{\Pi}^1_4$, and
hence absolute to all small generic extensions.
Proof. For instance the statement that for all countable iteration trees there exists a
full wellfounded branch is transparently $\Ubf{\Pi}^1_4$, while the requirement that
wellfoundedness be maintained under direct limits is $\Ubf{\Pi}^1_3$. \hfill $\Box$
\bigskip
\no{\bf 1.7 Lemma} The rank of $R_n$ is less than $u_{n+2}$.
Proof. In $V^{{\rm Coll}(\omega, u_{n+1})}$ let us define an auxiliary relation
$\hat{R}_n$ as follows. First we let $\hat{X}_n$ be the collection of all sequences
\[(\t_0, L_{u_{n+1}}(M^1_{\delta_1}), ...,
\t_{k-1}, L_{u_{n+1}}(M^k_{\delta_k}), \alpha)\]
where:
\leftskip 0.4in
\no (i) setting $ L_{u_{n+1}}(M^0_{\delta_0})= L_{u_{n+1}}(M_{\delta})$,
each $\t_{i}$ is a countable iteration tree on
$ L_{u_{n+1}}(M^i_{\delta_i})$ with final model $ L_{u_{n+1}}(M^{i+1}_{\delta_{i+1}})$;
\no (ii) for
$$\rho_{\t_i}: L_{u_{n+1}}(M^i_{\delta_i})\rightarrow L_{u_{n+1}}(M^{i+1}_{\delta_{i+1}})$$
the embedding provided by the main branch we have that for each $j\leq n$
$$\rho_{\t_i}(u_j)=u_j;$$
\no (iii) $\alpha$ is less than the sup of the
ordinals below $\delta_k$ definable from $z$, $\rho_{\t_0\t_1...\t_{k-1}}({\cal F})$, and
$u_1, u_2, ..., u_n$ in $ L_{u_{n+1}}(M^k_{\delta_k})$.
\leftskip 0in
For
$$x=(\t_0, L_{u_{n+1}}(M^0_{\delta_0}), ..., \t_{k-1}, L_{u_{n+1}}(M^k_{\delta_k}), \alpha)$$
and
$$y=(\s_0, L_{u_{n+1}}(N^0_{\gamma_0}), ..., \s_{l-1}, L_{u_{n+1}}(N^l_{\gamma_k}), \beta)$$
two elements in $\hat{X}_n$ we set $x\hat{R}_n y$
if $l\leq k$, $\t_i=\s_i$ for $i< k$,
and $\rho_{\s_{k}...\s_{l-1}}(\alpha)>\beta.$
Thus $\hat{X}_n$ and $\hat{R}_n$ are like $\Ubf{\Sigma}^1_1$ analogs of $X_n$ and
$R_n$ in $V^{{\rm Coll}(\omega, u_{n+1})}$. The main issue is to show that $\hat{R}_n$ is
again wellfounded.
The next two claims observe that we can try to copy over an infinite sequence in $\hat{R}_n$
to an infinite sequence in $R_n$. It may not quite work, since the ``copied" tree may diverge
at the very last place in the choice of the cofinal branch. However this is the only possible
failure,
and it can only happen in a manner material to the embedding below $v_n(\m, z)$
finitely many times.
\medskip
Claim(1). Let $(\t_0, L_{u_{n+1}}(M^1_{\delta_1}),
..., \t_{i-1}, L_{u_{n+1}}(M^i_{\delta_i}),...)$ be
an infinite sequence such that at each $k\in\o$
$$(\t_0, L_{u_{n+1}}(M^1_{\delta_1}), ..., \t_{k-1}, L_{u_{n+1}}(M^k_{\delta_k}), 0)\in \hat{X}_n.$$
Let $\theta_i+1$ equal the length of each $\t_i$.
Then there are trees $\s_0, \s_1, ..., \s_i,...$, models $\n_0=\m, \n_1, ....\n_i,...$,
$\n_i=L(N^i_{\kappa_i})$, and inclusion maps
$\tau_i: M^i_{\delta_i}\rightarrow N^i_{\kappa_i}$,
such that:
\leftskip 0.4in
\no (i) each
$\s_{i}$ is an iteration tree on $\n_i$ with final model
$\n_{i+1}$;
\no (ii) each $\s_i|_{\theta_i}$ equals the tree on $\n_i$ obtained by copying
$\t_i$ over using the inclusion map $\tau_i: M^i_{\delta_i}\rightarrow N^i_{\kappa_i}$,
and moreover $ M^i_{\delta_i}$ is a rank initial segment of
$N^i_{\kappa_i}$;
\no (iii) each $\s_i$ has length $\theta_i+1$.
\leftskip 0in
Proof of Claim: We obtain this by induction on $i$. So suppose inductively
that we have $\n_i=L(N_{\kappa_i}^i)$ and an inclusion map
$$\tau_i:M^i_{\delta_i}\rightarrow N_{\kappa_i}^i.$$
We then have $\t_i$ on $\m_i$ which we attempt to copy over to
a tree $\s_i$ on $\n_i$.
By elementarity of the maps $(\rho_{\t_j})_{j*\rho_{\t_{m(i)}}(\delta_{m(i)})$$
then
$$\rho_{m(i), \infty}(\delta_{m(i)})$$
provides an infinite descending chain in $\n_{\infty}$ contradicting that
$\m$ is an $M$-model in both $V$ and $V^{{\rm Coll}(\omega, u_{n+1})}$.
A similar contradiction arises from assuming that there is some $j$ for which
$$\rho_{\s_i}(u_j)>u_j=\rho_{\t_i}(u_j)$$
at infinitely many $i$.
\hfill (Claim $\Box$)
\medskip
Claim(3). Let $$(\t_0, L_{u_{n+1}}(M^1_{\delta_1}), ..., \t_{i-1},
L_{u_{n+1}}(M^i_{\delta_i}),...){\rm ,}$$
$$\s_0, \s_1, ..., \s_i,...{\rm ,}$$
$$\n_0, \n_1, ....\n_i,...{\rm ,}$$
be as in the previous claim.
Then for all but finitely many $i$
we have that if $\zeta<\delta_{i+1}$ is definable from
$u_1, u_2, ..., u_n$, $\rho_{\t_0...\t_i}({\cal F})$, and $z$ over
$L_{u_{n+1}}(M^{i+1}_{\delta_{i+1}})$ then
$$\rho_{\t_i}|_{\zeta}=\rho_{\s_i}|_{\zeta}.$$
Proof of claim. By the conclusion of claim(2) and 1.4. \hfill (Claim $\Box$)
\medskip
Claim(4). $\hat{R}_n$ is wellfounded.
Proof of claim. Otherwise we may choose
$(\t_0, L_{u_{n+1}}(M^1_{\delta_1}), ..., \t_{i},
L_{u_{n+1}}(M^{i+1}_{\delta_{i+1}}),...)$,
non-decreasing $k(i)$ in $\omega$ such that
$$k(i)\rightarrow \infty$$
as $i\rightarrow \infty$ and
ordinals $\alpha_i$ less than the sup of the ordinals below
$\delta_{k(i)+1}$ definable from
$u_1, u_2, ..., u_n$, $\rho_{\t_0...\t_{k(i)}}({\cal F})$,
and $z$ over $L_{u_{n+1}}(M^{k(i)+1}_{\delta_{k(i)+1}})$,
coordinated so that for
$$x_i=(\t_0, L_{u_{n+1}}(M^0_{\delta_0}), ...,
\t_{k(i)}, L_{u_{n+1}}(M^{k(i)+1}_{\delta_{k(i)+1}}),\alpha_i)$$
we have $x_i\hat{R}_nx_{i+1}$, and hence
$$\rho_{\t_{k(i)+1}\t_{k(i)+2}...\t_{k(i+1)}}(\alpha_i)>\alpha_{i+1}.$$
Following claims (1) to (3) we may find a corresponding
$\s_0, \s_1, ..., \s_i,...$ and $\n_0, \n_1, ....\n_i,...$ such that
at all but finitely many $i$
$$\rho_{\s_{k(i)+1}\s_{k(i)+2}...\s_{k(i+1)}}(\alpha_i)
=\rho_{\t_{k(i)+1}\t_{k(i)+2}...\t_{k(i+1)}}(\alpha_i)>\alpha_{i+1}.$$
This gives that $${\rm DirLim}(\n_i, \rho_{\s_i})$$ is illfounded, with a
contradiction to the absoluteness between $V$ and
$V^{{\rm Coll}(\omega, u_{n+1})}$ of being an $M$-model. \hfill (Claim$\Box$)
\medskip
Claim(5) The rank of $\hat{R}_n$ in $V^{{\rm Coll}(\omega, u_{n+1})}$ is at least as great as
$R_n$ in $V$.
Proof of claim. We can embed $\hat{R}_n$ into $R_n$, and the prove the
claim by transfinite induction on the ranking
functions. \hfill (Claim $\Box$)
\medskip
But now note that $\hat{R}_n$ is $\Sigma^1_1(w)$ for some $w$ appearing in
$L(\m)^{{\rm Coll}(\omega, u_{n+1})}$. Since it is wellfounded it has rank
less $\omega_1^{{\rm ck}(w)}$, which in turn is below $u_{n+2}$. \hfill $\Box$
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\empty
\bigskip
\bigskip
\newpage
\no{\large {\bf 2. Prewellorderings definable from the uniform indiscernibles}}
\bigskip
\no{\bf 2.1 Definition} Let us say that a subset $C$ of some finite
product $(\o^\o)^m$ of Baire space is
$\Gamma_{1, n}(z)$ if for some formula $\psi$ we have that
for all $x\in \o^\o$
$$x\in C$$
if and only if
$$L[x, z]\models \psi(x, z, u_1, u_2, ..., u_n).$$
Then bolderize as usual, with
$$\Ubf{\Gamma}_{1, n}=\bigcup_{z\in \o^\o} \Gamma_{1, n}(z).$$
\bigskip
Thus the $\Ubf{\Gamma}_{1, 0}$ prewellorders include all the $\Ubf{\Pi}^1_2$
prewellorders, as well as those prewellorderings defined by sets appearing in the
$\sigma$-algebra generated by $\Ubf{\Pi}^1_2$.
The next result can be proved from $\Ubf{\Pi}^1_2$ determinacy using
Hugh Woodin's fine analysis of the connections between determinacy and large
cardinals; to avoid distracting details I will merely argue for it under
large cardinal hypotheses; the assumptions of 2.2 {\it do} imply
$\Ubf{\Pi}^1_2$ determinacy in light of \cite{ms0}.
\bigskip
\def\l{\leq_{1, n}}
\no {\bf 2.2 Theorem} Assume there is a Woodin cardinal and a measurable above.
Then any $\Ubf{\Gamma}_{1, n}$ prewellorder has rank less than $u_{n+2}$.
Proof. Let $\leq_{1, n}$ be a $\Ubf{\Gamma}_{1, n}$ prewellorder.
For $x\in \o^\o$ let Rk$_{\l}(x)$ denote the rank of
$x$ with respect to $\l$. We will milk out a contradiction from the
proposition that $\{$ Rk$_{\l}(x): x\in \o^\o\}$ includes $u_{n+2}$.
Let $(\tau_m)_{m\in\o}$ be some reasonable enumeration
of the Skolem functions definable over a class model constructed from a real.
Appealing to the coding lemma for the point class $\Ubf{\Sigma}^1_3$ we may find
some $z_0\in \o^\o$ and some $\Sigma^1_3(z_0)$ set $A$ such that:
\leftskip 0.4in
\no (i) for all $\beta\beta.$$
But then if we let $(i_n)_{n\in\omega}$ enumerate the $i$'s at
which this takes place, and let
$$\rho_{i_n, \infty}: \m_{i_n}\rightarrow {\rm \: Dir \: Lim}
(\m_i, \rho_i)$$ be the
direct limit, then in $\m_\infty=_{df} {\rm \: Dir \: Lim}
(\m_i, \rho_i)$ we have that
$(\rho_{i_n,\infty}(\beta))_{n\in\omega}$ is an infinite descending
chain, contradicting 1.1(vi)'s assertion that this direct limit
$\m_\infty$ should be wellfounded.
\bigskip
Claim(2). If $\beta\zeta$ a corresponding
prewellorder on $\omega\times$ WO as follows: For $n, m\in \omega$,
$w, v$ in WO, with $||w||=\alpha$, $||v||=\beta$, we set
$$(n, w)<_{x^\sharp, 0} (m, v)$$
if
$$\tau_n^{L[x]}(\alpha, u_1, ..., u_{a(n)-2}, x)<
\tau_m^{L[x]}(\beta, u_1, ..., u_{a(m)-2}, x),$$
where $a(n)$ and $a(m)$ are the respective arities of the
Skolem functions $\tau_n$ and $\tau_m$. From the point of view of
calculating the inequality on display, the $u_1, u_2, ...$ can just
be treated like any other arbitrarily large $L[x]$-indiscernibles, and
hence the relation
$(n, w)<_{x^\sharp, 0} (m, v)$ is uniformly definable over
$L[x^\sharp, w, v]$ using $x^\sharp, w, v, n, m$ as
parameters. Since every subset of $(\omega_1^V)^{L[x]}$ is
definable from some $\alpha<\omega_1^V$ and finitely many
indiscernibles, this prewellorder has rank equal to at least
$((\omega_1^V)^+)^{L[x]}$.
More generally if $\zeta\zeta$. We then define $<_{x^\sharp, n+1}$ on
$\omega^2\times \{y^\sharp: y\in 2^\N\}$ by
$$(m_1, m_2, y^\sharp)<_{x^\sharp, n+1} (k_1, k_2, z^\sharp)$$ if for
$$\alpha=\tau_{m_1}^{L[y]}(u_1, u_2, ..., u_{n+1}, y),$$
$$\beta=\tau_{k_1}^{L[z]}(u_1, u_2, ..., u_{n+1}, z),$$ we have
$$\tau_{m_2}^{L[x]}(\alpha, u_{n+2}, u_{n+3}, ..., x)
<\tau_{k_2}^{L[x]}(\beta, u_{n+2}, u_{n+3}, ..., x).$$
In this way we obtain a prewellorder on a $\Pi^1_2$ set
which can be uniformly calculated from $x^\sharp$ and $u_1, u_2, ..., u_{n+1}$.
Again the indiscernibles $u_{n+2}, u_{n+3},...$ vanish into insignificance from
the point of view of complexity, since we simply substitute for them any
sufficiently large indiscernible over $L[x]$.
%The method of 2.2 can also be used to show under appropriate large cardinal
%assumptions that there is no $u_{n+2}$-sequence of distinct
%$\Ubf{\Gamma}^1_n$ sets in $L(\R)$. Assuming for a contradiction that
%$(C_{\beta})_{\beta0$. The further evolution of this problem would seem to require
%Jackson's deep analysis of the projective ordinals.
%A rather different direction in which this might lead is towards a finer analysis
%of HOD$^{L(\R)}$. It is known from \cite{steel} that HOD$^{L({\R})}$ is a core model, obtained
%by a direct limit of countable fine structural mice. Lemma 1.7 can obviously be
%used to provide some information regarding the large cardinal properties of
%the first $\o$ many uniform indiscernibles in HOD$^{L({\R})}$, but seems on its
%own to fall short of providing a complete characterization in terms of their
%properties over this inner model.
\bigskip
\noindent {\bf 3 Addendum: How far will this go?}
\bigskip
The method of 2.2 can be used to show under appropriate large cardinal
assumptions that there is no $u_{n+2}$-sequence of distinct
$\Ubf{\Gamma}^1_n$ sets in $L(\R)$. Assuming for a contradiction that
$(C_{\beta})_{\beta0$. The further evolution of this problem would seem to require
Jackson's deep analysis of the projective ordinals.
A rather different direction in which this might lead is towards a finer analysis
of HOD$^{L(\R)}$. It is known from \cite{steel} that HOD$^{L({\R})}$ is a core model, obtained
by a direct limit of countable fine structural mice. Lemma 1.7 can obviously be
used to provide some information regarding the large cardinal properties of
the first $\o$ many uniform indiscernibles in HOD$^{L({\R})}$, but seems on its
own to fall short of providing a complete characterization in terms of their
properties over this inner model.
\bigskip
\begin{thebibliography}{99}
\bibitem{hj1} G. Hjorth, {\it Some applications of coarse inner model
theory,} {\bf Journal of Symbolic Logic,} vol. 62(1997), pp. 337-365.
\bibitem{hj2} G. Hjorth, {\it Two applications of inner model theory
to $\Ubf{\Sigma}^1_2$ sets,}
{\bf Bulletin of Symbolic Logic,} vol. 2(1996),
pp. 94-107.
\bibitem{jackson} S. Jackson, {\it Partition properties and well-ordered
sequences,}
{\bf Annals of Pure and Applied Logic,} vol. 48(1990), pp. 81-101.
\bibitem{kms} A.S. Kechris, D.A. Martin, R.M. Solovay,
{\it Introduction to $Q$-theory,} {\bf Cabal seminar 79-81}, pp. 199-282,
Lecture Notes in Mathematics, 1019, Springer, Berlin, 1983
\bibitem{ms0} D.A. Martin, J.R. Steel,
{\it A proof of projective determinacy,}
{\bf Journal of the American Mathematical Society,} vol. 2(1989),
pp. 71-125.
\bibitem{ms} D.A. Martin, J.R. Steel, {\it Iteration trees,}
{\bf Journal of the American Mathematical Society,} vol. 7(1994), pp. 1-73
\bibitem{neeman} I. Neeman, {\it Optimal proofs of determinacy,}
{\bf Bulletin of Symbolic Logic,} vol. 1(1995), 327-339.
\bibitem{steel} J.R. Steel, {\it HOD$^{L({\R})}$ is a core model below $\Theta$,}
{\bf Bulletin of Symbolic Logic,} vol. 1(1995), 75-84.
\end{thebibliography}
\bigskip
6363 MSB
Mathematics
UCLA
CA90095-1555
greg@math.ucla.edu
www.math.ucla.edu/\~{}greg
\end{document}
*