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\title{An introduction to $\Pmax$ forcing
%\\ \small Appalachian Set Theory Workshop \\ Carnegie Mellon University \\ 9 September 2006
}
\author{Paul B. Larson\thanks{Research supported by NSF grants DMS-0401603 and DMS-0801009; conference expenses supported by
NSF grant DMS-0631446}
%\\ \small Transcribed by Peter LeFanu Lumsdaine and Yimu Yin
}

\maketitle

\subsection{Pre-introduction}

These notes are an account of a six-hour lecture series I presented
at Carnegie Mellon University on September 9, 2006, at the inaugural
meeting of the Appalachian Set Theory series. My remarks
(occasionally improvised) were faithfully transcribed by Peter
LeFanu Lumsdaine and Yimu Yin, who then presented me with a rough
draft of this article. Their account aimed to capture the feel of
the discussions (including some direct quotations), and I've tried
to preserve that as much as possible.

%Rather than attempt to make these notes precise and formal, I've
%tried to replicate the informal communicativity of the lecture by
%sticking as closely as possible to a verbatim transcript of Yimu's
%and my handwritten notes. As such, I doubt we've managed to catch
%all our errors, for which my apologies; I hope however that these
%notes manage to retain and convey the spirit of the original
%lecture.

\section{Introduction}

The forcing construction $\Pmax$ was invented by W. Hugh Woodin in
the early 1990's in the wake of his result that the saturation of
the nonstationary ideal on $\omega_{1}$ ($\NS_{\omega_1}$)
plus the existence of a measurable cardinal implies
that there is a definable counterexample to the Continuum Hypothesis
(in particular, it implies that $\bolddelta^1_2 = \omega_2$, which
implies $\lnot \CH$). These notes outline a proof of the $\Pi_{2}$
maximality of the $\Pmax$ extension, which we can state as follows.

\begin{theorem}[\cite{Woo99}]\label{goal1} Suppose that there exist
proper class many Woodin cardinals,
$A \sse \R$, $A \in L(\R)$, $\varphi$ is $\Pi_2$ in the extended
language containing two additional unary predicates, and in some set
forcing extension $$\la H(\omega_2), \in, \NS_{\omega_1}, A^* \ra
\believes \varphi$$ (where $A^*$ is the reinterpretation of $A$ in
this extension). Then $$L(\R)^\Pmax \believes \left[\la H(\omega_2)
, \in, \NS_{\omega_1}, A \ra \believes \varphi \right].$$
\end{theorem}




Forcing with $\Pmax$ does not add reals, so there is no need to
reinterpret $A$ in the last line of the theorem. The theorem says that any such $\Pi_2$ statement that we can force
in any extension must hold in the $\Pmax$ extension of $L(\R)$, so
$H({\omega_2})$ of $L(\R)^\Pmax$ is maximal, or complete, in a
certain sense; among other things, it models ZFC, Martin's Axiom,
certain fragments of Martin's Maximum \cite{FoMagSh}, and the
negation of the Continuum Hypothesis. The reinterpretation $A^*$
will be defined later, in terms of tree representations for sets of
reals. We will not give the definition of Woodin cardinals (but
see \cite{Kan}).

We have reworked the standard proof of Theorem \ref{goal1} in order
to minimize the prerequisites. In particular, the need for
(mentioning) sharps has been eliminated.
%Prerequisites are minimal. Sharps have been eliminated as far as possible;
However, they and much more will need to be reintroduced to go any
further than the material presented here.

%is a partial order in $L(\mathbb{R})$ which produces a $\Pi_{2}$
%maximal model for the structure $H(\omega_{2})$ with predicates for
%the nonstaitonary ideal on $\omega_{1}$ and set of reals in
%$L(\mathbb{R})$.

%This account of $\Pmax$ has been reworked to minimize the
%prerequisites.

%The invention of $\Pmax$ directly followed Woodin's proof that



 ``Woodin's book on $\Pmax$, \emph{The axiom of determinacy, forcing axioms and the nonstationary ideal}
 \cite{Woo99} runs to around 1000 pages. My article for the Handbook of Set Theory \cite{Lxx}, introducing
 $\Pmax$, has about 65. The advance notes for these lectures are about 30 pages, and previous lecture courses
 have taken about 12-15 hours to cover $\Pmax$; so today will, of course, have to be brief\ldots''



\section{Setup: iterations and the definition of $\Pmax$}

Suppose that $M \believes \ZFC$ and than $I \in M$ is a normal ideal on
${\omega_1^{M}}$ in $M$. Force over $M$ with $((\P({\omega_1})
\setminus I)/ I)^M$. The resulting generic $G$ is now an $M$-normal
ultrafilter on ${\omega_1}^M$; so we may form the corresponding
ultrapower and elementary embedding $j: M \to \Ult(M,G) :=
\{f:{\omega_1}^{M} \to M\ |\ f \in M\} / =_G$. (``We'll use this a
thousand times today.'') Note that $\crit(j) = {\omega_1}^M$,
$j({\omega_1}^M) \geq {\omega_2}^M$, $\Ord^{\Ult(M,G)} = \Ord^M$,
and for $A \in \P({\omega_1})^M$, $A \in G \iff {\omega_1}^M \in
j(A)$. There is no need to assume that $A$ is transitive, though it will
be in the cases were are interested in. When an ultrapower is well-founded, we identify it with its
transitive collapse.

 \begin{definition} $I$ is \emph{precipitous} if $\Ult(M,G)$ thus constructed is well-founded from the point of view of $M[G]$, for all $M$-generic $G$. (N.B.\ this is definable in $M$ via forcing.)
\end{definition}

 %\piccie

 \paragraph{} We need a pair of theories satisfying the
 following conditions.

\begin{itemize}
\item $T_0$,
%& \multicolumn{2}{l}{ZFC} \\
a theory consistent with ZFC and strong enough to make sense of the
generic ultrapower construction above and
 prove that $j : M \to \Ult(M,G)$ is elementary.
\item $T_1$,
%\multicolumn{2}{l}{ZFC + $\E$ a proper class of strong inaccessibles}
a theory consistent with ZFC and at least as strong as $T_0$ +
``every set lies in some $H(\kappa) \believes T_0$."
%\item $T_2$,
%%& \multicolumn{2}{l}{ZFC + $\E$ a proper class of Woodins} \\
%a theory consistent with ZFC and at least as strong as $T_1$ +
%``every set lies in some $H(\kappa) \believes T_1$."
\end{itemize}

In \cite{Lxx}, we take T$_{0}$ (which we call ZFC$^{\circ}$) to be
$\ZFC$ - Replacement - Powerset plus $``\P(\P(\omega_1))$ exists"
plus the scheme saying that definable trees of height $\omega_1$
have maximal branches. Then we let $T_1 = T_{0}$ + Powerset + Choice
+ $\Sigma_{1}$-Replacement 
%and $T_{2} = ZFC$ 
(though we don't express it in these terms). In
\cite{Woo99}, Woodin has an even weaker fragment of ZFC (which he
calls ZFC$^{*}$) playing the role of $T_{0}$. Today we may as well
let $T_{0}$ be ZFC and $T_{1}$ be ZFC plus the existence of a proper
class of strongly inaccessible cardinals. 
%and $T_{2}$ be ZFC plus the existence of a proper class of Woodin cardinals. 
>From now on we will just use the terms $T_{0}$ and $T_{1}$.

%The proof that these fragments are sufficient as claimed is left as
%an exercise for the reader.

\paragraph{} Our basic construction is the generic ultrapower. We now extend to iterated ultrapowers.
Suppose we have $(M_0,I_0)$, $G_0 \sse (\P({\omega_1})/I_0)^{M_0}$,
$j_0:(M_0,I_0) \to \Ult(M_0,G_0)$, all as before; let $M_1 =
\Ult(M_0,G_0)$, $I_1 = j_0(I_0)$.
% assume
%for now that this is the case at each stage.)
Now we can take the generic ultrapower of $M_1$ by $I_1$, and
iterate. At limit stages, we have a directed system of elementary
embeddings, so can just take the direct limit, so we can keep going
up to length ${\omega_1}$. (No further, as if we force again there,
we collapse ${\omega_1}^V$, so are back to countable length!) Note
that the final model of the iteration, $M_{\omega_1}$, is an element
of  $H(\omega_2)$.

\begin{definition} An \emph{iteration} of $(M,I)$ (as above; $M$ countable) of length $\gamma$ consists of $M_\alpha$, $I_\alpha$ ($\alpha \leq \gamma$), $G_\eta$ ($\eta < \gamma$), and $j_{\alpha,\beta}$ ($\alpha \leq \beta \leq \gamma$), satisfying

 \begin{itemize}
 \item $M_0 = M$, $I_0 = I$
 \item $G_\eta$ is $M_\eta$-generic for $(P({\omega_1})/I_\eta)^{M_\eta}$
 \item $j_{\eta,\eta+1}$ is the canonical embedding of $M_\eta$ into $\Ult(M_\eta,G_\eta) = M_{\eta+1}$
 \item $j_{\alpha,\beta}: M_\alpha \to M_\beta$ are a commuting family of elementary embeddings
 \item $I_\beta = j_{0,\beta}(I)$
 \item For limit $\beta$, $M_\beta$ is the direct limit of
 $\{M_\alpha\ |\ \alpha < \beta\}$ under the embeddings $j_{\alpha,\eta}$ $(\ \alpha \leq \eta < \beta)$.
\end{itemize}
\end{definition}

In practice, we almost always have $\gamma = {\omega_1}^N$ for some
larger $N \ess M$. We will generally write $\la M_\alpha, I_\alpha,
G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma,\ \eta <
\gamma \ra$ for the iteration, or just ``$j$ is an iteration'' to
mean that $j$ is the $j_{0,\gamma}$ of an iteration, with $j(M)$ for
$M_\gamma$. (As we will see, in some circumstances, if we know
$M_0$, $M_\gamma$, $j_{0,\gamma}$, we can (with slight assumptions)
recover the full iteration.) We say that the $M_\alpha$'s are
\emph{iterates} of $(M,I)$; $(M,I)$ is $\emph{iterable}$ if all
iterates are well-founded; and $(M,I)$ is an \emph{iterable pair} if
$M$ is a countable transitive model of $T_{0}$, $I$ a normal ideal
on $\P({\omega_1})$ in $M$, and $(M,I)$ is iterable.

If $M$ is well-founded and $M \believes ``I$ is precipitous," then certainly $(M,I)$ is
finitely iterable (i.e., its finite-length iterations produce
wellfounded models); and in fact, we will show that in this case
$(M,I)$ is iterable to any $\alpha \in \Ord^M$.

The proof of the following lemma is left an exercise (the proof is
by induction on the length of the iteration). In a typical
application, $M$ is $H(\kappa)^{N}$ for some suitable $\kappa$.

\begin{lemma}\label{lemma4} Suppose that $M \in N$ are models of $T_{0}$, $M$
is closed under ${\omega_1}$-sequences from $N$,
%(e.g.\ $M$ is some $H(\kappa)^N$),
and $\P(\P({\omega_1}))^M = \P(\P({\omega_1}))^N$. Let $I \in M$ be
an $M$-normal ideal on ${\omega_1}^M$. Then the following hold.
\begin{itemize}
\item for each iteration $\la M_\alpha,
I_\alpha, G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq
\gamma,\ \eta < \gamma \ra$ of $(M,I)$ there
is a unique iteration $\la N_\alpha, I_\alpha, G_\eta,
j^*_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma,\ \eta < \gamma
\ra$ of $(N,I)$ such that $\A \beta \leq \gamma$, $j^*_{0,\beta}(M)
= M_\beta$, $M_\beta$ is closed under ${\omega_1}$-sequences from
$N_\beta$, $\P(\P({\omega_1}))^{M_\beta} =
\P(\P({\omega_1}))^{N_\beta}$, and $j^*_{\alpha,\beta}
\rest_{M_\alpha} = j_{\alpha,\beta}$.
\item for each iteration $\la N_\alpha, I_\alpha, G_\eta,
j^*_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma,\ \eta < \gamma
\ra$ of $(N,I)$ there
is a unique iteration $\la M_\alpha,
I_\alpha, G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq
\gamma,\ \eta < \gamma \ra$ of $(M,I)$ such that $\A \beta \leq \gamma$, $j^*_{0,\beta}(M)
= M_\beta$, $M_\beta$ is closed under ${\omega_1}$-sequences from
$N_\beta$, $\P(\P({\omega_1}))^{M_\beta} =
\P(\P({\omega_1}))^{N_\beta}$, and $j^*_{\alpha,\beta}
\rest_{M_\alpha} = j_{\alpha,\beta}$.
\end{itemize}
\end{lemma}

 %\begin{proof} By induction; details left as exercise. \end{proof}

\begin{corollary} In the context of Lemma \ref{lemma4},
if $(M,I)$ has an ill-founded iterate by an iteration of length
$\alpha$, then so does $(N,I)$.
\end{corollary}

Lemma \ref{leastcon} below then shows that $(M,I)$ is iterable if
$N$ contains $\omega_{1}$ (recall that iterations can have length
at most $\omega_{1}$, and note that an illfounded iteration of length
$\omega_{1}$ must be illfounded at some countable stage).

First we fix a coding of elements of $H(\omega_1)$ by reals. Fix a
recursive bijection $\pi: \omega \times \omega \to \omega$, and say
$X \sse \omega$ \emph{codes} $a \in H(\omega_1)$ if
$$(\tc(\{a\}),\in) \iso (\omega, \{(i,j)\ |\ \pi(i,j) \in X\}),$$
where $tc(b)$ denotes the transitive closure of $b$. Then $\in$ and
$=$ are $\Sigma_1^1$ (as permutations of $\omega$ induce different
codes for the same object).

%Now ``$(M,I)$ is iterable'' is $\boldPi^1_2$ (``if the final model
%of an iteration has an infinite descending sequence, then so do the
%indices of the iteration''). But an iterable $(N,I)$ is
%$\boldPi^1_2$-correct.


\begin{lemma}\label{leastcon}
Suppose that $N$ is a transitive model of $T_{1}$, $\gamma \in
\Ord^N$, and $I$ is a normal precipitous ideal on ${\omega_1}^N$ in
$N$. Then any iterate of $(N,I)$ by an iteration of length $\gamma$
is well-founded.

\begin{proof} It suffices to prove that iterations of the form $(H(\kappa)^N,I)$ produce wellfounded
models for all $\kappa \in N$ such that $H(\kappa)^N \believes
T_{0}$; for if any iterate of $N$ is ill-founded, then some ordinal
in $N$ is large enough to witness this (i.e.\ $\sup (\rge (f))$,
where $f$ witnesses ill-foundedness) and by assumption (as $N
\believes T_1$) this is contained in some $H(\kappa)^N$ that models
$T_{0}$.

Let $(\bar{\gamma}, \bar{\kappa}, \bar{\eta})$ be the
lexicographically minimal triple $(\gamma, \kappa, \eta)$ satisfying
(with $N$) the formula $\varphi(N, \gamma, \kappa,
\eta)$ defined by ``$H(\kappa)^{N} \believes T_{0}$ and there
exists an iteration of $(H(\kappa)^{N}, I)$ of length $\gamma$ which
is ill-founded below the image of $\eta$''.

Using our fixed coding of elements of $H(\omega_{1})$ by reals there
is a $\Sigma^{1}_{1}$ formula $\varphi'(x,y,z)$  saying ``$x$ codes
a model of $T_{0}$ and a normal ideal in the model on the
$\omega_{1}$ of the model and there exists an iteration of this pair
whose length is coded by $y$ and which is illfounded below the image
of the element of this model coded by $z$."

For all cardinals $\kappa, \rho \in N$ and all ordinals $\gamma,
\eta \in N$, if $\rho \in N$ is larger than $|H(\kappa)|^N$,
$|\eta|^N$ and $|\gamma|^N$, then there exist reals coding
$H(\kappa)^N$, $\eta$, and $\gamma$ in any forcing extension of $N$
by $\Coll(\omega, \rho)$. Such an extension is correct about whether
these reals satisfy $\varphi'$.
%$\varphi$ is logically equivalent to some $\boldSigma_1^1$ formula
%in $N^{\Coll(\omega,\rho)}$.
However, this is a homogeneous forcing extension of $N$; so there is
a formula $\psi(\gamma,\kappa,\eta)$ saying that in every forcing
extension in which $H(\kappa)$ (of the ground model), $\eta$ and
$\gamma$ are all countable there exist reals coding $H(\kappa)$,
$\eta$ and $\gamma$ which satisfy $\varphi'$. It follows that
 that $N \believes \psi (\gamma,\kappa,\eta)$ if and only if
$\varphi(N, \gamma,\kappa,\eta)$ holds, and furthermore, for all
well-founded iterates $N^*$ of $N$, and all $\gamma, \kappa, \eta
\in N^*$, $N^* \believes \psi (\gamma,\kappa,\eta) $ if and only if
$\varphi(N^{*}, \gamma,\kappa,\eta)$ holds.

Since $I$ is precipitous in $N$, $\bar{\gamma}$ is a limit ordinal,
and clearly $\bar{\eta}$ is a limit ordinal as well. Fix an
iteration $\langle M_{\alpha}, G_{\beta}, j_{\alpha\delta} : \beta <
\alpha \leq \delta \leq \bar{\gamma} \rangle$ of
$(H(\bar{\kappa})^{N}, I)$ such that $j_{0\bar{\gamma}}(\bar{\eta})$
is not wellfounded, and let $\langle N_{\alpha}, G_{\beta},
j'_{\alpha\delta} : \beta < \alpha \leq \delta \leq \bar{\gamma}
\rangle$ be the corresponding iteration of $N$ as in Lemma
\ref{lemma4}. By the minimality of $\bar{\gamma}$ we have that
$N_{\alpha}$ is wellfounded for all $\alpha < \bar{\gamma}$. Since
$M_{\bar{\gamma}}$ is the direct limit of the iteration leading up
to it, we may fix $\gamma^{*} < \bar{\gamma}$ and $\eta^{*} <
j_{0\gamma^{*}}(\bar{\eta})$ such that
$j_{\gamma^{*}\bar{\gamma}}(\eta^{*})$ is not wellfounded. By Lemma
\ref{lemma4}, $j'_{\gamma^{*}, \bar{\gamma}}(\eta^{*}) =
j_{\gamma^{*}, \bar{\gamma}}(\eta^{*})$ and $j'_{\gamma^{*},
\bar{\gamma}}(\bar{\eta}) = j_{\gamma^{*},
\bar{\gamma}}(\bar{\eta})$.

\vspace{\baselineskip}


\begin{center}
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\begin{picture}(24,9)
\put(2,2){\line(0,1){8}} \put(12,2){\line(0,1){8}}
\put(22,2){\line(0,1){8}} \put(1,0){$M_{0}$}
\put(11,0){$M_{\gamma^{*}}$} \put(21,0){$M_{\bar{\gamma}}$}
\put(1.5,4.5){\_\_} \put(11.5,5.5){\_\_}
\put(2,4.5){\vector(4,1){20}} \put(12,5.5){\vector(4,1){10}}
\put(0,3.5){$\bar{\eta}$} \put(10,4.5){$\eta^{*}$}
\end{picture}
\end{center}

%Since $M_{\bar{\gamma}}$ is illfounded below $j_{0,\bar{\gamma}}$,
%there exists an $\eta' < j_{0,\bar{\gamma}} (\bar{\eta})$ such that
%$M_{\bar{\gamma}}$ is ill-founded below $\eta'$. \query

%Note that $\bar{\gamma}$, $\bar{\eta}$ are limits ($\bar{\gamma}$ by
%the fact that precipitousness of an ideal is a first-order
%property); so there exist $\gamma^* < \bar{\gamma}$, $\eta^* \in
%N_{\bar{\gamma}}$ s.t.\ $\eta' = j_{\gamma^*,\bar{\gamma}}
%(\eta^*)$.

But now, $N_{\gamma^*} \believes \psi(\bar{\gamma}-\gamma^*,
j_{0,\gamma^*}(\bar{\kappa}), \eta^*)$, $\bar{\gamma} - \gamma^* \leq
\bar{\gamma}$, and $\eta^* < j_{0,\gamma^*}(\bar{\eta})$,
contradicting minimality of
$(j_{0,\gamma^*}(\bar{\gamma}),j_{0,\gamma^*}((\bar{\kappa}),j_{0,\gamma^*}(\bar{\eta}))$
in $N_{\gamma^*}$.
\end{proof}
\end{lemma}


We note that ZFC does not imply the existence of iterable pairs. However, by Lemma \ref{leastcon}, if there exist a normal, precipitous
ideal $J$ on $\omega_{1}$, and a measurable cardinal $\kappa$ with a $\kappa$-complete ultrafilter $\mu$,
then there exist iterable
pairs. The main point here is that if $\theta > \kappa$ is a regular cardinal and
$X$ is a countable elementary submodel of $H(\theta)$ with $\kappa, J \in X$, then
$X$ can be end-extended below $\kappa$ by taking $\gamma$ to be any member of
$\bigcap (X \cap \mu)$, and letting $X[\gamma]$ be the set of values $f(\gamma)$ for
all functions $f$ in $X$ with domain $\kappa$. Applying this fact $\omega_{1}$ many times,
we get that the transitive collapse $M$ of $X \cap V_{\kappa}$ is a countable model which
is a rank initial segment of a model containing $\omega_{1}$. Letting $I$ be the image of $J$
under the transitive collapse, then, $(M,I)$ is an iterable pair. This is a special case of the
proof of Lemma \ref{extend}, and a key point in Woodin's proof (which appears in Chapter 3 of \cite{Woo99})
that if there exists a measurable cardinal and the nonstationary ideal on $\omega_{1}$ is saturated, then CH fails.

If there is a precipitous ideal on $\omega_1$, then sharps exist for
subsets of ${\omega_1}$, and a countable iterable model will be
correct about these sharps. We will work around this today to avoid
having to talk about sharps.
%The significance of the following point
%is essentially notational.

\begin{lemma}\label{o1correct} If $(M,I)$ is an iterable
pair and $A$ is an element of $\mathcal{P}(\omega_{1})^{M}$, then
$(\omega_{1}^{L[A]})^{M} = \omega_{1}^{L[A]}$.
\end{lemma}

\begin{proof} Let $\la M_\alpha,
I_\alpha, G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq
\omega_{1},\ \eta < \omega_{1} \ra$ be an iteration of $(M,I)$. The
ordinals of $M$ and $M_1$ are the same, so $L[A]^{M} =
L[A]^{M_{1}}$. The critical point of $j_{1\omega_{1}}$ is greater
than $\omega_{1}^{M}$, and thus greater than the $\omega_{1}$ of
$L[A]^{M}$. The restriction of $j_{1\omega_{1}}$ to $L[A]^{M}$ embeds
$L[A]^{M}$ elementarily into $L[A]^{M_{\omega_{1}}}$, which means
that $L[A]^{M}$ and $L[A]^{M_{\omega_{1}}}$ have the same
$\omega_{1}$. Since $\omega_{1} \subset M_{\omega_{1}}$,
$(\omega_{1}^{L[A]})^{M_{\omega_{1}}} = \omega_{1}^{L[A]}$.
\end{proof}

%\begin{lemma} Suppose that $(M,I)$ is an iterable pair, $X \sse \omega$, $X \in M$.
%Then $\E$ some elementary embedding $j: M \to M^*$,
%$\crit(j) = {\omega_1}^M$, $\omega_1 \in M^*$;
%indeed, $\A A \in \P({\omega_1})^{M}$, ${\omega_1}^{L[A]} = ({\omega_1}^{L[A]})^M$. \Query{\ not quite sure if we transcribed this right - I can't quite follow it in my notes.}

%\begin{proof} ${\omega_1}^{L[X]}$ countable in $M$. $L[X]^{M} = L[X]^{M^*}$, so
%$M \believes {\omega_1}^{L[X]} < \omega_1$. So $M, M^* \believes {\omega_1}^{L[X]} < {\omega_1}$;
%but as $\omega_1 \sse M^*$, $({\omega_1}^{L[X]}) = ({\omega_1}^{L[X]})^{M^*} = ({\omega_1}^{L[X]})^M$.

% Thus for $A \in \P({\omega_1})^{M}$, we have $A = j_0(A) \cap {\omega_1}^{M}$, $A \in M^*$, $A$
% countable in $M^*$. $(M_1,I_1)$ is an iterable pair, so $({\omega_1}^{L[A]})^{M_0} = {\omega_1}^{L[A]}$.
% But $\Ord^{M_0} = \Ord^{M_1}$, so $({\omega_1}^{L[A]})^{M_0} = ({\omega_1}^{L[A]})^{M_1}$.
%\end{proof}
%\end{lemma}

%This fact is included only to avoid talking about sharps.

\paragraph{} Now we can define $\Pmax$.

\begin{definition} The partial order $\Pmax$ is the set of
pairs $\la (M,I),a \ra$ such that
\begin{enumerate}
\item $M$ is a countable transitive model of $T_{0}$ + $\MA_{\aleph_1}$
\item $(M,I)$ is an iterable pair
\item $a \in \P(\omega_1)^M$ and $\E x \in \P(\omega)^M$ such that $\omega_1^{L[x,a]} = \omega_1^M$
\end{enumerate}
ordered by: $p < q$ (where $p = \la (M,I), a \ra$, $q = \la (N,J),b \ra$) if there is some iteration $j: (N,J) \to (N^*,J^*)$ such that
\begin{enumerate}
\item $j \in M$
\item $j(b) = a$
\item $J^* = N^* \cap I$ ( and hence $j(\omega_1^N) = \omega_1^M$)
\item $q \in H(\omega_1)^M$
\end{enumerate}
\end{definition}

Note that since $j \in M$ in definition of the $\Pmax$ order above, $N$ and $N^{*}$ are both in $M$ as well.

\begin{definition} We say that $(M,I)$ is a $\Pmax$ \emph{precondition} if there
exists an $a$ such that $\la(M,I),a\ra \in \Pmax$, or equivalently
just if $(M,I)$ satisfies conditions 1 and 2 in the definition of
$\Pmax$ conditions above.
\end{definition}

Suppose that $(M,I)$ is an iterable pair, and $j \colon (M,I) \to (M', I')$ is an iteration.
Then for any $A \in \mathcal{P}(\omega_{1})^{M}$ which is bounded in $\omega_{1}^{M}$, $j(A) = A$.
By Lemma \ref{o1correct}, it follows then that $\omega_{1}^{L[A]} < \omega_{1}^{M}$, since $j(\omega_{1}^{M})
> \omega_{1}^{M}$ if $j$ is nontrivial.
Therefore, the set $a$ from a $\Pmax$ condition
$\langle (M,I), a \rangle$ must always be unbounded in $\omega_1^M$
to make ${\omega_1}^{L[x,a]} = {\omega_1}^M$ possible.

If $p_0 < p_1 < p_2$ ($p_i = \la (M_i, I_i), a_i \ra $), and these
are witnessed by $j_{1,0}$, $j_{2,1}$, then $p_0 < p_2$ is witnessed
by $j_{1,0}(j_{2,1})$: $j_{1,0} \in H(\omega_2)^{M_0}$, $j_{2,1} \in
H(\omega_2)^{M_1}$; $j_{2,1}$ is an iteration of $(M_2,I_2)$, and
$j_{1,0}((M_2,I_2)) = (M_2,I_2)$.






%For if we fix a coding elements of $H(\omega_1)$ by reals, i.e.\ fix
%a recursive bijection $\pi: \omega \times \omega \to \omega$, and
%say $X \sse \omega$ codes $a \in H(\omega_1)$ if $(\tc(\{a\}),\in)
%\iso (\omega, \{(i,j)\ |\ \pi(i,j) \in X\}$. Then $\in$ and $=$ are
%$\Sigma_1^1$ (as can permute $\omega$, etc.).

Under our fixed coding, ``$(M,I)$ is iterable'' is $\Pi^1_2$ in
a code for $(M,I)$ : roughly, ``for anything satisfying the first-order properties of
being an iteration, either there is no infinite descending sequence in the ordinals of the final model,
or there is an infinite descending sequence in the indices of the iteration."
Since iterable models embed elementarily into models
containing $\omega_{1}$, they are $\boldPi^1_2$-correct. It follows
that ``$(M,I)$ is iterable" is absolute to iterable models
containing a code for $(M,I)$.

So now we see that $\Pmax \in L(\R)$ --- all constructions involved
are nicely codable.

\section{First properties of $\Pmax$}


The requirement that the models in $\Pmax$ conditions satisfy
MA$_{\aleph_{1}}$ is used for a particular consequence of
MA$_{\aleph_{1}}$ known as \emph{almost disjoint coding}
\cite{JenSo}. That is, it follows from MA$_{\aleph_{1}}$ that if $Z
= \{ z_{\alpha} : \alpha < \omega_{1}\}$ is a collection of infinite
subsets of $\omega$ whose pairwise intersections are finite (i.e.,
$Z$ is an \emph{almost disjoint family}), then for each $B \subseteq
\omega_{1}$ there exists a $y \subseteq \omega$ such that for all
$\alpha < \omega_{1}$, $\alpha \in B$ if and only if $y \cap
z_{\alpha}$ is infinite. This is used to show that if $\langle
(M,I), a \rangle$ is a $\Pmax$ condition, then any iteration of
$(M,I)$ is uniquely determined by the image of $a$ (Lemma
\ref{uniqlem} below), so the order on each comparable pair of
conditions is witnessed by a unique iteration.

%The use of Martin's Axiom is to ensure the uniqueness of $j$
%witnessing each $p \leq q$ (see Lemma \ref{uniqlem}; we will see
%that this is important in later contexts.

\begin{lemma}\label{uniqlem} Let $\langle (M,I), a \rangle$ be
a $\Pmax$ condition and let $A$ be a subset of $\omega_{1}$. Then
there is at most one iteration of $(M,I)$ for which $A$ is the image
of $a$.
%Furthermore, this iteration is in $L[\langle(M,I), a\rangle,
%A]$, if it exists.
\end{lemma}

\begin{proof}
Fix a real $x$ in $M$ such that $\omega_{1}^{M} = \omega_{1}^{L[a,
x]}$, and let $Z = \langle z_{\alpha} : \alpha < \omega_{1}^{M}
\rangle$ be the almost disjoint family defined recursively from the
constructibility order in $L[a, x]$ on $\mathcal{P}(\omega)^{L[a,
x]}$ (using $a$ and $x$ as parameters) by letting $\langle z_{i} : i < \omega \rangle$ be the constructibly least
%(in $L[a,x]$, using $a$ and $x$ as parameters)
partition of $\omega$ into infinite pieces, and, for
each $\alpha \in [\omega, \omega_{1}^{M})$, letting $z_{\alpha}$ be the constructibly
least infinite $z \subset \omega$ almost disjoint from each
$z_{\beta}$ $(\beta < \alpha)$. Suppose that
$$\mathcal{I} = \langle M_{\alpha}, G_{\beta}, j_{\alpha\delta} :
\beta < \alpha \leq \delta \leq \gamma \rangle$$ and
$$\mathcal{I}' = \langle M'_{\alpha}, G'_{\beta}, j'_{\alpha\delta}
: \beta < \alpha \leq \delta \leq \gamma' \rangle$$ are two
iterations of $(M,I)$ such that $j_{0\gamma}(a) = A
 = j'_{0\gamma'}(a)$. Then $$j_{0\gamma}(\omega_{1}^{M}) = \omega_{1}^{L[x,A]}
 = j_{0\gamma}(\omega_{1}^{M'})$$ and $j_{0\gamma}(Z) = j'_{0\gamma'}(Z)$
(this uses Lemma \ref{o1correct} to see that the constructibility
order on reals in $L[A,x]$ is computed correctly in $M_{\gamma}$ and
$M'_{\gamma'}$). Let $\langle z_{\alpha} : \alpha <
j_{0\gamma}(\omega_{1}^{M}) \rangle$ enumerate $j_{0\gamma}(Z)$.

Without loss of generality, $\gamma \leq \gamma'$. We show by
induction on $\alpha < \gamma$ that, for each such $\alpha$,
$G_{\alpha} = G'_{\alpha}$. This will suffice. Fix $\alpha$ and
suppose that
$$\{ G_{\beta} : \beta < \alpha \} = \{ G'_{\beta} : \beta < \alpha
\}.$$ Then $M_{\alpha} = M'_{\alpha}$. For each $B \in
\mathcal{P}(\omega_{1})^{M_{\alpha}}$, $B \in G_{\alpha}$ if and
only if $\omega_{1}^{M_{\alpha}} \in j_{\alpha(\alpha + 1)}(B)$, and
$B \in G'_{\alpha}$ if and only if $\omega_{1}^{M_{\alpha}} \in
j'_{\alpha(\alpha + 1)}(B)$. Applying almost disjoint coding, fix $x
\in \mathcal{P}(\omega)^{M_{\alpha}}$ such that for all $\eta <
\omega_{1}^{M_{\alpha}}$, $\eta \in B$ if and only if $x \cap
z_{\eta}$ is infinite. Then $B \in G_{\alpha}$ if and only if $x
\cap z_{\omega_{1}^{M_{\alpha}}}$ is infinite if and only if $B \in
G'_{\alpha}$.
%For the last part of the lemma, note that the argument just given
%gives a definition for each $G_{\alpha}$ in terms of $A$, $x$ and
%the iteration up to $\alpha$.
\end{proof}

%\begin{lemma}\label{uniqlem} Suppose $j$ is an iteration of a condition $\la (M,I),a \ra$;
%let $A = j(a)$. Then $j$ is the unique iteration of $(M,I)$ sending $a$ to $A$.

%\begin{proof} By induction on the length of $j$. It's clear at limits; but in the successor case, $j$ is the unique iteration satisfying the following definition:

% Fix $x \in \P(\omega)$ s.t.\ $\omega_1^{L[x,a]} = {\omega_1}^M$. Let $Z = \{z_\alpha\ |\ \alpha < \omega_1\}$ be the ``least almost disjoint family in $L[x,a]$'' --- i.e.\ $z_\alpha$ is the $L[x,a]$-constructibly-least infinite $z \sse \omega$ s.t.\ $z \cap z_\beta$ is finite for $\beta < \alpha$ and $\E$ some infinite $y \sse \omega$ s.t.\ $\A \beta < \alpha$ $y \cap (z \cup z_\beta)$ is finite.

% But recall: $MA_{\aleph_1}$ $\Imp$ almost disjoint coding of $\omega_1$, i.e.\ for any almost disjoint family $\{w_\alpha\ |\ \alpha < \omega_1\}$, any $B \sse \omega_1$, there is $e \sse \omega$ s.t.\ $\A \alpha < \omega_1$ $\Imp$ $e \cap \omega_\alpha$ infinite.

% So here, for all $B \in \P(\omega_1)^M$, $\E e_B$ s.t.\ $\A \alpha < \omega_1^M$ $\alpha \in B \Iff z_\alpha \cap e_B$. Also, $B \in G_0 \Iff \omega_1^M \in j_{(0,1)}(B)$, so $B \in G_0$ $\Iff$ $\omega_1^M \in j_{(0,1)}(B)$ $\Iff$ $z_{\omega_1} \cap e_B$ infinite.

% So recap: let $j(z) = \{z_\alpha\ |\ \alpha < j({\omega_1}^M)\}$. $j(z)$ is in $L[(M,I),A]$ and is uniquely determined by $j(a)$.

% Thus we are done: for suppos $\la M_\alpha, I_\alpha, G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma,\ \eta < \gamma \ra$ and $\la M^*_\alpha, I^*_\alpha, G^*_\eta, j^*_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma,\ \eta < \gamma^* \ra$ are iterations of $(M,I)$ satisfying the above.

% If they agree below some $\delta$ limit then they certainly agree at $\delta$, and at successors: suppose $\la M_\alpha, \ldots \ra \rest_{\gamma'} = \la M^*_\alpha, \ldots \ra \rest_{\gamma'}$ (i.e.\ $M_\alpha, I_\alpha$ for $\alpha \leq \gamma'$, $G_\eta$ for $\eta < \gamma'$); we'll show $G_{\gamma'} = G^*_{\gamma'}$.

% We have $M_{\gamma'} = M^*_{\gamma'}$. Fix $B \in \P(\omega_1)^{M_\gamma}$, $e_B$ s.t.\ $\A \alpha < {\omega_1}^{M_{\gamma'}}$, $\alpha \in B \Iff z_\alpha \cap e_B$ infinite.

% Then $B \in G_{\gamma'} \Iff (z_{{\omega_1}^M})^{M_\gamma}$ $\cap e_B$ infinite, and similarly $B \in G^*_{\gamma'} \Iff (z_{{\omega_1}^M})^{M^*_{\gamma'}} \cap e_B$ infinite.

% But $j_{0,\gamma}(a) = j^*_{0,\gamma^*}(a)$, so $j_{0,\gamma'}(Z) = j^*_{0,\gamma^*}(Z)$, so the R.H.S.\ in the previous paragraph are equivalent, so $G_{\gamma'} = G^*_{\gamma'}$ as desired.

%\end{proof}
%\end{lemma}

\begin{lemma}\label{eleven} ($T_{0}$) Suppose that $(M,I)$ is an iterable pair, and
$J$ is a normal ideal on $\omega_1$. Then there exists an iteration
$j\colon (M,I) \to (M^*,I^*)$ of length $\omega_1$ such that $I^* =
M^* \cap J$.

\begin{proof} Note that $I^* \sse M^* \cap J$ holds for any such $\omega_1$-length iteration.
To see this, first note that the critical sequence of an iteration of length $\omega_1$ is a
club.
%whose intersection with each member of $I^{*}$ is a bounded subset of $\omega_{1}$.
Every element $B$ of $I^*$ is $j_{\alpha,\omega_1}(b)$ for
some $\alpha < \omega_1$ and $b \in I_\alpha$. Then for all $\beta
\in [\alpha,\omega_1)$, $j_{\alpha,\beta}(b) \nin G_\beta$, so
${\omega_1}^{M_\beta} \nin j_{\alpha,\omega_1}(b) = B$; thus $B \in
\NS_{\omega_1}$, but $J$ is normal, so $NS_{\omega_1} \sse J$.

Conversely, for $\ess$: as $J$ is normal, we may let $\la E_i^\alpha\
|\ \alpha < \omega_1, i < \omega \ra$ be a partition of $\omega_1$
into $J$-positive pieces. Now, as we construct an iteration, let
$\{e_i^\alpha\ |\ i < \omega\}$ enumerate $\P(\omega_1)^{M_{\alpha}}
\setminus I_\alpha$, and build each $G_\beta$ in such a way that
if ${\omega_1}^{N_\beta} \in E_i^\alpha$ for some $\alpha \leq
\beta$ and $i < \omega$, then $j_{\alpha,\beta}(e_i^\alpha)$ is in
$G_\beta$.

%(No other extra restrictions.)

%\piccie

Now, for all $B \in \P(\omega_1)^{M_{\alpha}} \setminus
I_{\omega_1}$, $\E \alpha < \omega_1$, $i < \omega$ such that $B =
j_{\alpha,\omega_1}(e_i^\alpha)$ and for all $\beta \in
[\alpha,\omega_1)$, ${\omega_1}^{M_\beta} \in E_i^\alpha$ $\Imp$
${\omega_1}^{M_\beta} \in j_{0,\beta+1}(e_i^\alpha)$ $\Imp$
${\omega_1}^{M_\beta} \in B$.

So in particular, we have a club $C \sse \omega_1$ such that $C \cap
E_i^\alpha \sse B$, so $B \nin J$.\end{proof}
\end{lemma}

We may consider this as an \emph{iteration game} $G((M,I),J,B)$: two players collaborate on building an iteration of $(M,I)$, and play is as follows at each round  $\alpha$:
\begin{itemize}
\item if $\alpha \nin B$, player I does nothing, and player II chooses $G_\alpha$;
\item if $\alpha \in B$, player I specifies some element for $G_\alpha$, and player II must choose some $G_\alpha$ containing it.
\end{itemize}

Player I wins if $I_{\omega_1} = M_{\omega_1} \cap J$. The above
proof shows that player I has a winning strategy iff $B \nin J$.
(More precisely, it shows $\Pmi$; $\Imp$ is because if $B \in J$ then II may
choose some $I$-positive set and keep its images out of every
$G_{\alpha}$.)

The following lemma shows that $\Pmax$ satisfies a homogeneity property strong enough to imply
that the theory of the generic extension can be computed in the ground model. Since the existence of 
a proper class of Woodin cardinals implies that the theory of $L(\mathbb{R})$ is generically absolute,
it implies that the theory of the $\Pmax$ extension of $L(\mathbb{R})$ is generically absolute as well.

\begin{lemma} Suppose for each $x \in H(\omega_1)$ there exists a $\Pmax$
precondition $(M,I)$ such that $x \in M$. Then $\A  p_0,p_1 \in
\Pmax$, $\E q_0,q_1 \in \Pmax$ such that each $q_i \leq p_i$, and
$\Pmax \rest {q_0} \iso \Pmax \rest {q_1}$.

\begin{proof} Take $p_i = \la(M_i,I_i),a_i\ra$. Then let $(N,J)$ be a $\Pmax$ precondition such that
$p_0,p_1 \in H(\omega_1)^N$. Now take $j_i\colon(M_i,I_i) \to
(M_i^*,I_i^*)$ to be iterations in $N$ such that $I_i^* = M_i^* \cap
J$ (we may do so, by the previous theorem applied in $N$).

Now set $q_i = \la(N,J),j_i(a_i)\ra \in \Pmax$. Certainly these
satisfy $q_i < p_i$ as desired. (To see that these $q_i$ are indeed
conditions, note that the witnessing $x_i$ for $p_i$ (i.e., the $x
\in \mathcal{P}(\omega)^{M_{i}}$ such that $\omega_{1}^{L[x, a_{i}]}
= \omega_{1}^{M_{i}}$) still works for
$q_i$.)
But now $\Pmax \rest {q_0} \iso \Pmax \rest {q_1}$, for given any $r_0 = \la(N',I'),b \ra < q_0$, there is unique $j:(N,J) \to (N^*,J^*)$ witnessing this (and we have $b = j(j_0(a_0))$); now take $r_0$ to $r_1 := \la(N',J'),j(j_1(a_1))\ra < q_{1}$, also witnessed by $j$.
\end{proof}
\end{lemma}

Given $\gamma \in [\omega_1,\omega_2)$, a \emph{canonical function}
for $\gamma$ is a function $f\colon \omega_1 \to \omega_1$ such that
for some (equivalently, every) bijection $\pi:\omega_1 \to \gamma$,
$\{\alpha < \omega_1\ |\ \ot(\pi[\alpha]) = f(\alpha)\}$ contains a
club. In a normal ultrapower context, we then have: $f\colon
\omega_1 \to \Ord$, $[f]_\Ult = j(f)(\omega_1) = \ot(\pi[\omega_1])
= \gamma$.

Suppose that $\la M_\alpha, I_\alpha, G_\eta, j_{\alpha,\beta}\ |\
\alpha \leq \beta \leq \omega_{1},\ \eta < \omega_{1} \ra$ is an
iteration of length $\omega_1$ of some $(M,I)$, and let $\pi:
\omega_1 \to \Ord^{M_{\omega_1}}$ be a bijection; then $\pi$ induces
a canonical function $g$. For club-many $\alpha < \omega_1$,
${\omega_1}^{M_\alpha} = \alpha$, and also $\pi[\alpha] =
j_{\alpha,\omega_1}[\Ord^{M_\alpha}]$ (recall that we take direct limits at limit stages of iterations);
so there is a club $C \subset
\omega_{1}$ such that for all $\alpha \in C$, $\ot(\pi[\alpha]) =
\Ord^{M_\alpha}$. If $f \in ({\omega_1}^{\omega_1})^{M_\beta}$, for some $\beta < \alpha$,
then for any $\alpha \in [\beta, \omega_{1})$,
$$j_{\beta,\alpha+1}(f)({\omega_1}^{M_\alpha}) <
{\omega_1}^{M_{\alpha+1}} < \Ord^{M_{\alpha+1}} = \Ord^{M_\alpha},$$
which equals $g(\alpha)$ if $\alpha \in C$.
%So
%let $g$ be the canonical function as$\pi$; then
It follows that for any $f \in
({\omega_1}^{\omega_1})^{M_{\omega_1}}$, $g(\alpha)
> f(\alpha)$ for club-many $\alpha$.
We will use this fact to show that $\Pmax$ $\sigma$-closed (this is
an alternate proof avoiding sharps; some of what follows can be done
more easily and in more generality with sharps).

\begin{lemma}\label{oclose} Suppose that for each $x \in H(\omega_1)$ there
exists a $\Pmax$ precondition $(M,I)$ such that $x \in M$. If $p_i
\in \Pmax$ ($i < \omega$) are such that $\A i\ p_{i+1} < p_i$, then
$\E q \in \Pmax$ such that $\A i\ q < p_i$.
\end{lemma}

The proof of Lemma \ref{oclose} involves some new notions. Say that
$p_i = \la (M_i,I_i),a_i \ra$, and for each $j < i < \omega$, let
$k_{i,j}: M_j \to M_j^* \in M_i$ be the unique witness for $p_i <
p_j$. By the uniqueness of witnesses, the $k_{i,j}'s$ commute, so let
$N_i$, $J_i$ be the images of $M_i$, $I_i$ in the limit of the
directed system given by the embeddings $k_{i,j}$. Each $(N_{i}, J_{i})$ is an
iterate of the corresponding $(M_{i}, I_{i})$ by an iteration of length $\sup\{ \omega_{1}^{M_{i}} :
i < \omega\}$, so each $N_{i}$ is wellfounded.
%\Query{\ originally ``...under the directed system''};
Let $b = \bigcup_{i < \omega} a_i$. Then:
\begin{enumerate}
\item\label{one} Each $(N_{i}, J_{i})$ is iterable;
 \item for all $i$, ${\omega_1}^{N_i} = \sup_{j < \omega}
 {\omega_1}^{M_j}$;
 \item $i < j$ $\Imp$ $N_i \in H(\omega_2)^{N_j}$;
 \item for all $i$, $J_i = J_{i+1} \cap N_i$;
 \item\label{five} for each $i$ there exists some iteration $j_i:(M_i,I_i) \to (N_i,J_i)$ in $N_{i+1}$ such that $j_1(a_i) = b$
(and so in $N_{i+1}$, there is a canonical function for $\Ord^{N_i}$
that dominates on a club every member of
$(\omega_1^{\omega_1})^{N_i}$).
\end{enumerate}

We call a sequence $\la (N_i, J_i) : i < \omega \ra$ satisfying
(\ref{one}) to (\ref{five}) above a \emph{$\Pmax$ limit sequence}.
An \emph{$\la (N_i,J_i)\ |\ i < \omega \ra$-normal ultrafilter} is a
filter $G \sse \bigcup_{i} (\P(\omega_1)^{N_i} \setminus J_i)$ such
that for all $i < \omega$, and for all regressive $f \in
({\omega_1}^{\omega_1})^{N_i}$ $\E e \in G$ such that $f$ is
constant on $e$. Then we have $\Ult(\la (N_i, J_{i}) : i < \omega
\ra, G)$, i.e.\ a sequence of models whose $j$th model $\left[ \Ult(\la
(N_i,J_{i}) : i < \omega \ra,G) \right]_j$ is $$ \{ f:
{\omega_1}^{N_0} \to N_j\ |\ f \in \bigcup \{N_i\ |\ i < \omega\}\}/
=_{G}.$$ Now we will iterate this operation.


 \begin{definition} An iteration of $\la (N_i, J_{i}) : i < \omega \ra$ of length $\gamma$ is some
 $$\la \langle (N_i^\alpha, J_i^\alpha) : i < \omega \ra, G_\eta, j_{\alpha,\beta}\ |\ \alpha \leq \beta \leq \gamma, \eta < \gamma \ra$$ in which:
\begin{itemize}
\item each $G_\eta$ is a $\\langle (N^\eta_i, J^{\eta}_{i}) \mid i < \omega \rangle$-normal ultrafilter contained in $\bigcup_i (\P(\omega_1)^{N_i^\eta} \setminus J_i^\eta)$ such that for
each $i< \omega$, $j_{\eta,\eta+1} \rest N_i^\eta \to
N_i^{\eta+1}$ is the induced ultrapower
\item the $j_{\alpha,\beta}$ commute, and for limit $\beta$, $N_i^\beta$ is the direct limit of
$N_i^\alpha$ ($\alpha <\beta$)  under $j_{\alpha,\rho} \rest
N_i^\alpha$ ($\alpha \leq \rho < \beta$).
\end{itemize}
\end{definition}

In an iteration of this form, for each pair $i,\alpha$, there is in
$N_{i+1}^\alpha$ an iteration of $(M_i,I_i)$ of length
${\omega_1}^{N_{0}^\alpha}$, with final model $N_i^\alpha$.
%\Query{\ originally just for $\gamma$, not general $\alpha$}
Since each $(M_{i}, I_{i})$ is iterable, the wellfoundedness of each
$N^{\alpha}_{i}$ will follow from the wellfoundedness of
$\omega_{1}^{N^{\alpha}_{0}}$.

For each $\alpha <\gamma$, $$\omega_1^{N_0^{\alpha + 1}} = \sup \{
\Ord^{N_i^{\alpha}}\ |\ i < \omega\}.$$ To see this, fix for each $i \in \omega$ a canonical
function $g_i \in N^{\alpha}_{i+1}$ for $\Ord^{N_i}$. Then each
$g_i$ dominates on a club every member of
$(\omega_1^{\omega_1})^{N^{\alpha}_i}$.
%represents $\Ord^{N_i}$ in the ultrapower.
The $g_i$ are cofinal under mod-$NS_{\omega_1}$ domination in
$\bigcup_i ({\omega_1}^{\omega_1})^{N^{\alpha}_i}$, and each $g_{i}$
represents an ordinal in $N^{\alpha}_{i+1}$ in this ultrapower,
which shows that $\omega_1^{N_0^{\alpha + 1}} = \sup \{
\Ord^{N_i^{\alpha}}\ |\ i < \omega\}$. For limit $\beta$,
$${\omega_1}^{N_0^\beta} = \sup \{{\omega_1}^{N_0^\alpha}\ |\ \alpha
< \beta\}.$$ It follows that each $N^{\alpha}_{0}$ is wellfounded.

%Finally, for any $\alpha < \gamma$, if $\omega_{1}^{N^{\alpha}_{0}}$
%is wellfounded, then so is each $N^{\alpha}_{i}$, since
%$N^{\alpha}_{i}$ is the final model of an iteration of the iterable
%pair $(M_{i}, I_{i})$ of length $\omega_{1}^{N^{\alpha}_{0}}$.

We have shown the following.

\begin{fact} All iterations of $\Pmax$ pre-limit sequences give well-founded models.
\end{fact}

Again this can be rephrased in terms of games. Let $G_{\omega} (\la
(N_i,J_i)\ |\ i < \omega \ra, I, B)$ be the game of length
$\omega_1$, in which I and II collaborate to build an iteration of
$\la(N_i,J_i) : i < \omega\ra$ of length $\omega_1$, in which at
stage $\alpha$:
\begin{itemize}
\item if $\alpha \in B$, player I chooses $e \in \bigcup_i (\P(\omega_1)^{N_i^\alpha}\setminus J_i^\alpha)$, and
player II chooses $G_\alpha$, a $\la (N_i^\alpha, J^{i}_{\alpha}) \mid i < \omega \rangle$-normal
ultrafilter contained in $ \bigcup_i (\P(\omega_1)^{N_i^\alpha}\setminus
J_i^\alpha)$, with $e \in G_{\alpha}$;
\item if $\alpha \nin B$, player I does nothing, and player II chooses any suitable $G_\alpha$.
\end{itemize}
Player I wins if $\A i$ $j_{0,\omega_1}(J_i) = I \cap
M_i^{\omega_1}$. The argument just given (along with the argument for Lemma \ref{eleven}) shows the following.

%NB some of this is more easily done using sharps.

\begin{lemma} Suppose $\la (N_i,J_i)\ |\ i < \omega \ra$ is a $\Pmax$ pre-limit-sequence,
$I$ is a maximal ideal on $\omega_1$ and $B \sse \omega_1$. Then
Player I has a winning strategy in $G_{\omega} (\la (N_i,J_i)\ |\ i
< \omega \ra, I, B)$ if and only if $B$ is not in $I$.
\end{lemma}

%which again is possible iff $B \nin I$.

We now return to the proof of Lemma \ref{oclose}. We have that the
limit sequence $\langle (N_{i},J_{i}) : i < \omega \rangle$ induced
by the descending sequence $p_{i}$ $(i \in \omega)$ is iterable. Fix a
$\Pmax$ precondition $(M',I')$ such that this sequence is in
$H(\omega_{1})^{M'}$. Apply a winning strategy for player I in $M'$
for $G_{\omega_1}( \la (N_i, J_i)\ |\ i < \omega \ra, I', \omega_1)$
to get an iteration $j$ of $\langle (N_{i}, J_{i}) : i <
\omega\rangle$ of length $\omega_{1}^{M'}$.
%(all these are
%embeddings $j^*_i : (N_i, J_i) \to (N_i^*, J_i)$).
Then for all $i < \omega$,
%$j^*_{i+1} (J_i)$
$j(j_{i})$ witnesses that
$p_i
> \la (M',I'), j(b) \ra$.
%> \la (M',I'), j_0^*(b) \ra$.

%\begin{lemma}[Lemma 9 cont'd] \ldots suppose also that
%$\A x \in H(\omega_1)$ $\E$ some $\Pmax$ precondition $(M',I')$, $x \in M'$;
%then for every descending sequence $\la p_i\ |\ i < \omega \ra$ in $\Pmax$,
%there is some $q \in \Pmax$ such that $\A i$ $q < p_i$.

%\begin{proof} The induced sequence is iterable.
%\end{proof}
%\end{lemma}

Thus $\Pmax$ forcing is $\sigma$-closed, so it does not add any reals;
so $L(\R)^{V^\Pmax} = L(\R)^V$.


\section{Existence of $\Pmax$ conditions}

\begin{definition}Given $A \sse \R$, and an iterable pair $(M,I)$, we say $(N,I)$ is \emph{$A$-iterable}
if $A \cap M \in M$ and for any iteration $j:(M,I) \to (M^*,I^*)$, $j(A \cap M) = A \cap M^*$.
\end{definition}

In this section we will work through a proof of the following
existence theorem for $\Pmax$ conditions.

\begin{lemma}[Main existence lemma] Suppose there are infinitely many Woodin cardinals below some
measurable cardinal, and let $A \in \P(\R) \cap L(\R)$. Then there
exists an $A$-iterable $\Pmax$ precondition $(M,I)$ such that for
every set forcing extension $M^+$ of $M$ and every precipitous ideal
$I^+ \in M^+$ on $\omega_1^{M^+}$, $(M^+,I^+)$ is $A$-iterable.
\end{lemma}

%\begin{proof} First, recall some combinatorics.

We need to introduce towers of measures and homogeneous tree.
See \cite{MarSt89} for a detailed discussion of this material

\begin{definition} Given $Z \neq \phi$, a \emph{tower of measures} on $Z$ is
a sequence $\la \mu_i\ |\ i < \omega \ra$ such that each $\mu_i \sse
\P(Z^i)$ is an ultrafilter, and for all $k<i<j$ and all $A \in \mu_i$, we
have $\{b \in Z^j\ |\ b \rest i \in A \} \in \mu_j$ and $\{b \rest
k\ |\ b \in A\} \in \mu_k$.

Such a tower is \emph{countably complete} if whenever $\la A_i\ |\ i
< \omega \ra$ is such that eaach $A_i \in \mu_i$, there is $a \in
Z^\omega$ such that $\A i$ $a \rest i \in A_i$. \end{definition}

We note briefly that countable completeness is equivalent to: the
direct limit of $\Ult(V,\mu_i)$ is well-founded.

\begin{definition} A \emph{tree on $\omega \times Z$} is a set $T \sse (\omega \times Z)^{<\omega}$
such that for all $i < \omega, t \in T$ we have $t \rest i \in T$. The
projection of $T$ is $p[T] := \{ y \in \omega^\omega\ |\ \E c \in
Z^\omega\ \A i < \omega\ (y\rest i,c \rest i) \in T \}$.

%Such a tree is \emph{weakly $\kappa$-homogeneous} ( for $\kappa$ a
%cardinal) if there exist $\kappa$-complete ultrafilters $\mu_{a}
%\sse \P(Z^{|a|})$ for each $a \in \omega^{\less\omega}$ such that,
%for each such $a$, $\{c \in Z^{|a|}\ |\ (a,c) \in T\} \in \mu_{a}$,
%and such that $\A x \in p[T]$, $\la \mu_{x \rest i}\ |\ i <
%\omega\}$ is a countably complete tower.
%(Here
%a measure just means
%a $\kappa$-complete ultrafilter, and $p[T] := \{ y \in
%\omega^\omega\ |\ \E \bar{c} \in Z^\omega\ \A i < \omega\ (y\rest
%i,c \rest i) \in T \}$.)

Such a tree is \emph{weakly $\kappa$-homogeneous} (for $\kappa$ a
cardinal) if there exist $\kappa$-complete ultrafilters $\mu_{a,b}
\sse \P(Z^{|a|})$ such that $\A a,b \in \omega^{<\omega}$ with $|a|
= |b|$, $$\{c \in Z^{|a|}\ |\ (a,c) \in T\} \in \mu_{a,b},$$ and
such that for each $x \in p[T]$ there exists a $b \in \omega^\omega$
such that $\la \mu_{x \rest i, b \rest i}\ |\ i < \omega\}$ is a
countably complete tower.
%(Here a measure just means
%a $\kappa$-complete ultrafilter, and $p[T] := \{ y \in
%\omega^\omega\ |\ \E \bar{c} \in Z^\omega\ \A i < \omega\ (y\rest
%i,c \rest i) \in T \}$.)
\end{definition}

Weakly homogeneous trees originated from work of Kechris, Martin and
Solovay. The following fact is due to Woodin. A proof appears in
\cite{L}.

\begin{fact} If $\delta$ is a limit of Woodin cardinals and there is a measurable cardinal above $\delta$, then for each $A \in \P(\R) \cap L(\R)$ and $\gamma < \delta$,
there exists a $\gamma$-weakly-homogeneous tree $T$ such that $p[T] = A$.
\end{fact}

\begin{lemma}\label{extend} Suppose that $T \subset (\omega \times Z)^{\less\omega}$ is a $\gamma^+$-weakly-homogeneous tree, $\theta > (2^{|T|})^+$ is regular, $X \elem H(\theta)$, $T, \gamma
\in X$, $|X| < \gamma$, and $\bar{a} \in p[T]$. Then there exists $Y
\elem H(\theta)$ with $X \sse Y$, $X \cap \gamma = Y \cap \gamma$,
$|X| = |Y|$, and $\bar{a} \in p[T \cap Y]$.

\begin{proof} Fix $\la \mu_{a,b}\ |\ a,b \in \omega^{<\omega} \ra$ in $X$ witnessing the $\gamma^+$-weak-homogeneity of T. Since $\bar{a} \in p[T]$, $\E \bar{b}$ such that $\la \mu_{\bar{a} \rest i, \bar{b} \rest i}\ |\ i < \omega \ra$ is a countably complete tower.
Now let $A_i = \bigcap (X \cap \mu_{\bar{a} \rest i,
\bar{b} \rest i})$. Each $A_i$ is in $\mu_{\bar{a} \rest i, \bar{b}
\rest i}$; so there is $\bar{c} \in Z^\omega$ such that $\A i$
$\bar{c} \rest i \in A_i$. Take $Y = X[\{\bar{c}\rest i\ |\ i <
\omega\}] := \{f(\bar{c}\rest i)\ |\ f \in X, \dom(f) = Z^{<\omega},
i < \omega\}$.



%Think of this as a sort of Skolem Hull; we show elementariness as in
%the proof of \L{}\'o{}s' Theorem.

Elementarity of $Y$ follows from an argument similar to the proof of
\L{}\'o{}s's Theorem (see Theorem 1.1.13 of \cite{L}). To see that
$Y \cap \gamma = X \cap \gamma$, note that if $\alpha \in Y \cap
\gamma$, then $\alpha = f(c \rest i)$ for some $f \in X$, $\dom f =
Z^i$; but $\mu_{a \rest i,b \rest i}$ is $\gamma^+$-complete, so $f$
is constant on a set in $\mu_{a \rest i,b \rest i}$. But then this
constant value is in $X$, and $f(c\rest i)$ is this value, since $c
\rest i = \bigcap (\mu_{a \rest i, b \rest i} \cap X)$.
\end{proof}
\end{lemma}

Note that $Y$ in the proof above is in some sense a limit ultrapower
of the transitive collapse of $X$.

The following was first proved by Foreman, Magidor and Shelah from a
supercompact cardinal, and later improved by Woodin.

\begin{fact} If $\delta$ is Woodin, then $\Coll(\omega_1,<\delta)$ forces that $\NS_{\omega_1}$ is
presaturated, and hence precipitous.
\end{fact}

Recall that an ideal $I$ on $\omega_{1}$ is \emph{presaturated} if
for every sequence of maximal antichains $\{Q_i\ |\ i < \omega\} \subset
\mathcal{P}(\omega_{1})\setminus I$, $\A A \in I^+$, $\E B \sse A$,
$B \in I^+$ such that for all $i < \omega$, $$|\{E \in Q_i\ |\ E
\cap B \in I^+\}| \leq \aleph_1.$$

The following was proved by Kakuda and Magidor independently
\cite{Kak, Mag80}.

\begin{fact}Any c.c.c.\ forcing preserves that $\NS_{\omega_1}$ is precipitous.
\end{fact}

Recall the hypotheses of main existence lemma: $\delta$ is a limit
of Woodin cardinals, there exists a measurable cardinal greater than
$\delta$, and $A$ is in $\P(\R) \cap L(\R)$. To prove the lemma, let
$\kappa$ be the least Woodin cardinal, and $\gamma$ the least strong
inaccessible above $\kappa$. Fix $\gamma^+$-weakly-homogeneous trees
$S,T$, with $p[S] = A$, $p[T] = \R \setminus A$. Fix a regular
$\theta > (2^{|S|})^+, (2^{|T|})^+$. Let $X$ be a countable
elementary submodel of $H(\theta)$, with $S,T,\gamma, \kappa \in X$.
Repeatedly apply Lemma \ref{extend} above to obtain $Y \elem
H(\theta)$ such that $X \sse Y$, $X \cap \gamma = Y \cap \gamma$, $A
= p[S \cap Y]$, $\R \cap A = p[T \cap Y]$. (Then $|Y \cap \Ord| = 2^\omega$.)
Now let $N$ be the transitive collapse of $Y$, and let $\bar{S},
\bar{T}, \bar{\gamma}, \bar{\kappa}$ be the images of
$S,T,\gamma,\kappa$ therein. Let $h$ be $N$-generic for
$\Coll(\omega_1,<\bar{\kappa})$ followed by a c.c.c.\ poset of size
$2^{\omega_1}$ to make $\MA_{\aleph_1}$ hold. Then $N[h] \believes
\MA_{\aleph_1}$ + ``$\NS_{\omega_1}$ is precipitous''. Then $N[h]$ is iterable,
by Lemma \ref{leastcon}. Let $M$ be
$(V_{\bar{\gamma}})^{N[h]}$, and let $j:(M,\NS_{\omega_1}^M) \to
(M^*,\NS_{\omega_1}^*)$ be an iteration. By Lemma \ref{lemma4}, this induces an iteration of
$(N[h],\NS_{\omega_1}^{N[h]})$ with final model $(N^*, I^*)$ (which
we'll also call $j$). Now, $N^*$ is well-founded, and $p[\bar{S}]
\sse p[j(\bar{S})]$ and $p[\bar{T}] \sse p[j(\bar{T})]$. But by
elementarity, $N^* \believes p[j(\bar{S})] \cap p[j(\bar{T})] =
\emptyset$, and since $N^{*}$ is well-founded it is correct about
this. Then $p[\bar{S}] = p[j(\bar{S})]$ and $p[\bar{T}] =
p[j(\bar{T})]$, so $j(A \cap M) = p[j(\bar{S})] \cap M^* = A \cap
M^*$.
%\end{proof}
%\end{theorem}

\begin{remark} Instead of $\Coll(\omega_1,<\bar{\kappa})$, we could have taken
$h$ to be $N$-generic for any poset in $V_{\bar{\gamma}}^N$ such
that $N[h] \believes$ ``$\E$ precipitous $I$ on $\omega_1$'', and
the rest of the proof would have still gone through.
\end{remark}

The main existence lemma gives not only $A$-iterable preconditions
for any $A\in \P(\mathbb{R}) \cap L(\mathbb{R})$, but also
$A$-iterable preconditions containing any given real $x$, for any
$A\in \P(\mathbb{R}) \cap L(\mathbb{R})$, applying the lemma to the
set  $\{y \oplus x\ |\ y \in A\}$. Thus we have shown: if there
exist infinitely many Woodin cardinals below a measurable, then $\A
x \in \R$, $\A A \in \P(\R) \cap L(\R)$, there exists some $\Pmax$ condition
$\la (M,I),a \ra$, with $x \in M$, and $(M,I)$ $A$-iterable.

We didn't quite show $\la H(\omega_1)^M, \in, A \cap M \ra \elem \la
H(\omega_1), \in, A\ra$. We can do this using $A^\sharp$, or by
using not just $S,T$ as above but similar trees for all sets
projective in $A$. We omit this for now.

Given a filter $G \subset \Pmax$, $A_{G}$ denotes the set $\bigcup
\{e\ |\ \E \la (N,J),e \ra \in G \}$. We also omit a proof of the
following:

\begin{fact}[``The combinatorial heart of the $\Pmax$
analysis'']\label{heart} Suppose that for each $A \in \P(\R) \cap
L(\R)$ there exists an $A$-iterable $\Pmax$ precondition $(N,I)$
such that
$$\la H(\omega_1)^M, \in, A \cap M \ra \elem \la H(\omega_1), \in, A
\ra,$$ and suppose that $G \sse \Pmax$ is an $L(\R)$-generic filter.
Then $\A B \in \P(\omega_1)^{L(\R)[G]}$, $\E \la (M,I), a \ra \in G$
such that $B = j(b)$ for some $b \in \P(\omega_1)^M$, where $j$ is
the unique iteration of $(M,I)$ satisfying $j(a) = A_G$.
% := \bigcup \{e\ |\ \E \la (N,J),e \ra \in G \}$
\end{fact}

In other words, all subsets of $\omega_1$ in extensions come from
models in the conditions, and $L(\R)[G] = L(\R)[A_G]$.
%and $\P(\omega_1) = \P(\omega_1)_G$.

\begin{corollary}\label{heartcor}
Suppose that for each $A \in \P(\R) \cap L(\R)$ there exists an
$A$-iterable $\Pmax$ precondition $(N,I)$ such that $$\la
H(\omega_1)^M, \in, A \cap M \ra \elem \la H(\omega_1), \in, A
\ra,$$ and suppose that $G \sse \Pmax$ is an $L(\R)$-generic filter.
Then $\NS_{\omega_1}^{L(\R)[G]}$ is the collection of all sets of
the form $j(e)$, where for some $\la (M,I),a \ra \in G$, $e \in I$,
and $j$ is the iteration of $(M,I)$ sending $a$ to $A_G$.
\end{corollary}

Woodin has shown that the hypotheses of Fact \ref{heart} are
equivalent to the assertion that AD holds in $L(\mathbb{R})$.


 \section{$\Pi_2$ maximality}

\emph{Proof of Goal 1} So now fix some $\Pi_2$ sentence $\varphi =
\A x \E y \psi(x,y)$ (in the extended language with two new unary
predicates), and some $A \in \P(\R) \cap L(\R)$. To show that $$\la
H(\omega_2), \in, A, \NS_{\omega_1} \ra ^ {L(\R)^\Pmax} \believes
\varphi,$$ it is sufficient to show that for each $\la (M,I),a \ra
\in \Pmax$ and each $b \in H(\omega_2)^M$, there exist $\la
(N,\NS_{\omega_1}^N), e \ra \in \Pmax$ and $j:(M,I) \to (M^*, I^*)$
in $N$ such that $j(a) = e$, $I^* = M^* \cap \NS_{\omega_1}^N$, and
$$\la H(\omega_2)^N, \in, A \cap N, \NS_{\omega_1}^N \ra \believes
\E d\ \psi(j(b),d).$$ The argument is like the one for existence of
conditions.

So suppose $\la (M,I), a \ra$ is given. Fix $P$ forcing $\varphi$.
Let $\delta$ be the least Woodin cardinal with $p \in V_\delta$; let
$\kappa$ be the least strong inaccessible above $\delta$. Let $S,T$
be $\kappa^+$-weakly-homogeneous trees projecting to $A$, $\R
\setminus A$. Let $\theta > (2^{|S|})^+,(2^{|T|})^+ $ be regular.
Fix $Y \elem H(\theta)$ with $Y \cap \kappa$ countable, $p[S \cap Y]
= A$, $p[T \cap Y] = \R \setminus A$ and $\la (M,I),a\ra \in Y$.

Let $N$ be the transitive collapse of $Y$, and let
$\bar{P}$,$\bar{S}$,$\bar{\delta}$,$\bar{\kappa}$ be the respective
images of $P$,$S$,$\delta$,$\kappa$ under this collapse. Let $h_0$
be $\bar{P}$-generic for $N$. Note that since $P \in V_{\delta}$,
$\bar{\delta}$ remains Woodin in $N[h_{0}]$. The reinterpretation of
$A$ is the projection of $\bar{S}$ in the extension. Thus $$\la
H(\omega_2)^{N[h_0]}, \in, (p[\bar{S}])^{N[h_0]},
\NS_{\omega_1}^{N[h_0]} \ra \believes \varphi.$$ Pick an iteration
$j$ of $(M,I)$ in $N$ such that $j(I) = j(M) \cap
\NS_{\omega_1}^{N[h_0]}$. Then there exists a $d \in
H(\omega_2)^{N[h_0]}$ such that $$\la H(\omega_2)^{N[h_0]}, \in,
(p[\bar{S}])^{N[h_0]}, \NS_{\omega_1}^{N[h_0]} \ra \believes \psi
(j(b),d).$$ Let $h_1$ be $N[h_0]$-generic for $\Coll(\omega_1,
<\bar{\delta})^{N[h_0]}$ followed by some c.c.c.\ forcing making
$\MA_{\aleph_1}$ hold.
%$\bar{P}$ preserved Woodin-ness of
%$\bar{\delta}$.)
Now $\la ((V_{\bar{\kappa}})^{N[h_0][h_1]},
\NS_{\omega_1}^{N[h_0][h_1]}), j(a) \ra$ is the desired condition.
$\blacksquare$

 \section{Discussion}

\begin{question} You've shown that under these conditions, any forceable $\Pi_2$
statement must hold in the $\Pmax$ extension.
Can you give us some cool examples?
\end{question}

\emph{Answer.} One example is \ $\varphi_\AC$: ``For every
stationary, costationary $A,B \sse \omega_1$, there is some $\gamma
\in [\omega_1, \omega_2)$, some bijection $\pi: \omega_1 \to \gamma$
such that $$\{\alpha < \omega_1\ |\ \alpha \in A \iff (\pi[\alpha])
\in B\}$$ contains a club.'' This can be used to get an injection
$\P(\omega_1) \into \omega_2$, which shows that the Axiom of Choice
holds in the $\Pmax$ extension of $L(\mathbb{R})$.

Also, in some cases one can use $\Pmax$ to get $\Pi_2$ maximality
relative to a given $\Sigma_2$ statement; that is, for a given
$\Sigma_2$ statement for $H(\omega_{2})$, you can simultaneously get
all $\Pi_2$ statements forceably consistent with it.

Another useful aspect: often, the combinatorics of forcing to kill
off one thing while preserving another are not clear; the
combinatorics of doing the same by an iteration may be much clearer.
For instance, an analysis of iterations may help answer the question
of whether there exists a Dowker space on $\omega_{1}$.


\bibliographystyle{plain}
\bibliography{pmax}


\noindent Department of Mathematics\\
Miami University\\
Oxford, Ohio 45056\\United States\\
larsonpb\@@muohio.edu

\end{document}
