# The Mouse Set Theorem Just Past Projective

Mitch Rudominer

###### Abstract.

We identify a particular mouse, $M^{\text{ld}}$, the minimal ladder mouse,
that sits in the mouse order just past $M_{n}^{\sharp}$ for all $n$, and we
show that $\mathbb{R}\cap M^{\text{ld}}=Q_{\omega+1}$, the set of reals that
are $\Delta^{1}_{\omega+1}$ in a countable ordinal. Thus $Q_{\omega+1}$ is a
mouse set.

This is analogous to the fact that $\mathbb{R}\cap M^{\sharp}_{1}=Q_{3}$ where
$M^{\sharp}_{1}$ is the the sharp for the minimal inner model with a Woodin
cardinal, and $Q_{3}$ is the set of reals that are $\Delta^{1}_{3}$ in a
countable ordinal.

More generally $\mathbb{R}\cap M^{\sharp}_{2n+1}=Q_{2n+3}$. The mouse
$M^{\text{ld}}$ and the set $Q_{\omega+1}$ compose the next natural pair to
consider in this series of results. Thus we are proving the mouse set theorem
just past projective.

Some of this is not new. $\mathbb{R}\cap M^{\text{ld}}\subseteq Q_{\omega+1}$
was known in the 1990’s. But $Q_{\omega+1}\subseteq M^{\text{ld}}$ was open
until Woodin found a proof in 2018. The main goal of this paper is to give
Woodin’s proof.

###### Key words and phrases:

large cardinals, descriptive set theory, inner model theory

###### Contents

  1. 1 Introduction
  2. 2 Projective-like Pointclasses Just Past Projective
  3. 3 Ladder Mice
  4. 4 A Suslin Representation for $\Pi^{1}_{\omega}$
  5. 5 Main Proof
  6. 6 Beyond the first pointclass
  7. 7 Eliminating the ordinal parameter

##  1\. Introduction

Throughout this paper we write $\mathbb{R}$ to mean ${}^{\omega}{\omega}$ and
we call elements of ${}^{\omega}{\omega}$ _reals_.

In the 1990’s Martin, Steel and Woodin proved that, assuming large cardinals
or determinacy, $\mathbb{R}\cap M_{n}=$ the set of reals that are
$\Delta^{1}_{n+2}$ in a countable ordinal, where $M_{n}$ is the standard,
minimal proper-class inner model with $n$ Woodin cardinals (including the case
$M_{0}=L$.) For even $n$, $\mathbb{R}\cap M_{n}=C_{n+2}$, the largest
countable $\Sigma^{1}_{n+2}$ set. For odd $n$, $\mathbb{R}\cap M_{n}=Q_{n+2}$,
the largest countable $\Pi^{1}_{n+2}$ set closed downwards under
$\Delta^{1}_{n+2}$ degrees. See [Ste95].

Recall the definition of $n$-small (Definition 1.1 from [Ste95].) A premouse
$M$ is $n$-small above $\delta$ iff whenever $\kappa$ is the critical point of
an extender on the $M$-sequence, and $\delta<\kappa$, then

$\mathcal{J}^{M}_{\kappa}\not\models\text{there are }n\text{ Woodin cardinals
}>\delta.$

We say that M is $n$-small iff $M$ is $n$-small above $0$.

In the above-mentioned theorems, if one wishes to work with countable mice
rather than proper class models, one may substitute $M_{n}^{\sharp}$ for
$M_{n}$ since the two models have the same reals. Here $M_{n}^{\sharp}$ is the
least sound, iterable mouse that is not $n$-small.

In [Rud95] and [Rud97] we explored an extension of this theory to projective-
like pointclasses beyond the projective and achieved partial results. But even
at the very first step past projective the full mouse set theorem was left
open.

Let $\Sigma^{1}_{\omega}$ be the pointclass consisting of recursive unions of
Projective sets. Consider the projective hierarchy built over
$\Sigma^{1}_{\omega}$: $\Pi^{1}_{\omega}=\neg\Sigma^{1}_{\omega}$,
$\Sigma^{1}_{\omega+1}=\exists^{\mathbb{R}}\Pi^{1}_{\omega}$, etc. By [Ste83],
$\Sigma^{1}_{\omega+n}=\operatorname{\mathcal{P}}(\mathbb{R})\cap\Sigma_{n+1}(J_{2}(\mathbb{R}))$
and we get a Second Periodicity Theorem: $\Sigma^{1}_{\omega+2n}$ and
$\Pi^{1}_{\omega+2n+1}$ are scaled pointclasses. $\Sigma^{1}_{\omega}$ is not
closed under $\forall^{\omega}$ and so is not a perfect analog of
$\Sigma^{1}_{1}$. But $\Pi^{1}_{\omega+1}$ is closed under $\exists^{\omega}$
and is a good analog of $\Pi^{1}_{1}$ or $\Pi^{1}_{3}$. Let $Q_{\omega+1}$ be
the set of reals that are $\Delta^{1}_{\omega+1}$ in a countable ordinal.

In [Rud95], we defined the minimal ladder mouse $M^{\text{ld}}$ and showed
that $\mathbb{R}\cap M^{\text{ld}}\subseteq Q_{\omega+1}$. ($M^{\text{ld}}$ is
the minimal iterable mouse $M$ such that for each $n\in\omega$ there is a
cardinal $\delta_{n}$ of $M$ and an initial segment $P_{n}\trianglelefteq M$
such that $\delta_{n}$ is Woodin in $P_{n}$ and $P_{n}$ is not $n$-small above
$\delta_{n}$.)

But the other direction, $Q_{\omega+1}\subseteq M^{\text{ld}}$, remained open
until Woodin found a proof in the summer of 2018. Woodin transmitted the proof
to Steel who fleshed it out and extended it in some hand-written notes
[Ste18]. The author refined the relevant sections of those notes into this
paper. Woodin and Steel have elected to not appear as co-authors of this
paper, but the mathematical provenance of the ideas contained in the paper
should be clear from this paragraph.

No new technology is involved in the proof and it could have been discovered
in the 1990’s. The main ingredients of the proof are stationary tower forcing,
genericity iterations, and a lemma stating that all small generic extensions
of a sufficiently correct, countable, transitive model of Set Theory meet the
same equivalence classes of a thin, sufficiently definable equivalence
relation. (See Lemma 5.2.)

One of the key technical themes of the paper is that, even though the
$\delta_{n}$ of $M^{\text{ld}}$ are only Woodin over an initial segment
$P_{n}$ of $M^{\text{ld}}$ as opposed to all of $M^{\text{ld}}$, because they
are cardinals in $M^{\text{ld}}$ we can recover some of the same properties
that we would have if $\left\langle\delta_{n}\right\rangle$ were an
$\omega$-sequence of Woodins in $M^{\text{ld}}$. A good example of this theme
is Lemma 5.5 which gives us a filter $G$ for the countable stationary tower of
$M^{\text{ld}}$ such that for all $n$, $G\cap J^{M^{\text{ld}}}_{\delta_{n}}$
is $\mathbb{Q}_{<\delta_{n}}^{M}$-generic over $P_{n}$, and this implies that
the ultrapower $\operatorname{Ult}(M^{\text{ld}},G)$ has some of the same
properties as an ultrapower by the stationary tower up to the supremum of
$\omega$ Woodin cardinals.

Beyond $\operatorname{ZFC}$, there are two necessary hypotheses for this
paper:

  1. (1)

that determinacy holds for the pointclasses we consider (roughly,
$\boldsymbol{\Pi}_{2}(J_{2}(\mathbb{R}))$),

  2. (2)

and that the mice we consider exist and are sufficiently iterable.

We will not attempt to articulate optimal hypotheses. Instead, for the
official hypothesis for this paper we will use: $\operatorname{ZFC}+$ “there
exists $\omega$ Woodin cardinals with a measurable cardinal above them all.”
If we don’t explicitly state otherwise, this is the assumption of all theorems
in this paper.

This large cardinal hypothesis implies $\operatorname{AD}^{L(\mathbb{R})}$
(see Theorem 8.24 of [Nee10]) and it implies that $M_{\omega}^{\sharp}$ exists
and is $(\omega,\omega_{1},\omega_{1}+1)$-iterable (see Theorem 7.2 of
[Ste10].) So in particular the hypothesis gives us more than enough
determinacy and iterable-mice existence for the purposes of this paper. But
the reader should keep in mind that the assumption of actual large cardinal in
$V$ is not really an essential ingredient here, it is just a convenient way of
arranging the assumptions we really need: sufficient determinacy and mice.

##  2\. Projective-like Pointclasses Just Past Projective

######  Definition 2.1.

Let $\Pi^{1}_{\omega}$ be the pointclass of recursive intersections of
infinitely many (lightface) projective sets.

In more detail, fix for the remainder of this paper, for $n\in\omega$,
$G^{n}\subset\omega\times\mathbb{R}$ a universal $\Pi^{1}_{2n+1}$ set,
uniformly in $n$. (“Uniformly in $n$” here and below means there is a
recursive sequence of formula with the $n$-th formula defining the $n$-th
relation.) A $\Pi^{1}_{\omega}$ _code_ is a total recursive function
$h:\omega\to\omega$. If $h$ is a $\Pi^{1}_{\omega}$ code then
$A^{h}=\left\\{\,x\in\mathbb{R}\,\left|\,(\forall
n)\,G^{n}(h(n),x)\,\right.\right\\}$. Say $A\subseteq\mathbb{R}$ is
$\Pi^{1}_{\omega}$ iff $A=A^{h}$ for some $\Pi^{1}_{\omega}$ code $h$. In this
case we will also say that $h$ is a $\Pi^{1}_{\omega}$ code _for_ $A$.

Similarly we define $\Pi^{1}_{\omega}$ codes and $\Pi^{1}_{\omega}$ subsets of
$\omega^{s}\times\mathbb{R}^{t}$ for $s,t\in\omega$.

Then we define $\Sigma^{1}_{\omega}=\neg\Pi^{1}_{\omega}$,
$\Sigma^{1}_{\omega+1}=\exists^{\mathbb{R}}\Pi^{1}_{\omega}$,
$\Pi^{1}_{\omega+1}=\forall^{\mathbb{R}}\Sigma^{1}_{\omega}$ and
$\Delta^{1}_{\omega+1}=\Sigma^{1}_{\omega+1}\cap\Pi^{1}_{\omega+1}$.

Finally we define as usual the relativized and bold-face pointclasses
$\Pi^{1}_{\omega}(x)$ and $\boldsymbol{\Pi}^{1}_{\omega}$ etc.

The following remarks follow from [Ste83] and [Rud97]. Here we are following
[Ste83] in defining $\Sigma_{n}(J_{2}(\mathbb{R}))$ to mean
$\Sigma_{n}(J_{2}(\mathbb{R}),\left\\{V_{\omega+1}\right\\})$. Thus we are
always allowed $V_{\omega+1}$ (but not its elements) as a parameter.

######  Remarks 2.2.

  1. (1)

$\Sigma^{1}_{\omega}=\Sigma_{1}(J_{2}(\mathbb{R}))\cap\operatorname{\mathcal{P}}(\mathbb{R})$

  2. (2)

$\Pi^{1}_{\omega}=\Pi_{1}(J_{2}(\mathbb{R}))\cap\operatorname{\mathcal{P}}(\mathbb{R})$

  3. (3)

$\Sigma^{1}_{\omega+1}=\Sigma_{2}(J_{2}(\mathbb{R}))\cap\operatorname{\mathcal{P}}(\mathbb{R})$

  4. (4)

$\Pi^{1}_{\omega+1}=\Pi_{2}(J_{2}(\mathbb{R}))\cap\operatorname{\mathcal{P}}(\mathbb{R})$

  5. (5)

$\Sigma^{1}_{\omega}$ and $\Pi^{1}_{\omega+1}$ are scaled pointclasses.

###### proof sketch.

(1) Let $A\subset\mathbb{R}$ be $\Sigma^{1}_{\omega}$. Then there is a
recursive sequence of formulae
$\left\langle\,\varphi_{n}\,\left|\,n\in\omega\,\right.\right\rangle$ such
that $x\in A\leftrightarrow\exists
n(V_{\omega+1},\epsilon)\models\varphi_{n}[x]$. Since the satisfaction
relation is $\Delta_{1}$ definable in any rudimentarily closed, transitive
set, $A\in\Sigma_{1}(J_{2}(\mathbb{R}))$.

That
$\Sigma_{1}(J_{2}(\mathbb{R}))\cap\operatorname{\mathcal{P}}(\mathbb{R})\subseteq\Sigma^{1}_{\omega}$
follows from Lemma 1.1 of [Rud97].

(3) follows from Lemma 2.6 of [Rud97] becase, using the terminology from that
paper, $0$ is a good parameter for $J_{2}(\mathbb{R})$. (As we elaborate on in
section 6, the $\Sigma^{1}_{\omega+1}$ of this paper is equal to the
$\Sigma_{(2,1)}$ of [Rud97].)

(5) follows from Lemma 2.11 of [Rud97]. (Again, the $\Sigma^{1}_{\omega}$ of
this paper is equal to the $\Sigma_{(2,0)}$ of [Rud97] and the
$\Pi^{1}_{\omega+1}$ of this paper is equal to the $\Pi_{(2,1)}$ of [Rud97].)
∎

######  Definition 2.3.

We say that $x\in\mathbb{R}$ is $\Delta^{1}_{\omega+1}$ in a countable ordinal
iff there is an $\alpha<\omega_{1}$ such that for all $w\in\operatorname{WO}$
with $|w|=\alpha$, $\left\\{x\right\\}$ is $\Delta^{1}_{\omega+1}(w)$. Let
$Q_{\omega+1}=\left\\{\,x\in\mathbb{R}\,\left|\,x\text{ is
}\Delta^{1}_{\omega+1}\text{ in a countable ordinal}\,\right.\right\\}$.

By way of intuition, we consider $\Pi^{1}_{\omega+1}$ to be analogous to
$\Pi^{1}_{1}$ or $\Pi^{1}_{3}$ and we consider $Q_{\omega+1}$ to be analogous
to $Q_{1}$ or $Q_{3}$. These types of sets have been much studied by
descriptive set theorists and have many interesting properties. For example
$Q_{3}$ is the largest $\Pi^{1}_{3}$ set of reals closed downwards under
$\Delta^{1}_{3}$-degrees. See [KMS83].

Recall that $Q_{3}=\mathbb{R}\cap M_{1}^{\sharp}$. In the next section we will
define a mouse $M^{\text{ld}}$ such that $Q_{\omega+1}=\mathbb{R}\cap
M^{\text{ld}}$.

##  3\. Ladder Mice

######  Definition 3.1.

Let $M$ be a premouse, in the sense of [MS94]. A _ladder over $M$_ is a
sequence of ordinals
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
such that for all $n$:

  1. (1)

$\delta_{n}<\gamma_{n}<\delta_{n+1}<\operatorname{ord}(M)$,

  2. (2)

$\delta_{n}$ is a cardinal of $M$,

  3. (3)

$\delta_{n}$ is Woodin in $J^{M}_{\gamma_{n}}$,

  4. (4)

$\gamma_{n}$ is the least $\gamma$ such that $\mathcal{J}^{M}_{\gamma}$ is not
$n$-small above $\delta_{n}$.

To remind the reader of some notation from [MS94] and [Ste93],
$\mathcal{J}^{M}_{\gamma}$ is the model that includes a predicate for the last
extender, if there is one indexed at $\gamma$ on the $M$-sequence, whereas
$J^{M}_{\gamma}$ does not include such a predicate. In the definition above we
use $\mathcal{J}^{M}_{\gamma}$ in (4) because, by definition of $n$-small, the
least mouse that is not $n$ small must be active.

A ladder
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
over $M$ is _cofinal_ iff the $\gamma_{n}$ are cofinal in
$\operatorname{ord}(M)$.

$M$ is a _ladder mouse_ iff there is a ladder over $M$ and a _cofinal ladder
mouse_ iff there is a cofinal ladder over $M$.

$M$ is a _minimal ladder mouse_ iff $M$ is a ladder mouse but no initial
segment of $M$ is.

$M^{\text{ld}}$ is the least fully-iterable, sound ladder mouse, if it exists.
If $M^{\text{ld}}$ exists it is obviously a minimal ladder mouse (and so a
cofinal ladder mouse), and $M^{\text{ld}}$ projects to $\omega$ and so it is
an $\omega$-mouse in the sense of [Ste95].

######  Definition 3.2.

Let $M$ be a transitive model of a sufficient amount of set theory.

  * •

$M$ is $\Sigma^{1}_{n}$-_correct_ iff for all $\Sigma^{1}_{n}$ formula
$\varphi$ and all reals $x\in M$, $M\models\varphi[x]$ iff $\varphi(x)$.

  * •

$M$ is _projectively correct_ just in case it is $\Sigma^{1}_{n}$-correct for
all $n\in\omega$.

  * •

$M$ is $\Pi^{1}_{\omega}$-_correct_ iff for all $\Pi^{1}_{\omega}$ codes $h$
and all reals $x\in M$, $M\models A^{h}(x)$ iff $A^{h}(x)$. Clearly $M$ is
$\Pi^{1}_{\omega}$-correct iff it is projectively correct.

  * •

$M$ is $\Sigma^{1}_{\omega+1}$-_correct_ iff for all $\Pi^{1}_{\omega}$ codes
$h$ for subsets of $\mathbb{R}^{2}$ and all reals $x\in M$, $(\exists
y\in\mathbb{R}\cap M)M\models A^{h}(x,y)$ iff $(\exists
y\in\mathbb{R})A^{h}(x,y)$.

######  Remarks 3.3.

  1. (1)

Because we are assuming there exists $\omega$ Woodin cardinals with a
measurale cardinal above them, $M^{\text{ld}}$ exists and is
$(\omega_{1}+1)$-iterable.

  2. (2)

For every $n\in\omega$, $M^{\text{ld}}$ has a rank initial segment that
satisfies $\operatorname{ZFC}$+$\exists n$ Woodin cardinals. By [Ste95],
$M^{\text{ld}}$ is projectively correct and so $\Pi^{1}_{\omega}$-correct.

  3. (3)

Let
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be a ladder over $M^{\text{ld}}$. For even $n$,
$J^{M^{\text{ld}}}_{\gamma_{n}}[g]$ is $\Sigma^{1}_{n+2}$-correct, where $g$
is $\operatorname{Coll}(\delta_{n},\omega)$-generic over
$J^{M^{\text{ld}}}_{\gamma_{n}}$.

  4. (4)

Let
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be a ladder over $M$. $\delta_{n}$ is not necessarily fully Woodin in $M$, but
$\delta_{n}$ is Woodin in $M$ with respect to functions in
$J^{M}_{\gamma_{n}}$.

  5. (5)

By the previous two items, we can, at least informally, think of
$M^{\text{ld}}$ as the least mouse $M$ such that for every $n\in\omega$ there
is a cardinal $\delta$ of $M$ such that $\delta$ is “$\Sigma^{1}_{n}$-Woodin”
in $M$.

We can now state the main theorem of the paper:

######  Theorem 3.4.

Assume there exists $\omega$ Woodin cardinals with a measurable cardinal above
them all. Then $Q_{\omega+1}=\mathbb{R}\cap M^{\text{ld}}$.

We divide the above theorem up into its two directions:

######  Theorem 3.5.

Assume there exists $\omega$ Woodin cardinals with a measurable cardinal above
them all. Then $\mathbb{R}\cap M^{\text{ld}}\subseteq Q_{\omega+1}$.

######  Theorem 3.6.

Assume there exists $\omega$ Woodin cardinals with a measurable cardinal above
them all. Then $Q_{\omega+1}\subseteq M^{\text{ld}}$.

Theorem 3.5 was proven more than 25 years ago in [Rud95] and [Rud97]. We give
a sketch of the proof later in Section 6. The proof follows the same line of
reasoning as the proof that every real in $L$ is $\Delta^{1}_{2}$ in a
countable ordinal and the proof that every real in $M_{1}$ is $\Delta^{1}_{3}$
in a countable ordinal. Namely we show that every initial segment of
$M^{\text{ld}}$ that projects to $\omega$ is $\Pi^{1}_{\omega}$ definable from
its ordinal height.

As mentioned in the introduction, the main goal of this paper is to give
Woodin’s proof of Theorem 3.6.

Theorem 3.6 will follow from a quasi-correctness theorem for $M^{\text{ld}}$.
$M^{\text{ld}}$ is $\Pi^{1}_{\omega}$-correct but not
$\Sigma^{1}_{\omega+1}$-correct. Let $A\subset\mathbb{R}^{2}$ be
$\Pi^{1}_{\omega}$ and let $x\in\mathbb{R}\cap M^{\text{ld}}$. It may happen
that there is a $y\in\mathbb{R}$ such that $A(x,y)$ but there is no such $y$
in $M^{\text{ld}}$. But we will show that even if
$M^{\text{ld}}\not\models(\exists y)A(x,y)$, $M^{\text{ld}}$ can still
determine whether or not the statement “$(\exists y)A(x,y)$” is true. This is
similar to the situation with $M_{1}$ and $\Sigma^{1}_{3}$. $M_{1}$ is not
$\Sigma^{1}_{3}$-correct, but $M_{1}$ can still tell whether or not
$\Sigma^{1}_{3}$ statements are true. Let $A\subset\mathbb{R}^{2}$ be
$\Pi^{1}_{2}$ and let $x\in M_{1}$. Then $(\exists y\in\mathbb{R})\,A(x,y)$
iff

$M_{1}\models
1\|\\!\underset{\\!\\!\\!\\!\\!\operatorname{Coll}(\delta,\omega)}{\\!\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar}(\exists
y)A(x,y),$

where $\delta$ is the Woodin of $M_{1}$.

Next we state the $\Sigma^{1}_{\omega+1}$-quasi-correctness for
$M^{\text{ld}}$ precisely and we show how Theorem 3.6 follows from it.

If $z$ is a real, then by a _mouse over $z$_ we mean a mouse in the sense of
the theory of [MS94] augmented to use $z$ as an additional predicate. (So this
will be a model of the form $J_{\alpha}[\vec{E},z]$.)

######  Theorem 3.7.

There is a formula $\psi$ in the language of Set Theory such that, for all
reals $z$, for all $M$ a countable, iterable, ladder-mouse over $z$, there is
an ordinal parameter $\theta<\omega_{2}^{M}$ such that for all reals $x$ in
$M$, for all $\Pi^{1}_{\omega}$ codes $h$,

$(\exists y)A^{h}(x,y)\Leftrightarrow
J^{M}_{\omega_{2}^{M}}\models\psi[\theta,h,x].$

######  Remark 3.8.

If $M=M^{\text{ld}}$ then the ordinal parameter $\theta$ in Theorem 3.7 can be
eliminated. We prove this in Section 7.

Now, assuming Theorem 3.7, we prove Theorem 3.6.

###### proof of Theorem 3.6.

Let $x\in Q_{\omega+1}$. We need to show that $x\in M^{\text{ld}}$. Fix
$\alpha<\omega_{1}$ such that for all $w\in\operatorname{WO}$ with
$|w|=\alpha$, $\left\\{x\right\\}$ is $\Delta^{1}_{\omega+1}(w)$.

Let $M^{\prime}$ be the $(\alpha+1)$-th iterate of $M^{\text{ld}}$ by its
least measurable cardinal. ($M^{\text{ld}}$ has many measurable cardinals
because, in the definition of a ladder over $M$, $\delta_{n}$ is Woodin in
$J^{M}_{\gamma_{n}}$, so $J^{M}_{\gamma_{n}}\models$ “$\delta_{n}$ is an
inaccessible limit of measurable cardinals”, but $\delta_{n}$ is also a
cardinal of $M$ and so, by strong-acceptability, $\delta_{n}$ is a limit of
measurable cardinals in $M$.) So $M^{\prime}$ is a countable, iterable ladder
mouse with a ladder whose $\delta_{0}$ is greater than $\alpha$. Let $g$ be
$\operatorname{Coll}(\alpha,\omega)$-generic over $M^{\prime}$. It suffices to
see that $x\in M^{\prime}[g]$.

We can rearrange $M^{\prime}[g]$ as a mouse over $z$, where $z$ is some real
coding $g$. Let $M$ be this mouse over $z$. $M$ is a countable, iterable
ladder-mouse over $z$ and there is a real $w\in M\cap\operatorname{WO}$ with
$|w|=\alpha$.

Since $\left\\{x\right\\}$ is $\Sigma^{1}_{\omega+1}(w)$, it is also true that
$x$ is a $\Sigma^{1}_{\omega+1}(w)$ subset of $\omega\times\omega$. Fix a
$\Pi^{1}_{\omega}$ code $h$ such that for all $i,j\in\omega$,

$x(i)=j\Leftrightarrow(\exists y\in\mathbb{R})A^{h}(i,j,w,y).$

Let $\psi$ and $\theta$ be the formula and parameter given by Theorem 3.7 for
$M$. Then for all $i,j\in\omega$,

$x(i)=j\Leftrightarrow(\exists y\in\mathbb{R})A^{h}(i,j,w,y)\Leftrightarrow
J^{M}_{\omega_{2}^{M}}\models\psi[\theta,h,i,j,w].$

So $x\in M$. ∎

We now develop the ideas that will allow us to prove theorem 3.7. We start in
the next section with a Suslin representation for $\Pi^{1}_{\omega}$ sets.

##  4\. A Suslin Representation for $\Pi^{1}_{\omega}$

Let $\boldsymbol{\delta}^{1}_{\omega}=\sup_{n}\boldsymbol{\delta}^{1}_{n}$. In
this section we show how to express a $\Pi^{1}_{\omega}$ set as the projection
of a tree on $\omega\times\boldsymbol{\delta}^{1}_{\omega}$.

######  Definition 4.1.

Let $h$ be a $\Pi^{1}_{\omega}$ code for a subset of $\mathbb{R}$. We here
define $T^{h}$, a natural tree on
$\omega\times\boldsymbol{\delta}^{1}_{\omega}$ that projects to $A^{h}$.

Recall that $G^{n}\subset\omega\times\mathbb{R}$ is a universal
$\Pi^{1}_{2n+1}$ set, uniformly in $n$.

For $n,e\in\omega$, let
$G^{n}_{e}=\left\\{\,x\in\mathbb{R}\,\left|\,G^{n}(e,x)\,\right.\right\\}$ and
let
$\varphi^{n}_{e}=\left\langle\,\varphi^{n}_{e,i}\,\left|\,i\in\omega\,\right.\right\rangle$
be a $\Pi^{1}_{2n+1}$ scale on $G^{n}_{e}$ uniformly in $n$ and $e$, with each
of the norms $\varphi^{n}_{e,i}$ being regular. Let $<^{n}_{e,i}$ be the
prewellorder on $\mathbb{R}$ associated with $\varphi^{n}_{e,i}$. So “$x,y\in
G^{n}_{e}$ and $x<^{n}_{e,i}y$” is $\Pi^{1}_{2n+1}$, uniformly in $n,e,i$, and
$\varphi^{n}_{e,i}(x)=$ the rank of $x$ in $<^{n}_{e,i}$.
$\varphi^{n}_{e,i}:G^{n}_{e}\rightarrow\boldsymbol{\delta}^{1}_{2n+1}$.

Let $T^{n}_{e}$ be the tree for the scale $\varphi^{n}_{e}$:

$T^{n}_{e}=\left\\{\,\left(x\\!\\!\upharpoonright\\!k,\left\langle\,\varphi^{n}_{e,i}(x)\,\left|\,i<k\,\right.\right\rangle\right)\,\left|\,x\in
G^{n}_{e}\text{ and }k\in\omega\,\right.\right\\}.$

Then $G^{n}_{e}=p[T^{n}_{e}]$. (This is part of what it means that
$\left\langle\,\varphi^{n}_{e,i}\,\left|\,i\in\omega\,\right.\right\rangle$ is
a scale on $G^{n}_{e}$. )

Fix a recursive bijection between $\omega$ and $\omega\times\omega$, written
as $i=\left\langle i_{0},i_{1}\right\rangle$, with the property that
$i_{0},i_{1}\leq i$ for $i\in\omega$.

Now fix a $\Pi^{1}_{\omega}$ code $h$ and we will define the tree $T^{h}$. Let
$(s,u)\in\omega^{n}\times(\boldsymbol{\delta}^{1}_{\omega})^{n}$ for some
$n\in\omega$. Then $(s,u)\in T^{h}$ iff there is an $x\in\mathbb{R}$ extending
$s$ such that $(\forall i<n)\,x\in G^{i}_{h(i)}$ and
$u(i)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(x)$. Intuitively, one can think of
$T^{h}$ as building a pair $(x,f)$ where $x$ is a real and $f$ is a
certificate verifying that $x\in p[T^{n}_{h(n)}]$ for all $n\in\omega$.

We say that $T$ is a $\Pi^{1}_{\omega}$-tree iff $T=T^{h}$ for some
$\Pi^{1}_{\omega}$ code $h$.

Similarly we define $\Pi^{1}_{\omega}$-trees $T^{h}$ for $h$ a
$\Pi^{1}_{\omega}$ code for a subset of $\mathbb{R}^{k}$ for $k\in\omega$.

Our coding of pairs of integers by integers induces a mapping from
$\omega^{\omega}$ to $(\omega^{\omega})^{\omega}$:
$f\mapsto\left\langle\,f_{n}\,\left|\,n\in\omega\,\right.\right\rangle$ given
by $f_{n}(i)=f(\left\langle n,i\right\rangle)$.

######  Proposition 4.2.

$A^{h}=p[T^{h}]$.

###### Proof.

Let $x\in A^{h}=\bigcap\limits_{n}G^{n}_{h(n)}$. Let
$f:\omega\to\boldsymbol{\delta}^{1}_{\omega}$ be given by
$f(i)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(x)$. Then $(x,f)\in[T^{h}]$.

Conversely, suppose $(x,f)\in[T^{h}]$. We claim that $\forall
n\,(x,f_{n})\in[T^{n}_{h(n)}]$. Let $m>k>n\in\omega$ be such that
$\left\langle n,i\right\rangle<m$ for all $i<k$. Let $x^{\prime}$ be a real
such that $x^{\prime}\\!\\!\upharpoonright\\!m=x\\!\\!\upharpoonright\\!m$ and
$x^{\prime}$ witness that
$(x\\!\\!\upharpoonright\\!m,f\\!\\!\upharpoonright\\!m)\in T^{h}$. Then
$x^{\prime}\in G^{n}_{h(n)}$ and $(\forall i<k)\,f_{n}(i)=f(\left\langle
n,i\right\rangle)=\varphi^{n}_{h(n),i}(x^{\prime})$. So $x^{\prime}$ witnesses
that $(x\\!\\!\upharpoonright\\!k,f_{n}\\!\\!\upharpoonright\\!k)\in
T^{n}_{h(n)}$. As $n,k$ were arbitrary, $\forall
n\,(x,f_{n})\in[T^{n}_{h(n)}]$.

∎

######  Remark 4.3.

The definition of the first $n$ levels of $T^{h}$ depends only on
$\left\langle\,G^{i}_{h(i)}\,\left|\,i<n\,\right.\right\rangle$ and the norms
$\left\langle\,\varphi^{i}_{h(i),j}\,\left|\,i,j<n\,\right.\right\rangle$, all
of which are $\Pi^{1}_{2n+1}$.

Next we want to consider the version of $T^{h}$ inside of a countable,
transitive model.

Let $M$ be a countable transitive model of $\operatorname{ZFC}$ that is
$\Pi^{1}_{2n+1}$-correct. Let $A\subset\mathbb{R}$ be $\Pi^{1}_{2n+1}$, and
let $\varphi:A\rightarrow\boldsymbol{\delta}^{1}_{2n+1}$ be a regular
$\Pi^{1}_{2n+1}$ norm on $A$ with associated pre-wellorder $<_{\varphi}$. Then
there is a natural way to interpret $\varphi$ in $M$. $\varphi^{M}:A\cap
M\rightarrow(\boldsymbol{\delta}^{1}_{2n+1})^{M}$ is given by
$\varphi^{M}(x)=$ the rank of $x$ in $<_{\varphi}\cap M$.

######  Definition 4.4.

In the situation described in the previous paragraph, we define
$\sigma^{M}_{\varphi}:\operatorname{ran}(\varphi^{M})\rightarrow\boldsymbol{\delta}^{1}_{2n+1}$
by $\sigma^{M}_{\varphi}(\varphi^{M}(x))=\varphi(x)$ for all $x\in A\cap M$.
This is well-defined and order preserving since for $x,y\in A\cap M$,
$\varphi^{M}(x)=\varphi^{M}(y)$ iff $\varphi(x)=\varphi(y)$ and
$\varphi^{M}(x)<\varphi^{M}(y)$ iff $\varphi(x)<\varphi(y)$.

$\sigma^{M}_{\varphi}$ is the inverse of the transitive collapse of
$\varphi[A\cap M]$.

If $M$ is projectively correct then $\sigma^{M}_{\varphi}$ does not depend on
the norm $\varphi$.

######  Lemma 4.5.

Let $M$ be a countable, transitive, projectively correct model of
$\operatorname{ZFC}$. Let $A,B$ be two $\Pi^{1}_{2n+1}$ sets of reals and let
$\varphi_{A}:A\to\boldsymbol{\delta}^{1}_{2n+1}$ and
$\varphi_{B}:B\to\boldsymbol{\delta}^{1}_{2n+1}$ be $\Pi^{1}_{2n+1}$ norms on
$A$ and $B$ respectively. Let
$\alpha<\min(\operatorname{ran}(\varphi^{M}_{A}),\operatorname{ran}(\varphi^{M}_{B}))$.
Then $\sigma^{M}_{\varphi_{A}}(\alpha)=\sigma^{M}_{\varphi_{B}}(\alpha)$.

###### Proof.

If $x\in A$ and $y\in B$ and $\varphi^{M}_{A}(x)=\varphi^{M}_{B}(y)$, then
$\varphi_{A}(x)=\varphi_{B}(y)$. This is because, by Theorem 3.3.2 of [HK81],
the relation “$\varphi_{A}(x)=\varphi_{B}(y)$” is $\Delta^{1}_{2n+3}$ and so
absolute for $M$. ∎

######  Definition 4.6.

Let $M$ be a countable, transitive, projectively correct model of
$\operatorname{ZFC}$. Then
$\sigma^{M}:(\boldsymbol{\delta}^{1}_{\omega})^{M}\to\boldsymbol{\delta}^{1}_{\omega}$
is the union of all functions $\sigma^{M}_{\varphi}$ where $\varphi$ is a
(lightface) projective norm on a (lightface) projective set.

Let $h$ be a $\Pi^{1}_{\omega}$ code. Then we define
$\sigma^{M,h}:(T^{h})^{M}\to T^{h}$ by

$\sigma^{M,h}\left((\left\langle
n_{0},\cdots,n_{k}\right\rangle,\left\langle\alpha_{0},\cdots,\alpha_{k}\right\rangle)\right)=(\left\langle
n_{0},\cdots,n_{k}\right\rangle,\left\langle\sigma^{M}(\alpha_{0}),\cdots,\sigma^{M}(\alpha_{k})\right\rangle).$

######  Lemma 4.7.

Let $M$ be a countable, transitive, projectively correct model of
$\operatorname{ZFC}$. Let $h$ be a $\Pi^{1}_{\omega}$ code. Then
$\sigma^{M,h}:(T^{h})^{M}\to T^{h}$ is a tree isomorphism onto a subtree of
$T^{h}$.

###### Proof.

If $(s,u)=(\left\langle
n_{0},\cdots,n_{k}\right\rangle,\left\langle\alpha_{0},\cdots,\alpha_{k}\right\rangle)\in(T^{h})^{M}$
then there is a real $x\in M\cap G^{0}_{h(0)}\cap\cdots\cap G^{k}_{h(k)}$ such
that $x(i)=n_{i}$ and $\alpha_{i}=(\varphi^{i_{0}}_{h(i_{0}),i_{1}}(x))^{M}$
for $0\leq i\leq k$. So
$\sigma^{M}(\alpha_{i})=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(x)$ for $0\leq i\leq
k$ and so $x$ witnesses that $\sigma^{M,h}(s,u)\in T^{h}$.

That $\sigma^{M,h}$ is injective, preserves the lengths of finite sequences
and sequence extension is obvious. ∎

######  Definition 4.8.

If $T$ is a tree on $\omega\times U$ and $x$ is a real then $T(x)$ is the tree
through $x$:

$T(x)=\left\\{\,u\,\left|\,(x\\!\\!\upharpoonright\\!\operatorname{lh}(u),u)\in
T\,\right.\right\\}.$

If $T$ is a $\Pi^{1}_{\omega}$ tree for a subset of $\mathbb{R}^{2}$ and
$x\in\mathbb{R}$ we say that $T(x)$ is a $\Pi^{1}_{\omega}(x)$-tree and a
$\boldsymbol{\Pi}^{1}_{\omega}$-tree.

######  Remarks 4.9.

Let $M$ be a countable, transitive, projectively correct model of
$\operatorname{ZFC}$. $M$ is $\Pi^{1}_{\omega}$-correct, but not necessarily
$\Sigma^{1}_{\omega+1}$-correct. Let $h$ be a $\Pi^{1}_{\omega}$ code for
$A\subset\mathbb{R}^{2}$. Let $x\in\mathbb{R}\cap M$.

  1. (1)

$\exists yA(x,y)\Leftrightarrow T^{h}(x)$ is illfounded.

  2. (2)

$M\models\exists yA(x,y)\Leftrightarrow(T^{h})^{M}(x)$ is illfounded.

  3. (3)

$M\models\exists yA(x,y)\Rightarrow\exists yA(x,y)$.

  4. (4)

$\exists yA(x,y)\not\Rightarrow M\models\exists yA(x,y)$.

  5. (5)

$(y,f)\in[(T^{h})^{M}]\Rightarrow(y,\sigma^{M}[f])\in[T^{h}]$.

######  Definition 4.10.

Let $M$ be a countable, transitive, projectively correct model of
$\operatorname{ZFC}$. Let $h$ be a a $\Pi^{1}_{\omega}$ code and let
$x\in\mathbb{R}\cap M$. Let $T=(T^{h}(x))^{M}$, so that $T\in M$ and
$M\models``T$ is a $\boldsymbol{\Pi}^{1}_{\omega}$ tree.” Then we say that
$(T,h,x,M)$ is _correctly wellfounded_ iff $(T^{h}(x))^{V}$ is wellfounded
(and consequently $T=(T^{h}(x))^{M}$ is wellfounded.) We say that $(T,h,x,M)$
is _incorrectly wellfounded_ iff $T$ is wellfounded but $(T^{h}(x))^{V}$ is
illfounded.

We will often write that $(T^{h}(x))^{M}$ is correctly or incorrectly
wellfounded as an abbreviation for the statement that
$\left((T^{h}(x))^{M},h,x,M\right)$ is.

We also extend these definitions to the case where $M$ itself is not a model
of all of $\operatorname{ZFC}$ but $M$ has a rank initial segment that is a
model of $\operatorname{ZFC}$. Further we extend these definitions to the case
where $M$ is not wellfounded but has a rank initial segment in its wellfounded
part that is a model of $\operatorname{ZFC}$.

The key to our proof that $M^{\text{ld}}$ is
quasi-$\Sigma^{1}_{\omega+1}$-correct is this: We will show that if $T$ is a
$\boldsymbol{\Pi}^{1}_{\omega}$-tree in $M^{\text{ld}}$ then $M^{\text{ld}}$
is able to determine whether or not $T$ is correctly wellfounded.

######  Theorem 4.11.

Let $z$ be a real and $M$ a countable, iterable, ladder-mouse over $z$.
Suppose $T_{1}=(T^{h_{1}}(x_{1}))^{M}$ and $T_{2}=(T^{h_{2}}(x_{2}))^{M}$ are
$\boldsymbol{\Pi}^{1}_{\omega}$-trees of $M$. Suppose $(T_{1},h_{1},x_{1},M)$
is correctly wellfounded and $(T_{2},h_{2},x_{2},M)$ is incorrectly
wellfounded. Then $\operatorname{rank}(T_{1})<\operatorname{rank}(T_{2})$.

######  Remark 4.12.

Let $M$ be as in the theorem and let $T$ be a
$\boldsymbol{\Pi}^{1}_{\omega}$-tree of $M$. It follows from the theorem that
we can make sense of saying that $T$ is correctly or incorrectly wellfounded
without reference to an $h$ or $x$. For if
$T=(T^{h_{1}}(x_{1}))^{M}=(T^{h_{2}}(x_{2}))^{M}$, by the theorem it is not
possible for $(T,h_{1},x_{1},M)$ to be correctly wellfounded and
$(T,h_{2},x_{2},M)$ to be incorrectly wellfounded.

Now, assuming Theorem 4.11, we prove Theorem 3.7.

###### proof of Theorem 3.7.

Let $z$ be a real and $M$ a countable, iterable, ladder-mouse over $z$. Let
$\theta$ be the minimum of the ranks of all $\boldsymbol{\Pi}^{1}_{\omega}$
trees in $M$ that are incorrectly wellfounded. If there are no incorrectly
wellfounded $\boldsymbol{\Pi}^{1}_{\omega}$ trees in $M$ then $M$ is
$\Sigma^{1}_{\omega+1}$-correct and in this case let $\theta=0$.

$M$ is a model of $\operatorname{PD}$ and also is a strongly acceptable model
of $\operatorname{GCH}$. So
$(\omega_{1})^{M}<(\boldsymbol{\delta}^{1}_{\omega})^{M}<(\omega_{2})^{M}$. So
all $\boldsymbol{\Pi}^{1}_{\omega}$ trees in $M$ are in
$J^{M}_{(\omega_{2})^{M}}$ and either have a branch in
$J^{M}_{(\omega_{2})^{M}}$ or have rank less than $(\omega_{2})^{M}$. So
$\theta<(\omega_{2})^{M}$.

Let $x\in\mathbb{R}\cap M$ and let $h$ be a $\Pi^{1}_{\omega}$ code. Then

$(\exists y)A^{h}(x,y)\Leftrightarrow J^{M}_{(\omega_{2})^{M}}\models
T^{h}(x)\text{ is
illfounded}\,\vee\,(\theta>0\,\wedge\,\operatorname{rank}(T^{h}(x))\geq\theta.)$

∎

Previously we proved Theorem 3.6 based on Theorem 3.7. But it is also possible
to give a proof of Theorem 3.6 based on Theorem 4.11 without going through
Theorem 3.7 and thus avoiding mentioning the parameter $\theta$.

###### alternate proof of Theorem 3.6.

Let $x\in Q_{\omega+1}$. We need to show that $x\in M^{\text{ld}}$.

For $i\in\omega$ let $\bar{i}=\left\langle
i,0,0,\cdots\right\rangle\in\mathbb{R}$. Thus we are representing integers as
reals. This allows us to avoid having to define what we mean by the tree
through $i$, $T(i)$, when $T$ is a tree and $i$ an integer.

Fix a countable ordinal $\alpha$ such that $x$ is $\Delta^{1}_{\omega+1}$ in
every representative of $\alpha$.

Let $M^{\prime}$ be the $(\alpha+1)$-th iterate of $M^{\text{ld}}$ by its
least measurable cardinal. So $M^{\prime}$ is a countable, iterable ladder
mouse with a ladder whose $\delta_{0}$ is greater than $\alpha$. Let $g$ be
$\operatorname{Coll}(\alpha,\omega)$-generic over $M^{\prime}$. It suffices to
see that $x\in M^{\prime}[g]$.

We can rearrange $M^{\prime}[g]$ as a mouse over $z$, where $z$ is some real
coding $g$. Let $M$ be this mouse over $z$. $M$ is a countable, iterable
ladder-mouse over $z$ and there is a real $w\in M\cap\operatorname{WO}$ with
$|w|=\alpha$.

Fix two $\Pi^{1}_{\omega}$ codes $h_{1},h_{2}$ such that for all
$i,j\in\omega$,

$x(i)=j\Leftrightarrow(\exists
y\in\mathbb{R})A^{h_{1}}(\bar{i},\bar{j},w,y)\Leftrightarrow(\forall
y\in\mathbb{R})\neg A^{h_{2}}(\bar{i},\bar{j},w,y).$

Claim. For all $i,j\in\omega$,

$x(i)=j\Leftrightarrow M\models T^{h_{1}}(\bar{i},\bar{j},w)\text{ is
illfounded
}\,\vee\,\operatorname{rank}(T^{h_{2}}(\bar{i},\bar{j},w))<\operatorname{rank}(T^{h_{1}}(\bar{i},\bar{j},w)).$

###### Proof of Claim..

First suppose $x(i)=j$. Then $T^{h_{1}}(\bar{i},\bar{j},w)$ is illfounded and
$T^{h_{2}}(\bar{i},\bar{j},w)$ is well-founded. By Remarks 4.9,
$(T^{h_{2}}(\bar{i},\bar{j},w))^{M}$ is correctly-wellfounded. If
$M\not\models T^{h_{1}}(\bar{i},\bar{j},w)\text{ is illfounded }$, then
$(T^{h_{1}}(\bar{i},\bar{j},w))^{M}$ is incorrectly-wellfounded and so by
Theorem 4.11,
$\operatorname{rank}(T^{h_{2}}(\bar{i},\bar{j},w))^{M}<\operatorname{rank}(T^{h_{1}}(\bar{i},\bar{j},w))^{M}$.

Conversely, suppose $x(i)\not=j$. Then $T^{h_{1}}(\bar{i},\bar{j},w)$ is well-
founded and $T^{h_{2}}(\bar{i},\bar{j},w)$ is illfounded. By Remarks 4.9,
$(T^{h_{1}}(\bar{i},\bar{j},w))^{M}$ is correctly well-founded and
$(T^{h_{2}}(\bar{i},\bar{j},w))^{M}$ is either illfounded or incorrectly well-
founded. By Theorem 4.11,
$M\not\models\operatorname{rank}(T^{h_{2}}(\bar{i},\bar{j},w))<\operatorname{rank}(T^{h_{1}}(\bar{i},\bar{j},w))$.
∎

So $x\in M$.

∎

In the next section we turn to the proof of Theorem 4.11.

##  5\. Main Proof

This section is devoted to a proof of Theorem 4.11.

Let $z$ be a real and $M$ a countable, iterable, ladder-mouse over $z$.
Suppose $M\models``T_{1}$ and $T_{2}$ are
$\boldsymbol{\Pi}^{1}_{\omega}$-trees” and suppose $T_{1}$ is correctly
wellfounded and $T_{2}$ is incorrectly wellfounded. We want to prove that
$\operatorname{rank}(T_{1})<\operatorname{rank}(T_{2})$. To simplify the
notation in this section we will assume $z=0$. The proof with arbitrary $z$ is
the same, but relativized to $z$.

Theorem 4.11 will follow from the three Lemmas below: Lemmas 5.1, 5.2 and 5.4.

######  Lemma 5.1.

Let $M$ be a countable, iterable, ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$. Let $y\in\mathbb{R}$. Then there is a non-dropping
iteration $\pi:M\to M^{\prime}$ such that, letting
$\left\langle\,\delta^{\prime}_{n},\gamma^{\prime}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be the image under $\pi$ of the ladder, we have that $M^{\prime}$ is a
countable, iterable ladder-mouse with
$\left\langle\,\delta^{\prime}_{n},\gamma^{\prime}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M^{\prime}$ and for each $n\in\omega$ there is a $g_{n}$ such
that $g_{n}$ is $\operatorname{Coll}(\delta^{\prime}_{n},\omega)$-generic over
$J^{M^{\prime}}_{\gamma^{\prime}_{n}}$ and $y\in
J^{M^{\prime}}_{\gamma^{\prime}_{n}}[g_{n}]$.

The iteration described in the previous lemma will come from performing a
Woodin genericity iteration over each of the levels of the ladder
sequentially.

######  Lemma 5.2.

Let $\Gamma=\Sigma^{1}_{n}$ for some $n>0$. Let $M$ be a countable, transitive
model of $\operatorname{ZFC}$ and $\delta>\omega$ a cardinal of $M$. Suppose
that whenever $\mathbb{P}\in V^{M}_{\delta+1}$ is a poset and $g$ is
$\mathbb{P}$-generic over $M$, then $M[g]$ is $\Gamma$-correct (in $V$). Let
$g_{1}$ and $g_{2}$ be $\operatorname{Coll}(\delta,\omega)$-generic over $M$.
Let $E\subset\mathbb{R}^{2}$ be a $\Gamma$, thin equivalence relation. Then
$M[g_{1}]$ and $M[g_{2}]$ meet the same $E$-equivalence classes.

######  Note.

In the above Lemma we could replace $\Sigma^{1}_{n}$ with any pointclass
$\Gamma$ for which we can make sense of the notion of $\Gamma$-correctness.

We will use Lemma 5.2 via the following corollary.

######  Corollary 5.3.

Fix $n\in\omega$. Let $M$ be a countable, transitive model of
$\operatorname{ZFC}$ and $\delta>\omega$ a cardinal of $M$. Suppose that
whenever $\mathbb{P}\in V^{M}_{\delta+1}$ is a poset and $g$ is
$\mathbb{P}$-generic over $M$, then $M[g]$ is $\Sigma^{1}_{2n+4}$-correct. Let
$g_{1}$ and $g_{2}$ be $\operatorname{Coll}(\delta,\omega)$-generic over $M$.

Let $h$ be a $\Pi^{1}_{\omega}$ code, let $y\in M[g_{1}]$ and suppose that
$y\in\bigcap\limits_{i<n}G^{i}_{h(i)}$. Then there is a $y^{\prime}\in
M[g_{2}]$ such that

  * •

$y^{\prime}\\!\\!\upharpoonright\\!n=y\\!\\!\upharpoonright\\!n$

  * •

$y^{\prime}\in\bigcap\limits_{i<n}G^{i}_{h(i)}$.

  * •

For $i<n$,
$\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y^{\prime})=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y)$

###### Proof.

Consider the binary relation $E$ on $\mathbb{R}^{2}$ given by

$\displaystyle E(z,w)\Leftrightarrow$ $\displaystyle
z,w\notin\bigcap\limits_{i<n}G^{i}_{h(i)}\vee\Bigg{(}z,w\in\bigcap\limits_{i<n}G^{i}_{h(i)}\,\wedge$
$\displaystyle\left(z\\!\\!\upharpoonright\\!n,\left\langle\,\varphi^{i_{0}}_{h(i_{0}),i_{1}}(z)\,\left|\,i<n\,\right.\right\rangle\right)=\left(w\\!\\!\upharpoonright\\!n,\left\langle\,\varphi^{i_{0}}_{h(i_{0}),i_{1}}(w)\,\left|\,i<n\,\right.\right\rangle\right)\Bigg{)}.$

Clearly $E$ is a $\Sigma^{1}_{2n+4}$ equivalence relation on $\mathbb{R}$.
($\Sigma^{1}_{2n+4}$ is more than we need—we are not claiming it is optimal.)

We claim that $E$ is thin. To see this, suppose towards a contradiction that
$S$ is a perfect set of $E$-inequivalent reals. There is a natural wellorder
$<$ on $S$ defined by

$\displaystyle z<w\Leftrightarrow$ $\displaystyle
z\in\bigcap\limits_{i<n}G^{i}_{h(i)}\wedge\Bigg{(}w\notin\bigcap\limits_{i<n}G^{i}_{h(i)}\,\vee\bigg{[}w\in\bigcap\limits_{i<n}G^{i}_{h(i)}\,\wedge$
$\displaystyle\left(z\\!\\!\upharpoonright\\!n,\left\langle\,\varphi^{i_{0}}_{h(i_{0}),i_{1}}(z)\,\left|\,i<n\,\right.\right\rangle\right)<_{\text{lex}}\left(w\\!\\!\upharpoonright\\!n,\left\langle\,\varphi^{i_{0}}_{h(i_{0}),i_{1}}(w)\,\left|\,i<n\,\right.\right\rangle\right)\bigg{]}\Bigg{)}.$

Since $<$ is projectively definable and $S$ is perfect, this gives a
projectively definable wellorder of $\mathbb{R}$. But that contradicts
$\operatorname{PD}$. So $E$ is thin.

By Lemma 5.2 with $\Gamma=\Sigma^{1}_{2n+4}$, there is a $y^{\prime}\in
M[g_{2}]$ such that $E(y_{1},y_{2})$. ∎

######  Lemma 5.4.

Let $M$ be a countable, iterable ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$ and
$\gamma=\sup\left\\{\,\gamma_{n}\,\left|\,n\in\omega\,\right.\right\\}$. Then
there is a countable, not-necessarily-wellfounded model $M^{*}$ and a
$\Sigma_{0}$-elementary-embedding

$\pi:M\to M^{*}$

such that

  * •

$\operatorname{crit}(\pi)=(\omega_{1})^{M}$.

  * •

$\pi((\omega_{1})^{M})=\gamma\in\operatorname{wfp}(M^{*})$.

  * •

$M^{*}$ is projectively correct.

  * •

$J^{M}_{\gamma_{n}}\in M^{*}$, for each $n$.

The embedding described in the previous lemma will come from taking a
stationary tower ultrapower.

Now, assuming Lemmas 5.1, 5.2 and 5.4, we will give a proof of Theorem 4.11.

###### proof of Theorem 4.11.

As mentioned at the top of the section, we will assume that the real parameter
$z=0$ to simplify the notation. Let $M$ be a countable, iterable ladder-mouse.

Suppose $T_{1}=(T^{h_{1}}(x_{1}))^{M}$ and $T_{2}=(T^{h_{2}}(x_{2}))^{M}$ are
$\boldsymbol{\Pi}^{1}_{\omega}$-trees of $M$. Suppose $(T_{1},h_{1},x_{1},M)$
is correctly wellfounded and $(T_{2},h_{2},x_{2},M)$ is incorrectly
wellfounded. We want to prove that
$\operatorname{rank}(T_{1})<\operatorname{rank}(T_{2})$.

To simplify the notation again, we will assume that $x_{1}=x_{2}=0$.

By assumption, in $V$, $T^{h_{1}}$ is wellfounded and $T^{h_{2}}$ is
illfounded. Let $y\in
p[T^{h_{2}}]=A^{h_{2}}=\bigcap\limits_{n}G^{n}_{h_{2}(n)}$. So then

$\left(y,\left\langle\,\varphi^{n_{0}}_{h_{2}(n_{0}),n_{1}}(y)\,\left|\,n\in\omega\,\right.\right\rangle\right)\in[T^{h_{2}}].$

Let
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be a ladder over $M$. By applying Lemma 5.1 and replacing $M$ with the
$M^{\prime}$ of that lemma, we may assume that we have, for each $n$, $g_{n}$
$\operatorname{Coll}(\delta_{n},\omega)$-generic over $J^{M}_{\gamma_{n}}$
such that $y\in J^{M}_{\gamma_{n}}[g_{n}]$.

Now we apply Lemma 5.4 to obtain $\pi:M\to M^{*}$ as in the lemma. Let
$T_{1}^{*}=\pi(T_{1}),T_{2}^{*}=\pi(T_{2})$. It suffices to see that
$M^{*}\models\text{``}\operatorname{rank}(T_{1}^{*})<\operatorname{rank}(T_{2}^{*})$”.

Since $M^{*}$ is projectively correct, it follows that
$(\boldsymbol{\delta}^{1}_{n})^{M^{*}}\in\operatorname{wfp}(M^{*})$ for each
$n$, and so
$(\boldsymbol{\delta}^{1}_{\omega})^{M^{*}}\in\operatorname{wfp}(M^{*})$. So
$T_{1}^{*},T_{2}^{*}\in\operatorname{wfp}(M^{*})$ and
$T_{1}^{*}=(T^{h_{1}})^{M^{*}}$ and $T_{2}^{*}=(T^{h_{2}})^{M^{*}}$.

In Definition 4.6 and Lemma 4.7 we assumed that $M$ was a countable,
transitive, projectively correct model of $\operatorname{ZFC}$. But the
definition and lemma work also for $M^{*}$ even though $M^{*}$ is not
necessarily wellfounded and $M^{*}$ is not necesarily a model of
$\operatorname{ZFC}$, but rather it has rank initial segments that are. Thus
we have
$\sigma^{M^{*}}:(\boldsymbol{\delta}^{1}_{\omega})^{M^{*}}\to\boldsymbol{\delta}^{1}_{\omega}$,
and we have that the extensions $\sigma^{M^{*},h_{1}}:T_{1}^{*}\rightarrow
T^{h_{1}}$ and $\sigma^{M^{*},h_{2}}:T_{2}^{*}\rightarrow T^{h_{2}}$ are tree
embeddings.

$M^{*}\models\text{``}T_{1}^{*}\text{ and }T_{2}^{*}$ are wellfounded”. But
$M^{*}$ is not necessarily wellfounded so it doesn’t necessarily follow that
$T_{1}^{*}$ and $T_{2}^{*}$ are really wellfounded. Now $T^{h_{1}}$ is
wellfounded and since we have $\sigma^{M^{*},h_{1}}:T_{1}^{*}\rightarrow
T^{h_{1}}$, we _do_ know that $T_{1}^{*}$ is wellfounded. This implies that
$\operatorname{rank}(T_{1}^{*})\in\operatorname{wfp}(M^{*})$. To see that
$M^{*}\models\text{``}\operatorname{rank}(T_{1}^{*})<\operatorname{rank}(T_{2}^{*})$”
it suffices to see that
$\operatorname{rank}(T_{2}^{*})\notin\operatorname{wfp}(M^{*})$, or
equivalently that $T_{2}^{*}$ is not wellfounded. We will show that $y\in
p[T_{2}^{*}]$. (Thus we are also showing that in the present circumstance,
$M^{*}$ is necessarily illfounded.) This follows from the following

Claim. For all $n>0$, there is a $y_{n}\in\mathbb{R}\cap M^{*}$ such that
$y_{n}\\!\\!\upharpoonright\\!n=y\\!\\!\upharpoonright\\!n$,
$y_{n}\in\bigcap\limits_{i<n}G^{i}_{h_{2}(i)}$, and for all $i<n$,
$\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y_{n})=\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y)$.

Assuming the claim,
$\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y)\in\operatorname{ran}(\sigma^{M^{*}})$
for all $i$ and letting $\alpha_{i}$ be such that
$\sigma^{M^{*}}(\alpha_{i})=\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y)$ we have
that for all $n$,

$\left(y\\!\\!\upharpoonright\\!n,\left\langle\,\alpha_{i}\,\left|\,i<n\,\right.\right\rangle\right)=\left(y_{n}\\!\\!\upharpoonright\\!n,\left\langle\,(\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y_{n}))^{M^{*}}\,\left|\,i<n\,\right.\right\rangle\right)\in[T^{*}_{2}]$

and so

$\left(y,\left\langle\,\alpha_{i}\,\left|\,i\in\omega\,\right.\right\rangle\right)\in[T^{*}_{2}].$

To finish the proof of Theorem 4.11 we give the

###### proof of the claim.

Fix $n$ and let $N=J^{M}_{\gamma_{2n+2}}$. $N$ has $2n+2$ Woodin cardinals
above $\delta_{2n+2}$, so if $\mathbb{P}\in N$ is a poset with
$\mathbb{P}\subset V^{N}_{\delta_{2n+2}}$ and $g$ is $\mathbb{P}$-generic over
$N$, then $N[g]$ is $\Sigma^{1}_{2n+4}$-correct.

Since $N$ is countable in $M^{*}$, there is a $g\in M^{*}$ that is
$\operatorname{Coll}(\delta_{2n+2},\omega)$-generic over $N$. By Corollary 5.3
there is a $y^{\prime}\in N[g]$ such that
$y^{\prime}\\!\\!\upharpoonright\\!n=y\\!\\!\upharpoonright\\!n$,
$y^{\prime}\in\bigcap\limits_{i<n}G^{i}_{h_{2}(i)}$, and for all $i<n$,
$\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y^{\prime})=\varphi^{i_{0}}_{h_{2}(i_{0}),i_{1}}(y)$.
Let $y_{n}=y^{\prime}$.

∎

And that completes the proof of Theorem 4.11.

∎

In the remainder of this section we discharge our obligation to give the
proofs of Lemmas 5.1, 5.2 and 5.4.

###### sketch of proof of Lemma 5.1.

Lemma 5.1 is not a new result: it was essentially proven in Section 4 of
[Rud97]. We give only a sketch here.

Since $\delta_{0}$ is Woodin in $J^{M}_{\gamma_{0}}$, we can perform the usual
genericity iteration on $J^{M}_{\gamma_{0}}$ (see Theorem 7.14 from [Ste10]),
yielding $\bar{\pi}_{0}:J^{M}_{\gamma_{0}}\to P_{1}$ such that there is a $g$,
$\operatorname{Coll}(\bar{\pi}_{0}(\delta_{0}),\omega)$-generic over $P_{1}$
such that $y\in P_{1}[g]$.

Since $\delta_{0}$ a cardinal in $M$, a total extender over
$J^{M}_{\delta_{0}}$ is total over $M$. A similar fact is true about the image
of $\delta_{0}$ under non-dropping iterations of $M$. Therefore the
$\bar{\pi}_{0}$-iteration of $J^{M}_{\gamma_{0}}$ induces a corresponding
iteration on all of $M$ yielding a non-dropping iteration $\pi_{0}:M\to
M^{\prime}_{1}$ such that letting
$\left\langle\,\delta^{(1)}_{n},\gamma^{(1)}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be the image under $\pi_{0}$ of the ladder of $M$, we have that
$M^{\prime}_{1}$ is a countable, iterable ladder-mouse with
$\left\langle\,\delta^{(1)}_{n},\gamma^{(1)}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M^{\prime}_{1}$ and there is a $g_{0}$ such that $g_{0}$ is
$\operatorname{Coll}(\delta^{(1)}_{0},\omega)$-generic over
$J^{M^{\prime}_{1}}_{\gamma^{(1)}_{0}}$ and $y\in
J^{M^{\prime}_{1}}_{\gamma^{(1)}_{0}}[g_{0}]$.

Now we repeat the argument of the previous two paragraphs using
$M^{\prime}_{1}$ instead of $M$ and using $\delta^{(1)}_{1}$ in place of
$\delta_{0}$. This yields a non-dropping iteration $\pi_{1}:M^{\prime}_{1}\to
M^{\prime}_{2}$. We can take the critical point of $\pi_{1}$ to be greater
than $\gamma^{(1)}_{0}$. Then, letting
$\left\langle\,\delta^{(2)}_{n},\gamma^{(2)}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be the image under $\pi_{1}$ of the ladder of $M^{\prime}_{1}$, we have that
$M^{\prime}_{2}$ is a countable, iterable ladder-mouse with
$\left\langle\,\delta^{(2)}_{n},\gamma^{(2)}_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M^{\prime}_{2}$ and for $n=0,1$ there is a $g_{n}$ such that
$g_{n}$ is $\operatorname{Coll}(\delta^{(2)}_{n},\omega)$-generic over
$J^{M^{\prime}_{2}}_{\gamma^{(2)}_{n}}$ and $y\in
J^{M^{\prime}_{2}}_{\gamma^{(2)}_{n}}[g_{n}]$.

We continue this process yielding $\pi_{n}:M^{\prime}_{n}\to M^{\prime}_{n+1}$
and let $\pi:M\to M^{\prime}$ be the direct limit.

∎

Next we turn to the proof of Lemma 5.2. The proof is based on similar
arguments by Greg Hjorth (see Lemma 2.2 from [Hjo97]) and Foreman and Magidor
(see Theorem 3.4 of [FM95].) See also Lemma 3.1 from [Sch14].

###### proof of Lemma 5.2.

Let $\Gamma=\Sigma^{1}_{n}$ for some $n>0$. Let $M$ be a countable, transitive
model of $\operatorname{ZFC}$ and $\delta>\omega$ a cardinal of $M$. Suppose
that whenever $\mathbb{P}\in V^{M}_{\delta+1}$ is a poset and $g$ is
$\mathbb{P}$-generic over $M$, then $M[g]$ is $\Gamma$-correct. Let $g_{1}$
and $g_{2}$ be $\operatorname{Coll}(\delta,\omega)$-generic over $M$. Let
$E\subset\mathbb{R}^{2}$ be a $\Gamma$, thin equivalence relation. We wish to
prove that $M[g_{1}]$ and $M[g_{2}]$ meet the same $E$-equivalence classes.

We may assume that $g_{1}\times g_{2}$ is
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$. This is because there is an h such that $h$ is
$\operatorname{Coll}(\delta,\omega)$-generic over both both $M[g_{1}]$ and
$M[g_{2}]$. If $M[g_{1}]$ and $M[h]$ meet the same $E$-equivalence classes and
$M[g_{2}]$ and $M[h]$ meet the same $E$-equivalence classes, then $M[g_{1}]$
and $M[g_{2}]$ meet the same $E$-equivalence classes.

So suppose towards a contradiction that $g_{1}\times g_{2}$ is
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$ and there is an $x\in\mathbb{R}\cap M[g_{1}]$ such that for all
$y\in\mathbb{R}\cap M[g_{2}]$, $\neg E(x,y)$.

Claim 1.  There is a name for a real $\tau\in M$ such that for all
$h_{1},h_{2}$ with $h_{1}\times h_{2}$
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$, $\neg E(\tau^{h_{1}},\tau^{h_{2}})$.

###### proof of Claim 1.

Let $\tau\in M$ be such that $x=\tau^{g_{1}}$. By hypothesis $M[g_{1}\times
g_{2}]$ is $\Gamma$-correct so $M[g_{1}\times g_{2}]\models\neg
E(\tau^{g_{1}},\tau^{g_{2}})$.

Let $\dot{g}_{\text{left}}$ and $\dot{g}_{\text{right}}$ be the canonical
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-names
in $M$ for the left and right component of the generic. Let $\check{\tau}$ be
the canonical
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-name
for $\tau\in M$. Then $\check{\tau}$, $\dot{g}_{\text{left}}$ and
$\dot{g}_{\text{right}}$ are all invariant under automorphisms of
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$.
Since
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$
is homogeneous

$1\overset{M}{\|\underset{\\!\\!\\!\\!\\!\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)}{\\!\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar\\!\\!\\!\relbar\\!\\!\relbar\\!\\!\relbar}}\neg
E(\check{\tau}^{\dot{g}_{\text{left}}},\check{\tau}^{\dot{g}_{\text{right}}}).$

Let $h_{1},h_{2}$ be such that $h_{1}\times h_{2}$ is
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$. Then $M[h\times h_{2}]\models\neg E(\tau^{h_{1}},\tau^{h_{2}})$ and
$M[h\times h_{2}]$ is $\Gamma$-correct so $\neg E(\tau^{h_{1}},\tau^{h_{2}})$.
∎

To obtain a contradiction and prove the lemma we will show

Claim 2.  There is a set $H$ such that for all $h_{1},h_{2}\in H$ with
$h_{1}\not=h_{2}$, $h_{1}\times h_{2}$ is
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$ and $\left\\{\,\tau^{h}\,\left|\,h\in H\,\right.\right\\}$ is a
perfect set of reals.

###### proof of Claim 2.

Let $\left\langle\,C_{n}\,\left|\,n=1,2,\cdots\,\right.\right\rangle$
enumerate the dense open subsets of
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$
in $M$ and let $D_{n}=\bigcap\limits_{i\leq n}C_{n}$. For $s\in 2^{<\omega}$
we define $p_{s}\in\operatorname{Coll}(\delta,\omega)$ such that

  1. (1)

$s\subset t\Rightarrow p_{s}\subset p_{t}$

  2. (2)

If $|s|=|t|=n$ and $s\not=t$, then $(p_{s},p_{t})\in D_{n}$

  3. (3)

If $|s|=n$ then $p_{s}$ decides $\tau\\!\\!\upharpoonright\\!n$.

It is easy to see that $p_{s}$ satisfying these conditions can be defined by
induction on the length of $s$. [Let $p_{\emptyset}=\emptyset$. Let $n\geq 1$
and suppose we have defined $p_{s}$ for all $s$ of length $n-1$. Let
$\left\langle\,(s,t)_{i}\,\left|\,i=1\cdots k\,\right.\right\rangle$ enumerate
all pairs $(s,t)\in{{}^{n}{2}\times{}^{n}{2}}$ such that $s\not=t$. Let
$p^{0}_{s}=p_{s\upharpoonright(n-1)}$ for all $s\in{}^{n}{2}$. For $i=1\cdots
k$, let $(s,t)=(s,t)_{i}$ and let $p^{i}_{s}$ and $p^{i}_{t}$ be chosen so
that $p^{i-1}_{s}\subset p^{i}_{s}$ and $p^{i-1}_{t}\subset p^{i}_{t}$ and
$(p^{i}_{s},p^{i}_{t})\in D_{n}$. For all $u\in{}^{n}{2}$ with $u\not=s$ and
$u\not=t$, let $p^{i}_{u}=p^{i-1}_{u}$. Then
$\left\\{\,p^{k}_{s}\,\left|\,s\in{}^{n}{2}\,\right.\right\\}$ maintains
induction hypotheses (1) and (2). For each $s$, choose $p_{s}$ extending
$p^{k}_{s}$ to also maintain induction hypothesis (3).]

For $x\in 2^{\omega}$ let $h_{x}$ be the filter on
$\operatorname{Coll}(\delta,\omega)$ generated by
$\left\\{\,p_{x\upharpoonright n}\,\left|\,n\in\omega\,\right.\right\\}$ and
let $H=\left\\{\,h_{x}\,\left|\,x\in 2^{\omega}\,\right.\right\\}$. If $x,y\in
2^{\omega}$ and $x\not=y$ then $h_{x}\times h_{y}$ is
$\operatorname{Coll}(\delta,\omega)\times\operatorname{Coll}(\delta,\omega)$-generic
over $M$, because for sufficiently large $n$,
$x\\!\\!\upharpoonright\\!n\not=y\\!\\!\upharpoonright\\!n$ so that
$(p_{x\upharpoonright n},p_{y\upharpoonright n})\in D_{n}$, and
$p_{x\upharpoonright n}\in h_{x}$ and $p_{y\upharpoonright n}\in h_{y}$.

Sine $x\to\tau^{h_{x}}$ is a continuous injection from Cantor space,
$\left\\{\,\tau^{h_{x}}\,\left|\,x\in 2^{\omega}\,\right.\right\\}$ is
perfect. ∎

By Claims 1 and 2 there is a perfect set of $E$-inequivalent reals,
contradicting the fact that $E$ is thin.

∎

Finally we turn to the proof of Lemma 5.4. This proof uses stationary tower
Forcing. See [Lar04] and page 653, Section 34 of [Jec03] for background.

Let $M$ be a countable ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$ and
$\gamma=\sup\left\\{\,\gamma_{n}\,\left|\,n\in\omega\,\right.\right\\}$.

For $n<\omega$, let

$\mathbb{Q}_{<\delta_{n}}^{M}=\left\\{\,a\in
J^{M}_{\delta_{n}}\,\left|\,M\models
a\subset\operatorname{\mathcal{P}}_{\omega_{1}}(J^{M}_{\delta_{n}})\text{ and
}a\text{ is stationary.}\,\right.\right\\}$

For $a,b\in\mathbb{Q}_{<\delta_{n}}^{M}$, say that $b\leq a$ iff $\cup
a\subseteq\cup b$ and $\bar{b}\subseteq a$ where $\bar{b}=\left\\{\,x\cap\cup
a\,\left|\,x\in b\,\right.\right\\}$.
$\big{(}\mathbb{Q}_{<\delta_{n}}^{M},<\big{)}$ is the countable stationary
tower forcing up to $\delta_{n}$ in $M$. Also let
$\mathbb{Q}_{<\gamma}^{M}=\bigcup\limits_{n<\omega}\mathbb{Q}_{<\delta_{n}}^{M}$.

######  Lemma 5.5.

Let $M$ be a countable ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$ and
$\gamma=\sup\left\\{\,\gamma_{n}\,\left|\,n\in\omega\,\right.\right\\}$. There
is a $G\subset\mathbb{Q}_{<\gamma}^{M}$ such that for $n\in\omega$, $G\cap
J^{M}_{\delta_{n}}$ is $\mathbb{Q}_{<\delta_{n}}^{M}$-generic over
$J^{M}_{\gamma_{n}}$.

First we show how Lemma 5.4 follows from Lemma 5.5

###### proof of Lemma 5.4..

Let $M$ be a countable, iterable ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$ and
$\gamma=\sup\left\\{\,\gamma_{n}\,\left|\,n\in\omega\,\right.\right\\}$. We
need to construct a countable, not-necessarily-wellfounded model $M^{*}$ and a
$\Sigma_{0}$-elementary-embedding

$\pi:M\to M^{*}$

such that

  * •

$\operatorname{crit}(\pi)=\omega_{1}^{M}$.

  * •

$\pi(\omega_{1}^{M})=\gamma\in\operatorname{wfp}(M^{*})$.

  * •

$M^{*}$ is projectively correct.

  * •

$J^{M}_{\gamma_{n}}\in M^{*}$, for each $n$.

$\pi:M\to M^{*}$ will be a stationary tower ultrapower.

Let $G$ be as in Lemma 5.5.

If $A$ is an infinite set in $J^{M}_{\gamma}$, let

$G\\!\\!\upharpoonright\\!A=G\cap\operatorname{\mathcal{P}}(\operatorname{\mathcal{P}}_{\omega_{1}}(A))^{M}=\left\\{\,a\in
G\,\left|\,\cup a=A\,\right.\right\\}.$

$G\\!\\!\upharpoonright\\!A$ is an $M$-ultrafilter on
$\operatorname{\mathcal{P}}_{\omega_{1}}(A)^{M}$.

We form the ultrapower $\operatorname{Ult}(M,G)$ by taking all functions $f\in
J^{M}_{\gamma}$ with
$\operatorname{dom}(f)=\operatorname{\mathcal{P}}_{\omega_{1}}(A)^{M}$ for
some infinite set $A\in J^{M}_{\gamma}$. If
$\operatorname{dom}(f)=\operatorname{\mathcal{P}}_{\omega_{1}}(A)^{M}$ and
$\operatorname{dom}(g)=\operatorname{\mathcal{P}}_{\omega_{1}}(B)^{M}$ we say
that $f\sim g$ iff

$\left\\{\,x\in\operatorname{\mathcal{P}}_{\omega_{1}}(A\cup
B)^{M}\,\left|\,f(x\cap A)=g(x\cap B)\,\right.\right\\}\in G$

and we say that $[f]E[g]$ iff

$\left\\{\,x\in\operatorname{\mathcal{P}}_{\omega_{1}}(A\cup
B)^{M}\,\left|\,f(x\cap A)\in g(x\cap B)\,\right.\right\\}\in G.$

We have $\Sigma_{0}$ Łos’s Theorem for $\operatorname{Ult}(M,G)$. We let
$M^{*}=\operatorname{Ult}(M,G)$, identifying the well-founded part of $M^{*}$
with the transitive set to which it is isomorphic.

Letting $\pi:M\to M^{*}$ be given by $\pi(x)=[c_{x}]$, where $c_{x}$ is the
constant $x$ function, we have that $\pi$ is $\Sigma_{0}$-elementary.

If $A$ is any infinite set in $J^{M}_{\gamma}$, the identity function on
$\operatorname{\mathcal{P}}_{\omega_{1}}(A)^{M}$ represents $\pi[A]$ and so
$A\in M^{*}$. Thus $J^{M}_{\gamma_{n}}\in M^{*}$, for each $n$ and
$\gamma\subseteq\operatorname{wfp}(M^{*})$.

If $\alpha<\gamma$ is any ordinal, the function $f$ with domain
$\operatorname{\mathcal{P}}_{\omega_{1}}(\alpha)^{M}$ given by
$f(x)=\operatorname{o.t.}(x)$ represents $\alpha$. It follows that $\alpha$ is
countable in $M^{*}$. So $\operatorname{crit}(\pi)=\omega_{1}^{M}$ and
$\pi(\omega_{1}^{M})\geq\gamma$.

Claim 1.  $\pi(\omega_{1}^{M})=\gamma$

###### Proof.

Let $f$ represent a countable ordinal in $\operatorname{Ult}(M,G)$. We will
show that $[f]_{G}<\gamma$. Let $A=\cup\operatorname{dom}(f)$ so that
$f:\operatorname{\mathcal{P}}_{\omega_{1}}^{M}(A)\to\omega_{1}^{M}$. We will
find a $\kappa<\gamma$ such that $f\in J^{M}_{\kappa}$, and such that

$\left\\{\,x\in\operatorname{\mathcal{P}}_{\omega_{1}}^{M}(J^{M}_{\kappa})\,\left|\,f(x\cap
A)\leq\operatorname{o.t.}(x\cap\kappa)\,\right.\right\\}\in G.$

Since the function $g$ given by $g(x)=\operatorname{o.t.}(x\cap\kappa)$
represents $\kappa$ in $\operatorname{Ult}(M,G)$, we will have that
$[f]_{G}\leq\kappa$.

Fix $n\in\omega$ so that $A,f,a\in J^{M}_{\delta_{n}}$. Work in
$J^{M}_{\gamma_{n}}$, where $\delta_{n}$ is Woodin. In particular,
$\delta_{n}$ is a limit of completely Jónsson cardinals.

Fix an arbitrary $a\in\mathbb{Q}^{M}_{\delta_{n}}$.

Let $\kappa<\delta_{n}$ be such that $\kappa$ is completely Jónsson in
$J^{M}_{\gamma_{n}}$ and $A,f,a\in J^{M}_{\kappa}$. Let $b$ be the set of
countable $X\subset J^{M}_{\kappa}$ such that

  * •

$X\cap(\cup a)\in a$

  * •

$f(X\cap A)\leq\operatorname{o.t.}(X\cap\kappa)$.

The proof of Theorem 2.7.8 of [Lar04] shows that $b$ is stationary. Thus
$b\leq a$.

Since $a$ was arbitrary in $\mathbb{Q}^{M}_{\delta_{n}}$ and $G\cap
J^{M}_{\delta_{n}}$ is $J^{M}_{\gamma_{n}}$-generic, there is such a $\kappa$
and $b$ with $b\in G\cap J^{M}_{\delta_{n}}$. For this $\kappa$,
$[f]_{G}\leq\kappa$. ∎

In particular, $\gamma$ is definable in $M^{*}$ and so
$\gamma\in\operatorname{wfp}(M^{*})$.

Claim 2.  $M^{*}$ is projectively correct.

###### Proof.

Let $x=[f]_{G}$ be a real in $\operatorname{Ult}(M,G)$ with $f\in
J^{M}_{\gamma}$. Let $A=\cup\operatorname{dom}(f)$ so that
$f:\operatorname{\mathcal{P}}_{\omega_{1}}^{M}(A)\to\mathbb{R}$. Let $n$ be
such that $f\in J^{M}_{\delta_{n}}$. Then $f$ also represents $x$ in
$\operatorname{Ult}(J^{M}_{\gamma_{n}},G\cap J^{M}_{\delta_{n}})$.

Since $\delta_{n}$ is Woodin in $J^{M}_{\gamma_{n}}$,

$\mathbb{R}\cap\operatorname{Ult}(J^{M}_{\gamma_{n}},G\cap
J^{M}_{\delta_{n}})=\mathbb{R}\cap J^{M}_{\gamma_{n}}[G\cap
J^{M}_{\delta_{n}}].$

So we have that

$\mathbb{R}\cap M^{*}=\bigcup\limits_{n}\mathbb{R}\cap
J^{M}_{\gamma_{n}}[G\cap J^{M}_{\delta_{n}}].$

Since $M$ is iterable, $J^{M}_{\gamma_{2n}}[G\cap J^{M}_{\delta_{2n}}]$ is
$\Sigma^{1}_{2n+1}$-correct.

Since correctness is a closure property, it follows that $\mathbb{R}\cap
M^{*}$ is projectively correct. ∎

This completes the proof of Lemma 5.4 assuming Lemma 5.5.

∎

Finally we give the proof of Lemma 5.5, which completes the proof of Lemma
5.4, and thus the proof of Theorem 4.11, and thus the proof of Theorem 3.7 and
thus the proof of Theorem 3.6.

Here we will use more details about stationary tower forcing. For the reader’s
convenience we copy here some of the definitions and results from [Lar04] and
Section 34 of [Jec03]

######  Definition 5.6.

Let $\kappa$ be inaccessible. Then

$\mathbb{Q}_{<\kappa}=\left\\{\,a\in V_{k}\,\left|\,a\text{ is stationary and
}\forall x\in a,x\text{ is countable.}\,\right.\right\\}$

######  Definition 5.7.

Let $\kappa$ be inaccessible and let $D\subset\mathbb{Q}_{<\kappa}$ be dense.
Let $y$ be countable. We say that $y$ _captures_ $D$ iff there is a $p\in D$
such that $p\in y$ and $y\cap\cup p\in p$.

######  Definition 5.8.

We say that $y$ _end-extends_ $x$ if $y\cap V_{\operatorname{rank}(x)}=x$.

######  Definition 5.9.

Let $\kappa$ be inaccessible and let $D\subset\mathbb{Q}_{<\kappa}$ be dense.
Then $\operatorname{sp}(D)=$ the set of countable $x\prec V_{\kappa+1}$ such
that there exists a countable $y\prec V_{\kappa+1}$ such that $x\subseteq
y,y\text{ end-extends }x\cap V_{\kappa}$ and $y\text{ captures }D.$

######  Definition 5.10.

Let $\kappa$ be inaccessible and let $D\subset\mathbb{Q}_{<\kappa}$ be dense.
We say that $D$ is _semi-proper_ if $\operatorname{sp}(D)$ contains a club.

######  Lemma 5.11.

Let $\kappa$ be inaccessible and let $D\subset\mathbb{Q}_{<\kappa}$ be dense
and semi-proper. Let $\mu>\kappa$ be inaccessible. Then for any countable
$x\prec V_{\mu}$ such that $D\in x$, $\exists y\prec V_{\mu}$ s.t. $y$ is
countable, $x\subseteq y,y\text{ end-extends }x\cap V_{\kappa}$ and $y\text{
captures }D$.

###### Proof.

(cf Lemma 2.5.4 of [Lar04] and Lemma 34.12 of [Jec03].) Let $x\prec V_{\mu}$
be countable with $D\in x$. Since $D$ is cofinal in $V_{\kappa}$, $\kappa\in
x$. Since $D$ is semi-proper, there is an $f:V_{\kappa+1}^{<\omega}\rightarrow
V_{\kappa+1}$ such that for all countable $z\subset V_{\kappa+1}$, if $z$ is
closed under $f$ then $z\in\operatorname{sp}(D)$. Since $x\prec V_{\mu}$,
there is such an $f\in x$. Fix such an $f$. Then $x\cap V_{\kappa+1}$ is
closed under $f$ so $x\cap V_{\kappa+1}\in\operatorname{sp}(D)$. Let $\bar{y}$
witness this, i.e. $\bar{y}$ is countable, $\bar{y}\prec V_{\kappa+1},x\cap
V_{\kappa+1}\subseteq\bar{y},\bar{y}\text{ end-extends }x\cap V_{\kappa}$ and
$\bar{y}\text{ captures }D$. Let

$y=\left\\{\,h(w)\,\left|\,\operatorname{dom}(h)=V_{\kappa}\wedge h\in x\wedge
w\in\bar{y}\cap V_{\kappa}\,\right.\right\\}.$

Then $y$ is countable, $y\prec V_{\mu}$ and $x\subset y$. Furthermore, since
$x\cap V_{\kappa+1}\subset\bar{y}$, $y\cap V_{\kappa}=\bar{y}\cap V_{\kappa}$.
So $y\text{ end-extends }x\cap V_{\kappa}$ and $y\text{ captures }D$. ∎

######  Lemma 5.12.

Let $\delta$ be Woodin. Let $D\subset\mathbb{Q}_{<\delta}$ be dense. Then
there are cofinally many inaccessible $\kappa<\delta$ such that $D\cap
V_{\kappa}$ is dense in $\mathbb{Q}_{<\kappa}$ and semi-proper.

###### Proof.

This is Lemma 34.13 of [Jec03] and also follows from Theorem 2.7.5 of [Lar04].
∎

###### proof of Lemma 5.5.

Let $M$ be a countable ladder-mouse with
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
a ladder over $M$ and
$\gamma=\sup\left\\{\,\gamma_{n}\,\left|\,n\in\omega\,\right.\right\\}$. We
must show that there is a $G\subset\mathbb{Q}_{<\gamma}^{M}$ such that for
$n\in\omega$, $G\cap J^{M}_{\delta_{n}}$ is
$\mathbb{Q}_{<\delta_{n}}^{M}$-generic over $J^{M}_{\gamma_{n}}$.

Claim 1.  Fix $n\in\omega$ and $a\in\mathbb{Q}_{<\delta_{n}}^{M}$. Let
$\lambda=\delta_{n}+\omega$. Working in $M$, let $b=\\{x\prec V_{\lambda}\mid
x$ is countable and $x\cap\cup a\in a$ and $\forall D\in x\cap
J^{M}_{\gamma_{n}}$ such that $D$ is a dense subset of
$\mathbb{Q}_{<\delta_{n}}^{M}$, $x$ captures $D\\}$. Then $M\models b$ is
stationary.

###### proof of Claim 1..

Work in $M$. Let $f:V_{\lambda}^{<\omega}\rightarrow V_{\lambda}$, and we will
construct an $x\in b$ closed under $f$.

Let $\mu$ be the least inaccessible greater than $\delta_{n}$. Since
$\delta_{n+1}$ is a limit of inaccessibles in $M$, we have
$\delta_{n}<\lambda<\mu<\delta_{n+1}$. We will find a countable $x\prec
V_{\mu}$ so that $f\in x$, $x\cap\cup a\in a$ and $\forall D\in x\cap
J^{M}_{\gamma_{n}}$ such that $D$ is a dense subset of
$\mathbb{Q}_{<\delta_{n}}^{M}$, $x$ captures $D$. Then $x\cap V_{\lambda}\in
b$ and $x\cap V_{\lambda}$ is closed under $f$.

We will construct $x$ as an increasing union $x=\bigcup\limits_{i}x_{i}$.

Let $x_{0}\prec V^{M}_{\mu}$ be countable, with $f\in x_{0}$ and such that
$x_{0}\cap\cup a\in a$.

Let $\left\langle\,D_{0}^{j}\,\left|\,j\in\omega\,\right.\right\rangle$
enumerate the dense subsets of $\mathbb{Q}_{<\delta_{n}}^{M}$ in $x_{0}\cap
J^{M}_{\gamma_{n}}$.

Temporarily working in $J^{M}_{\gamma_{n}}$ where $\delta_{n}$ is Woodin,
Lemma 5.12 gives us that there are cofinally many inaccessible cardinals
$\kappa<\delta_{n}$ such that $D^{0}_{0}\cap V_{\kappa}$ is dense and semi-
proper in $\mathbb{Q}_{<\kappa}$. Fix such a $\kappa_{0}$ with $a\in
V_{\kappa_{0}}$.

As the definition of semi-proper is local, it remains true in the full $M$
that $\kappa_{0}$ is inaccessible and that $D^{0}_{0}\cap V_{\kappa_{0}}$ is
dense and semi-proper in $\mathbb{Q}_{<\kappa_{0}}$. Now working in the full
$M$ we apply Lemma 5.11 to conclude that there is a countable $x_{1}\prec
V_{\mu}$ s.t. $x_{0}\subseteq x_{1},x_{1}\text{ end-extends }x_{0}\cap
V_{\kappa_{0}}$ and $x_{1}\text{ captures }D_{0}^{0}\cap V_{\kappa_{0}}$.

Let $\left\langle\,D_{1}^{j}\,\left|\,j\in\omega\,\right.\right\rangle$
enumerate the dense subsets of $\mathbb{Q}_{<\delta_{n}}^{M}$ in $x_{1}\cap
J^{M}_{\gamma_{n}}$.

Let $\pi(n)=(\pi_{0}(n),\pi_{1}(n))$ be a bijection from $\omega$ to
$\omega\times\omega$ such that for all $n$, $\pi_{0}(n)\leq n$, and
$\pi(0)=(0,0)$.

We will continue to define $x_{i},\kappa_{i}$ by induction.

Suppose that $t\geq 1$ and we have the following:

  1. (1)

$x_{0}\subseteq\cdots\subseteq x_{t}$, countable elementary submodels of
$V_{\mu}$

  2. (2)

$\left\langle\,D_{i}^{j}\,\left|\,j\in\omega\,\right.\right\rangle$ enumerates
the dense subsets of $\mathbb{Q}_{<\delta_{n}}^{M}$ in $x_{i}\cap
J^{M}_{\gamma_{n}}$, for $i=0,\cdots,t$.

  3. (3)

$\kappa_{i}<\delta_{n}$ is inaccessible and $D_{\pi_{0}(i)}^{\pi_{1}(i)}\cap
V_{\kappa_{i}}$ is dense and semi-proper for $i<t$.

  4. (4)

$x_{i+1}$ captures $D_{\pi_{0}(i)}^{\pi_{1}(i)}\cap V_{\kappa_{i}}$ for $i<t$.

  5. (5)

$x_{i+1}$ end-extends $x_{i}\cap V_{\kappa_{i}}$ for $i<t$.

  6. (6)

$\kappa_{i}>\kappa_{i-1}$ for $0<i<t$.

We will define $\kappa_{t}$, $x_{t+1}$, and
$\left\langle\,D_{t+1}^{j}\,\left|\,j\in\omega\,\right.\right\rangle$.

Temporarily working in $J^{M}_{\gamma_{n}}$ where $\delta_{n}$ is Woodin,
Lemma 5.12 gives us that there are cofinally many inaccessible cardinals
$\kappa<\delta_{n}$ such that $D_{\pi_{0}(t)}^{\pi_{1}(t)}\cap V_{\kappa}$ is
dense and semi-proper in $\mathbb{Q}_{<\kappa}$. Fix such a $\kappa_{t}$ with
$\kappa_{t}>\kappa_{t-1}$.

As the definition of semi-proper is local, it remains true in the full $M$
that $\kappa_{t}$ is inaccessible and that $D_{\pi_{0}(t)}^{\pi_{1}(t)}\cap
V_{\kappa_{t}}$ is dense and semi-proper in $\mathbb{Q}_{<\kappa_{t}}$. Now
working in the full $M$ we apply Lemma 5.11 to conclude that there is a
countable $x_{t+1}\prec V_{\mu}$ s.t. $x_{t}\subseteq x_{t+1},x_{t+1}\text{
end-extends }x_{t}\cap V_{\kappa_{t}}$ and $x_{t+1}\text{ captures
}D_{\pi_{0}(t)}^{\pi_{1}(t)}\cap V_{\kappa_{t}}$.

Let $\left\langle\,D_{t+1}^{j}\,\left|\,j\in\omega\,\right.\right\rangle$
enumerate the dense subsets of $\mathbb{Q}_{<\delta_{n}}^{M}$ in $x_{t+1}\cap
J^{M}_{\gamma_{n}}$.

Let $x=\bigcup\limits_{i}x_{i}$. $x$ is countable, $x\prec V_{\mu}$ and $f\in
x$. $x$ end-extends $x_{0}\cap V_{\kappa_{0}}$ so $x\cap\cup a\in a$. Let
$D\in x\cap J^{M}_{\gamma_{n}}$ be a dense subset of
$\mathbb{Q}_{<\delta_{n}}^{M}$. Then $D=D_{i}^{j}$ for some $i,j$. Let $t$ be
such that $\pi(t)=(i,j)$. Since $x_{t+1}\text{ captures }D\cap V_{\kappa_{t}}$
and $\kappa_{s}>\kappa_{t}$ for $s>t$ and $x_{s+1}$ end-extends $x_{s}\cap
V_{\kappa_{s}}$ for $s>t$, we have that $x_{s}$ captures $D\cap
V_{\kappa_{t}}$ for all $s\geq t$, so $x$ captures $D$. ∎

Claim 2.  For $n\in\omega$, for each $a\in\mathbb{Q}_{<\delta_{n}}^{M}$ there
is a $b\in\mathbb{Q}_{<\delta_{n+1}}^{M}$ such that $b<a$ and

$b\Vdash\dot{G}\cap\mathbb{Q}_{<\check{\delta}_{n}}\text{ is
}J^{M}_{\check{\gamma}_{n}}\text{ generic.}$

More precisely, for all $D\in J^{M}_{\gamma_{n}}$ such that $D$ is a dense
subset of $\mathbb{Q}_{<\delta_{n}}^{M}$, and for all
$c\in\mathbb{Q}_{<\gamma}^{M}$ with $c\leq b$, there is a $d\subseteq c$,
$d\in\mathbb{Q}_{<\gamma}^{M}$, and there is an $e\in D$ such that $d<e$.

###### proof of Claim 2..

Fix $n\in\omega$ and $a\in\mathbb{Q}_{<\delta_{n}}^{M}$. Let $b$ be as in
Claim 1. Then $b\in\mathbb{Q}_{<\delta_{n+1}}^{M}$ and $b<a$. Fix $D\in
J^{M}_{\gamma_{n}}$ such that $D$ is a dense subset of
$\mathbb{Q}_{<\delta_{n}}^{M}$, and fix $c\in\mathbb{Q}_{<\gamma}^{M}$ with
$c\leq b$.

Let $d^{\prime}=\left\\{\,x\in c\,\left|\,D\in x\,\right.\right\\}$. Then
$d^{\prime}$ is stationary and for all $x\in d^{\prime}$, since $x\cap
V^{M}_{\lambda}\in b$, (where $\lambda=\delta_{n}+\omega$) $x$ captures $D$.
There is an $e\in D$ such that, letting $d=\left\\{\,x\in
d^{\prime}\,\left|\,x\text{ captures }D\text{ at }e\,\right.\right\\}$, $d$ is
stationary. So $d<e$. ∎

To finish the proof of Lemma 5.5, let
$\left\langle\,D^{n}_{k}\,\left|\,k\in\omega\,\right.\right\rangle$ enumerate
the dense subsets of $\mathbb{Q}_{<\delta_{n}}^{M}$ in $J^{M}_{\gamma_{n}}$,
for each $n$. Define by induction on $n$ a sequence of conditions
$a_{n}\in\mathbb{Q}_{<\gamma}^{M}$ such that

  * •

$a_{n}\in\mathbb{Q}^{M}_{<\delta_{n}}$,

  * •

$a_{n+1}<a_{n}$,

  * •

For $m,k<n$, there is an $e\in D^{m}_{k}$ such that $a_{n+1}<e$.

Claim 2 lets us construct such a sequence. (Notice that when defining
$a_{n+1}$ we have an obligation to ensure that $a_{n+1}<e$ for some $e\in
D^{m}_{n}$, for $m<n$, not only for $m=n$. But Claim 2 is strong enough to
allow this.) Let $G=\left\\{\,a\in\mathbb{Q}_{<\gamma}^{M}\,\left|\,a\geq
a_{n}\text{ for some }n\,\right.\right\\}$. Then for $n\in\omega$, $G\cap
J^{M}_{\delta_{n}}$ is $\mathbb{Q}_{<\delta_{n}}^{M}$-generic over
$J^{M}_{\gamma_{n}}$.

∎

##  6\. Beyond the first pointclass

By combining the ideas of this paper with those from [Rud97], we believe that
it should be possible to prove the mouse set theorem for projective-like
pointclasses beyond $\Pi^{1}_{\omega+1}$. In this section we describe the
relationship between this paper and [Rud97] and we sketch a proof of Theorem
3.5.

Recall that
$\Sigma^{1}_{\omega+n}=\operatorname{\mathcal{P}}(\mathbb{R})\cap\Sigma_{n+1}(J_{2}(\mathbb{R}))$.
In [Rud97], the pointclasses were named differently. In that paper the
following pointclasses were defined. For $\alpha\geq 2$,

$\Sigma_{(\alpha,0)}=\operatorname{\mathcal{P}}(\mathbb{R})\cap\Sigma_{1}(J_{\alpha}(\mathbb{R})),$

and for $n\geq 0$

$\displaystyle\Pi_{(\alpha,n)}$ $\displaystyle=\neg\Sigma_{(\alpha,n)},$
$\displaystyle\Delta_{(\alpha,n)}$
$\displaystyle=\Sigma_{(\alpha,n)}\cap\Pi_{(\alpha,n)},$
$\displaystyle\Sigma_{(\alpha,n+1)}$
$\displaystyle=\exists^{\mathbb{R}}\Pi_{(\alpha,n)}.$

The $\Sigma^{1}_{\omega}$ of this paper is equal to the $\Sigma_{(2,0)}$ of
[Rud97], and the $\Pi^{1}_{\omega+1}$ of this paper is equal to the
$\Pi_{(2,1)}$ of [Rud97].

The pointclasses $\Sigma_{(\alpha,n)},\Pi_{(\alpha,n)},\Delta_{(\alpha,n)}$
are only interesting for $\alpha$ such that $\alpha$ begins a $\Sigma_{1}$-gap
(see [Ste83]) and $J_{\alpha}(\mathbb{R})$ is not admissible. In [Rud97] such
$\alpha$ were called _projective-like_.

For $\alpha$ projective-like and $n\geq 1$, [Rud97] defines

$A_{(\alpha,n)}=_{\text{def}}\left\\{\,x\in\mathbb{R}\,\left|\,\exists\xi<\omega_{1}\,x\in\Delta_{(\alpha,n)}(\xi)\,\right.\right\\}$

and defines

$A_{(\alpha,0)}=\bigcup\limits_{\beta<\alpha}\operatorname{OD}^{J_{\beta}(\mathbb{R})}.$

The $Q^{1}_{\omega+1}$ of this paper is equal to the $A_{(2,1)}$ of [Rud97].

In [Rud97] also the mice were named differently. In that paper a hierarchy of
mouse-smallness properties called $(\alpha,n)$-_petite_ , for $\alpha\geq 1$
and $n\in\omega$ was defined. The $M^{\text{ld}}$ of this paper is equal to
the least mouse $M$ that is not $(2,1)$-petite.

In [Rud97] we show that if $M$ is iterable and $(\alpha,n)$-petite, then
$\mathbb{R}\cap M\subseteq A_{(\alpha,n)}$. Taking $\alpha=2$ and $n=1$, and
translating into the language of the current paper, we get that
$\mathbb{R}\cap M^{\text{ld}}\subseteq Q^{1}_{\omega+1}$, which is Theorem 3.5
of this paper. Below we present a sketch of the proof of this theorem. The
interested reader can consult [Rud97] for details.

######  Definition 6.1.

A countable premouse $M$ is $n$-iterable iff player II wins the weak iteration
game on $M$ of length $n$. (See [Ste93].)

######  Definition 6.2.

An $\omega$-mouse is a sound premouse that projects to $\omega$.

######  Remarks 6.3.

  * •

Let $P(x)\leftrightarrow x\in\mathbb{R}$ codes a countable premouse $M$ and
$M$ is $n$-iterable. Then $P$ is projective.

  * •

By [Ste95], if $M$ and $N$ are $n$-small $\omega$-mice and $M$ is fully
iterable and $N$ is $n$-iterable, then $M\trianglelefteq N$ or
$N\trianglelefteq M$.

######  Definition 6.4.

A premouse $M$ is _$\Pi^{1}_{\omega}$ -iterable_ iff $M$ is $n$-iterable for
all $n$.

######  Remarks 6.5.

  * •

Let $P(x)\leftrightarrow x\in\mathbb{R}$ codes a countable premouse $M$ and
$M$ is $\Pi^{1}_{\omega}$-iterable. Then $P$ is $\Pi^{1}_{\omega}$.

  * •

$\Pi^{1}_{\omega}$-iterable was called $\Pi_{(2,0)}$-iterable in [Rud97].

######  Definition 6.6.

A premouse $M$ is _ladder-small_ iff no initial segment of $M$ is a ladder
mouse.

######  Remark 6.7.

Ladder-small is the same as $(2,1)$-petite from [Rud97].

######  Lemma 6.8.

Suppose $M$ and $N$ are countable $\omega$-mice and $M$ is ladder-small and
fully iterable and $N$ is $\Pi^{1}_{\omega}$-iterable. Then $M\trianglelefteq
N$ or $N\trianglelefteq M$.

###### sketch of proof.

This is essentially part (3) of Theorem 7.16 of [Rud97], with
$(\alpha,n)=(2,0)$ and $\delta=0$. Here we give only a sketch.

Compare $M$ and $N$ forming iteration trees $\mathcal{T}$ on $M$ and
$\mathcal{U}$ on $N$. On the $M$ side, at limits, choose branches by
iterability. On the $N$ side, at limits, choose the unique cofinal,
wellfounded branch $b$ such that $Q(b,\mathcal{U})$ is
$\Pi^{1}_{\omega}$-iterable above $\delta(b,\mathcal{U})$, if there is a
unique such branch. If it ever occurs that there is not a unique such branch,
then we can show that for each $n$, there must be a maximal branch $b_{n}$
such that $Q(b_{n},\mathcal{U})$ is not $n$-small above
$\delta(b_{n},\mathcal{U})$. Since $\delta(b_{n},\mathcal{U})$ remains a
cardinal, the process has generated a ladder. This is a contradiction since we
assumed that $M$ is ladder-small. ∎

###### Proof of Theorem 3.5.

Let $x\in\mathbb{R}\cap M^{\text{ld}}$ and we will show that $x\in
Q_{\omega+1}$. Let $\xi$ be the rank of $x$ in the order of construction of
$M^{\text{ld}}$. Let $N\trianglelefteq M^{\text{ld}}$ be the
$\trianglelefteq$-least initial segment of $M$ that contains $x$ and projects
to $\omega$. By Lemma 6.8, if $N^{\prime}$ is any $\omega$-mouse and is
$\Pi^{1}_{\omega}$-iterable, then $N\trianglelefteq N^{\prime}$ or
$N^{\prime}\trianglelefteq N$. Thus $x$ is the unique real such that there is
some $\Pi^{1}_{\omega}$-iterable $\omega$-mouse $N^{\prime}$ such that $x$ is
the $\xi$th real in the order of construction of $N^{\prime}$. This gives a
definition for $\left\\{x\right\\}$ that is $\Sigma^{1}_{\omega+1}$ in any
code for $\xi$. So $x$ is $\Delta^{1}_{\omega+1}(\xi)$, so $x\in
Q_{\omega+1}$. ∎

To formulate a mouse set theorem for larger pointclasses, one must give a
definition of the appropriate mouse for each pointclass. In [Rud97] that was
done via the recursive definition of $(\alpha,n)$-petite. [Rud97] develops the
recursive machinery only up through the ordinal
$\alpha=\omega_{1}^{\omega_{1}}$. We believe, but we haven’t checked
carefully, that by combining the ideas of this paper with those from [Rud97],
it should be possible to prove the following conjecture:

######  Conjecture 6.9.

Assume there are $\omega$ Woodin cardinals with a measurable cardinal above
them all. Suppose that
$(2,0)\leq_{\text{lex}}(\alpha,n)\leq_{\text{lex}}(\omega_{1}^{\omega_{1}},0)$.
Let $M$ be the least fully iterable mouse that is not $(\alpha,n)$-petite.
Then $\mathbb{R}\cap M=A_{(\alpha,n)}$.

Corollary 7.17 of [Rud97] gives $\mathbb{R}\cap M\subseteq A_{(\alpha,n)}$ and
Theorem 8.1 of [Rud97] gives $A_{(\alpha,n)}\subseteq\mathbb{R}\cap M$ in the
case that $\operatorname{cof}(\alpha)>\omega$ or $n=0$. What was missing from
the induction in [Rud97] was a proof that
$A_{(\alpha,n)}\subseteq\mathbb{R}\cap M$ in the case that
$\operatorname{cof}(\alpha)=\omega$ (or $\alpha$ a successor ordinal) and
$n>0$. The ideas of the current paper should be able to be generalized to fill
that gap.

A warning for readers of [Rud97]: That paper contains some material which,
with the benefit of hindsight, we now know is off-point. Because it was
unknown whether or not $A_{(\alpha,n)}\subseteq\mathbb{R}\cap M$, where $M$ is
the least mouse that is not $(\alpha,n)$-petite, [Rud97] defines a second
hierarchy of mouse smallness properties called $(\alpha,n)$-small. The least
mouse that is not $(\alpha,n)$-small is greater in the mouse hierarchy than
the least mouse that is not $(\alpha,n)$-petite. For example, the least mouse
that is not $(2,1)$-petite is $M^{\text{ld}}$, but the least mouse that is not
$(2,1)$-small is $M^{\text{lda}}$, the minimal _admissible_ ladder mouse, i.e.
the minimal mouse $M$ such that $M$ is a ladder-mouse and $M$ is admissible.
Thus $M^{\text{ld}}\lhd M^{\text{lda}}$.

In [Rud97] it was shown that $\mathbb{R}\cap M^{\text{ld}}\subseteq
Q_{\omega+1}\subseteq\mathbb{R}\cap M^{\text{lda}}$, and in general that
$\mathbb{R}\cap M\subseteq A_{(\alpha,n)}\subseteq\mathbb{R}\cap M^{\prime}$,
where $M$ is the least mouse that is not $(\alpha,n)$-petite and $M^{\prime}$
is the least mouse that is not $(\alpha,n)$-small. It was then conjectured
that $A_{(\alpha,n)}=\mathbb{R}\cap M^{\prime}$, where $M^{\prime}$ is the
least mouse that is not $(\alpha,n)$-small. But we now know that this
conjecture was incorrect. The current paper shows that
$Q_{\omega+1}=\mathbb{R}\cap M^{\text{ld}}$ and so the conjecture that
$Q_{\omega+1}=\mathbb{R}\cap M^{\text{lda}}$ is false. In this paper we
replace the false conjecture with Conjecture 6.9.

The reader of [Rud97] should be warned that they will encounter material about
$(\alpha,n)$-_big_ premice (i.e. premice which are not $(\alpha,n)$-small)
which should essentially be ignored as this material was introduced in support
of the false conjecture. In other words, because it was not known how to prove
that $Q_{\omega+1}$ was contained in the minimal ladder mouse, [Rud97]
introduced the notion of the minimal admissible ladder mouse and proved that
$Q_{\omega+1}$ was contained in that mouse. But the current paper gives a
proof that $Q_{\omega+1}$ is contained in the minimal ladder mouse and that
indicates that the introduction of the minimal admissible ladder mouse was an
unnecessary distraction.

##  7\. Eliminating the ordinal parameter

In this section we show that the ordinal parameter $\theta$ in Theorem 3.7 can
be eliminated in the case that $M=M^{\text{ld}}$.

The ideas and results in this section are due to John Steel.

In this section we will make use of some material from the previous section.
In particular to understand this section the reader should be familiar with
the Comparison Theorem for $\Pi^{1}_{\omega}$-iterable ladder-small mice. See
Definition 6.4, Definition 6.6 and Lemma 6.8.

To motivate this section, consider the following fact

######  Fact.

Assume there is a Woodin cardinal with a measurable cardinal above it. To
every $\Sigma^{1}_{3}$ formula $\varphi$ we can recursively associate a
formula $\sigma$ in the language of mice such that for all reals $x$ we have

$\displaystyle\varphi(x)\Leftrightarrow$ $\displaystyle\exists\text{ a
}\Pi^{1}_{2}\text{-iterable, sound, countable }x\text{-mouse }M\text{
projecting to $\omega$ s.t. }$ $\displaystyle M\models\sigma[x].$

The statement “$\exists\text{ a }\Pi^{1}_{2}\text{-iterable, sound, countable
}x\text{-mouse }M$ projecting to $\omega$ s.t. $M\models\sigma[x]$” is a
$\Sigma^{1}_{3}$ statement. So the above fact gives a canonical representation
of all $\Sigma^{1}_{3}$ statements in terms of mice.

###### sketch of proof of the fact.

Let $\sigma$ be the formula such that for all countable, sound $x$-mice $M$
projecting to $\omega$, $M\models\sigma[x]$ iff $M$ is not $1$-small but every
initial segment of $M$ is $1$-small and letting $\delta$ be the unique Woodin
cardinal of $M$ we have that $1$ forces that after collapsing $\delta$ to be
countable, $\varphi[x]$ holds.

The first clause of $\sigma(x)$ asserting that $M$ is not $1$-small but every
initial segment of $M$ is $1$-small can be thought of as the first-order
statement “I am $M_{1}^{\sharp}(x)$.” Any countable, sound $x$-mouse $M$ that
projects to $\omega$ such $M\models$“I am $M_{1}^{\sharp}(x)$” is called a
_putative_ $M_{1}^{\sharp}(x)$. If $M$ is a putative $M_{1}^{\sharp}(x)$ then
$M$ is equal to the real $M_{1}^{\sharp}(x)$ iff $M$ is fully iterable.

If $\varphi(x)$ holds in $V$ then $M_{1}^{\sharp}(x)\models\sigma[x]$.
Conversely, if $M$ is a $\Pi^{1}_{2}$-iterable, sound $x$-mouse that projects
to $\omega$ such that $M\models\sigma[x]$, then we can simply iterate the last
extender of $M$ $\omega_{1}$-times and apply Shoenfield absolutenss to
conclude that $\varphi(x)$ is true in $V$. ∎

We would like to have have an analogous result for $M^{\text{ld}}(x)$. That
is, we would like to have the following

######  Conjecture 7.1.

Assume there are $\omega$ Woodin cardinals with a measurable cardinal above
them all. To every $\Sigma^{1}_{\omega+1}$ formula $\varphi$ we can
recursively associate a formula $\sigma$ in the language of mice such that for
all reals $x$ we have

$\displaystyle\varphi(x)\Leftrightarrow$ $\displaystyle\exists\text{ a
}\Pi^{1}_{\omega}\text{-iterable, sound, countable }x\text{-mouse }M\text{
projecting to $\omega$ s.t. }$ $\displaystyle M\models\sigma[x].$

The formula $\sigma$ from the conjecture should assert that $M$ is a “putative
$M^{\text{ld}}(x)$” and that $M\models\psi[x]$ where $\psi$ is the formula
given by theorem 3.7. The problem with this is that theorem 3.7 includes a
non-uniform ordinal parameter which prevents us from giving a uniform
canonical representation of $\Sigma^{1}_{\omega+1}$ forumulae in terms of
mice.

In this section we will show how to eliminate that ordinal parameter. We will
not give a proof of the conjecture here.

######  Definition 7.2.

If $z\in\mathbb{R}$ then $M^{\text{ld}}_{z}$ refers to the $z$-mouse which is
like $M^{\text{ld}}$ but built over $z$. That is, $M^{\text{ld}}_{z}$ is the
$\trianglelefteq$-least fully-iterable $z$-mouse $M$ for which there is a
ladder over $M$.

######  Theorem 7.3.

Fix $z\in\mathbb{R}$, let $M=M^{\text{ld}}_{z}$, and let
$\theta<\omega_{2}^{M}$ be the minimum of the ranks of all
$\boldsymbol{\Pi}^{1}_{\omega}$ trees in $M$ that are incorrectly wellfounded.
Then $\theta$ is definable in $J^{M}_{\omega_{2}^{M}}$, uniformly in $z$,
without parameters (other than the extender sequence of $M$.)

######  Note.

See Remark 4.12 for what it means to say that a tree is correctly or
incorrectly wellfounded without reference to a $\Pi^{1}_{\omega}$-code or real
parameter.

The $\theta$ from Theorem 7.3 is exactly the ordinal parameter from the proof
of theorem 3.7.

For simplicity we will ignore the real parameter $z$ and prove the theorem for
$M=M^{\text{ld}}$.

######  Definition 7.4.

Let $\left\langle\,\sigma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
enumerate the sentences in the language of premice.

For $n\in\omega$ and $w\in\operatorname{WO}$, let
$S_{n,w}=\big{\\{}x\in\mathbb{R}\,\mid\,x$ codes a
$\Pi^{1}_{\omega}$-iterable, ladder-small $\omega$-mouse $M$ such that
$\operatorname{ord}(M)=|w|$ and $M\models\sigma_{n}\big{\\}}$.

$S_{n,w}$ is $\Pi^{1}_{\omega}(w)$.

For $n\in\omega$ and $w\in\operatorname{WO}$, uniformly, fix $h_{n,w}$ a
$\Pi^{1}_{\omega}(w)$-code for $S_{n,w}$ and let $T_{n,w}=T^{h_{n,w}}$.

######  Remark 7.5.

By Lemma 6.8, either $|w|<(\omega_{1})^{M^{\text{ld}}}$ in which case every
element of $S_{n,w}$ codes the same countable level of $M^{\text{ld}}$, or
else $|w|\geq(\omega_{1})^{M^{\text{ld}}}$ and every countable level of
$M^{\text{ld}}$ is an initial segment of every $M$ coded by the elements of
$S_{n,w}$.

######  Lemma 7.6.

Let $P\lhd J^{M^{\text{ld}}}_{\omega_{1}^{M^{\text{ld}}}}$ be an
$\omega$-mouse. Let $w\in\operatorname{WO}$ with $|w|=\operatorname{ord}(P)$.
Fix $n\in\omega$ and suppose that $P\models\neg\sigma_{n}$. Then $T_{n,w}$ is
wellfounded (in $V$.)

###### Proof.

$p[T_{n,w}]=S_{n,w}=\emptyset$ because if $M$ is a $\Pi^{1}_{\omega}$-iterable
$\omega$-mouse with $\operatorname{ord}(M)=\operatorname{ord}(P)$, then by
Lemma 6.8, $M=P$, so $M\not\models\sigma_{n}$. ∎

######  Definition 7.7.

For any projectively correct mouse $M$ such that $M\models\omega_{1}$ exists,
let $\operatorname{NSat}^{M}=\big{\\{}(n,w)\,\mid\,n\in\omega$ and
$w\in\operatorname{WO}$ with $|w|<\omega_{1}^{M}$ and for
$P=\mathcal{J}^{M}_{|w|}$, $P$ projects to $\omega$ and
$P\models\neg\sigma_{n}\big{\\}}$.

If $T=T^{M}_{n,w}$ for $(n,w)\in\operatorname{NSat}^{M}$ then we say that $T$
is an $\operatorname{NSat}$ tree of $M$.

By Lemma 7.6, all of the $\operatorname{NSat}$ trees of $M^{\text{ld}}$ are
correctly wellfounded. (Again, see Remark 4.12 for what it means to say that a
tree is correctly or incorrectly wellfounded without reference to a
$\Pi^{1}_{\omega}$-code or real parameter.) The key to the proof of Theorem
7.3 is that the ranks of the $\operatorname{NSat}$ trees in $M^{\text{ld}}$
are cofinal in the supremum of the ranks of all correctly wellfounded trees in
$M^{\text{ld}}$.

######  Lemma 7.8.

Let $T\in M^{\text{ld}}$ be a $\boldsymbol{\Pi}^{1}_{\omega}$-tree that is
correctly wellfounded. Then there is an
$(n,w)\in\operatorname{NSat}^{M^{\text{ld}}}$ such that
$\operatorname{rank}(T)<\operatorname{rank}(T^{M^{\text{ld}}}_{n,w})$.

First we give the proof of Theorem 7.3 assuming Lemma 7.8.

###### proof of Theorem 7.3.

As mentioned above, for simplicity we will ignore the real parameter $z$ and
prove the theorem for $M^{\text{ld}}$.

Let $\theta^{\prime}=$ the supremum of the ranks of all
$\boldsymbol{\Pi}^{1}_{\omega}$-trees in $M^{\text{ld}}$ that are correctly
wellfounded. By Lemmas 7.6 and 7.8, $\theta^{\prime}=$ the supremum of the
ranks of the $\operatorname{NSat}$ trees in $M^{\text{ld}}$. Since
$\operatorname{NSat}^{M^{\text{ld}}}$ is definable without parameters in
$J^{M^{\text{ld}}}_{\omega_{2}^{M^{\text{ld}}}}$, this gives a definition of
$\theta^{\prime}$ that is also so definable. Then $\theta^{M^{\text{ld}}}$ is
the minimum rank of a wellfounded $\boldsymbol{\Pi}^{1}_{\omega}$-tree $T\in
M^{\text{ld}}$ with $\operatorname{rank}(T)>\theta^{\prime}$. ∎

######  Definition 7.9.

Fix $z\in\mathbb{R}$ and let $G$ be given by Lemma 5.5 for
$M^{\text{ld}}_{z}$. That is, let
$\gamma=\operatorname{ord}(M^{\text{ld}}_{z})$ and let
$G\subset\mathbb{Q}_{<\gamma}^{M^{\text{ld}}_{z}}$ be such that for
$n\in\omega$, $G\cap J^{M^{\text{ld}}_{z}}_{\delta_{n}}$ is
$\mathbb{Q}_{<\delta_{n}}^{M^{\text{ld}}_{z}}$-generic over
$J^{M^{\text{ld}}_{z}}_{\gamma_{n}}$, where
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
is a ladder over $M^{\text{ld}}$. Let $\pi:M^{\text{ld}}_{z}\to M^{*}$ be the
stationary tower ultrapower. We will capture this setting by saying that
$\pi:M^{\text{ld}}_{z}\to M^{*}=\operatorname{Ult}(M^{\text{ld}}_{z},G)$ is a
_ladder stationary tower embedding_ (for
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$.)

Let $g$ be $\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}$
for some $\delta\in M^{\text{ld}}$. $M^{\text{ld}}[g]$ can be reorganized as
$M^{\text{ld}}_{z}$ for some $z\in\mathbb{R}$. Let $\psi:M^{\text{ld}}_{z}\to
N^{*}=M^{\text{ld}}_{z}[H]$ be a ladder stationary tower embedding. We will
capture this setting by saying that $\psi:M^{\text{ld}}[g]\to
N^{*}=\operatorname{Ult}(M^{\text{ld}}[g],H)$ is a ladder stationary tower
embedding _above $g$_.

Let $M^{*}$ be as in the above definition and let $T\in M^{*}$ be a
$\boldsymbol{\Pi}^{1}_{\omega}$-tree. In what follows it will be convenient
for us to use the terminology $T$ is _correctly-wellfound_ or _incorrectly-
wellfounded_. But because $M^{*}$ is not a wellfounded, iterable mouse, Remark
4.12 does not apply to $M^{*}$ and so we need to be careful about what these
terms mean. Notice that Definition 4.10 still applies to $M^{*}$.

######  Definition 7.10.

Let $M^{*}$ be a not-necessarily wellfounded model that has a rank initial
segment $(V_{\lambda})^{M^{*}}\in\operatorname{wfp}(M^{*})$ such that
$(V_{\lambda})^{M^{*}}\models\operatorname{ZFC}$ and $(V_{\lambda})^{M^{*}}$
is projectively-correct. Let $T\in(V_{\lambda})^{M^{*}}$ be such that
$M^{*}\models``T$ is a $\boldsymbol{\Pi}^{1}_{\omega}$-tree”. Then we will say
that $T$ is _correctly wellfounded_ iff there is a $\Pi^{1}_{\omega}$-code $h$
and a real $x\in(V_{\lambda})^{M^{*}}$ such that
$\left((T^{h}(x))^{M^{*}},h,x,M^{*}\right)$ is correctly wellfounded. This
means that $T$ is wellfounded (in $V$ and so in $M^{*}$) and $T^{h}(x)$ is
wellfounded in $V$.

We will say that $T$ is _incorrectly wellfounded_ iff there is a
$\Pi^{1}_{\omega}$-code $h$ and a real $x\in(V_{\lambda})^{M^{*}}$ such that
$\left((T^{h}(x))^{M^{*}},h,x,M^{*}\right)$ is incorrectly wellfounded. This
means that $T$ is wellfounded (in $V$ and so in $M^{*}$) and $T^{h}(x)$ is
illfounded in $V$.

######  Remark 7.11.

Since Remark 4.12 does not necessarily apply to $M^{*}$, we cannot rule out
the possibility that $T$ is both correctly and incorrectly wellfounded. By
Remark 4.12,this cannot happen if $M^{*}$ is an iterable ladder mouse.

Now we turn to a discussion of the proof of Lemma 7.8. Let $T\in
M^{\text{ld}}$ be a $\boldsymbol{\Pi}^{1}_{\omega}$-tree that is correctly
wellfounded. Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M,G)$ be a
ladder stationary tower embedding. Let $T^{*}=\pi(T)$. It suffices to show
that
$M^{*}\models\text{``}\operatorname{rank}(T^{*})<\operatorname{rank}(T_{n,w})$
for some $(n,w)\in\operatorname{NSat}^{M^{*}}$”.

Since $T$ is correctly wellfounded, $T^{*}$ is correctly wellfounded. Notice
that $M^{*}\models\text{``}T_{n,w}$ is wellfounded for all
$(n,w)\in\operatorname{NSat}$” since $M^{\text{ld}}$ satisfies that sentence.
To find an $(n,w)\in\operatorname{NSat}^{M^{*}}$ such that
$M^{*}\models\text{``}\operatorname{rank}(T^{*})<\operatorname{rank}(T_{n,w})$”,
we will look for an $(n,w)\in\operatorname{NSat}^{M^{*}}$ such that
$(T_{n,w})^{M^{*}}$ is incorrectly wellfounded or illfounded. We can
accomplish this because Lemma 7.6 is not true if we repalce $M^{\text{ld}}$
with $M^{*}$. That is, $M^{*}$ has non-standard countable initial segments.

Let $P^{*}$ be the least initial segment of $M^{*}$ such that
$\omega_{1}^{M^{\text{ld}}}\in P^{*}$ and $P^{*}$ projects to $\omega$. There
is such a $P^{*}$ since
$\operatorname{ord}(M^{\text{ld}})=\omega_{1}^{M^{*}}$. Notice that $P^{*}$ is
ladder-small and $\Pi^{1}_{\omega}$-iterable since every countable initial
segment of $M^{\text{ld}}$ has that property and $M^{*}$ is
$\Pi^{1}_{\omega}$-correct.

######  Lemma 7.12.

Let $P$ be a $\Pi^{1}_{\omega}$-iterable, ladder-small $\omega$-mouse with
$\omega_{1}^{M^{\text{ld}}}\in P$. Let $w\in\operatorname{WO}$ be such that
$|w|=\operatorname{ord}(P)$. Then there is an $n$ such that
$P\models\neg\sigma_{n}$ and $T_{n,w}$ is illfounded.

###### Proof.

If the statement of the lemma were not true, then we would have that for all
$n$, the following are equivalent

  1. (a)

$P\models\sigma_{n}$

  2. (b)

There is a $\Pi^{1}_{\omega}$-iterable, ladder small $\omega$-mouse $N$, with
$\operatorname{ord}(N)=|w|$, such that $N\models\sigma_{n}$.

  3. (b)

For all $\Pi^{1}_{\omega}$-iterable, ladder small $\omega$-mice $N$, with
$\operatorname{ord}(N)=|w|$, $N\models\sigma_{n}$.

This means that the theory of $P$ is $\Delta^{1}_{\omega+1}(w)$. By Theorem
3.6, the theory of $P$ is in $M^{\text{ld}}$. But $P$ is sound and projects to
$\omega$ so it is coded by its theory. So $P\in M^{\text{ld}}$. This is a
contradiction since $P$ would be countable in $M^{\text{ld}}$ but
$\omega_{1}^{M^{\text{ld}}}\in P$. ∎

Let $w\in\operatorname{WO}\cap M^{*}$ with $|w|=\operatorname{ord}(P^{*})$ and
let $n$ be given by Lemma 7.12. Then $(n,w)\in\operatorname{NSat}^{M^{*}}$ and
$(T_{n,w})^{M^{*}}$ is incorrectly wellfounded or illfounded. If we could
apply Theorem 4.11 to $M^{*}$ we could now conclude that
$\operatorname{rank}(T^{*})<\operatorname{rank}(T_{n,w})^{M^{*}}$. But the
proof of Theorem 4.11 does not apply to $M^{*}$ because it is not an iterable
ladder-mouse.

Suppose $T,S$ are $\boldsymbol{\Pi}^{1}_{\omega}$-trees in $M^{*}$ and $T$ is
correctly wellfounded and $S$ is incorrectly wellfounded. If
$T,S\in\operatorname{ran}(\pi)$, say $T=\pi(\bar{T}),S=\pi(\bar{S})$, then
$\bar{T}$ is correctly wellfounded and $\bar{S}$ is incorrectly wellfounded so
$\operatorname{rank}(\bar{T})<\operatorname{rank}(\bar{S})$ so then we can
conclude that $\operatorname{rank}(T)<\operatorname{rank}(S)$.

Now $T^{*}\in\operatorname{ran}(\pi)$ but
$T_{n,w}\not\in\operatorname{ran}(\pi)$ because $w\not\in M^{\text{ld}}$.
However $w$ can be chosen to be generic over $M^{\text{ld}}$ and it turns out
that is enough to see that
$\operatorname{rank}(T^{*})<\operatorname{rank}(T_{n,w})$.

######  Lemma 7.13.

Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M^{\text{ld}},G)$ be a
ladder stationary tower embedding. Let $T,S$ be
$\boldsymbol{\Pi}^{1}_{\omega}$-trees in $M^{*}$. Suppose that $T$ is
correctly wellfounded and $S$ is incorrectly wellfounded. Suppose further that
$T$ is $\Pi^{1}_{\omega}(x)$ and $S$ is $\Pi^{1}_{\omega}(y)$ for
$x,y\in\mathbb{R}\cap M^{\text{ld}}[g]$, where $g$ is
$\operatorname{Coll}(\delta,\omega)$ generic over $M^{\text{ld}}$ for some
$\delta\in M^{\text{ld}}$, and there are $\Pi^{1}_{\omega}$ codes
$h_{1},h_{2}$ such that $(T,h_{1},x,M^{*})$ is correctly wellfounded and
$(S,h_{2},y,M^{*})$ is incorrectly wellfounded. Then
$\operatorname{rank}(T)<\operatorname{rank}(S)$.

Towards a proof of Lemma 7.13, fix $\Pi^{1}_{\omega}$ codes $h_{1},h_{2}$ such
that $T=(T^{h_{1}}(x))^{M^{*}}$ and $S=(T^{h_{2}}(y))^{M^{*}}$ and
$(T,h_{1},x,M^{*})$ is correctly wellfounded and $(S,h_{2},y,M^{*})$ is
incorrectly wellfounded. Then $(T^{h_{1}}(x))^{M^{\text{ld}}[g]}$ is correctly
wellfounded and $(T^{h_{2}}(y))^{M^{\text{ld}}[g]}$ is incorrectly
wellfounded, or illfounded. Because $M^{\text{ld}}[g]$ can be rearranged as
$M^{\text{ld}}_{z}$ for some $z\in\mathbb{R}$, we can apply Theorem 4.11 to
conclude that
$M^{\text{ld}}[g]\models\text{``}\operatorname{rank}((T^{h_{1}}(x)))<\operatorname{rank}((T^{h_{2}}(y)))$”.
Let $\psi:M^{\text{ld}}[g]\rightarrow
N^{*}=\operatorname{Ult}(M^{\text{ld}}[g],H)$ be a ladder stationary tower
embeddding above $g$. Then
$N^{*}\models\text{``}\operatorname{rank}((T^{h_{1}}(x)))<\operatorname{rank}((T^{h_{2}}(y)))$”.
We want to see that
$M^{*}\models\text{``}\operatorname{rank}((T^{h_{1}}(x)))<\operatorname{rank}((T^{h_{2}}(y)))$”.
This turns out to be true because
$(T^{h_{1}}(x))^{N^{*}}=(T^{h_{1}}(x))^{M^{*}}$ and
$(T^{h_{2}}(y))^{N^{*}}=(T^{h_{2}}(y))^{M^{*}}$. To see that this is true, we
now turn to an alternative characterization of these trees that is independent
of stationary tower extensions and depends only on symmetric collapses.

######  Lemma 7.14.

Let $M$ be a countable, transitive model of $\operatorname{ZFC}$ and
$\delta>\omega$ a cardinal of $M$. Suppose that whenever $\mathbb{P}\in
V^{M}_{\delta+1}$ is a poset and $g$ is $\mathbb{P}$-generic over $M$, then
$M[g]$ is projectively-correct. Let $g_{1}$ and $g_{2}$ be
$\operatorname{Coll}(\delta,\omega)$-generic over $M$. Let $x\in\mathbb{R}\cap
M$.

Let $\varphi$ be a $\Sigma^{1}_{n}(x)$ norm for some $n$. Let
$y_{1}\in\mathbb{R}\cap M[g_{1}]$. Then there is a $y_{2}\in\mathbb{R}\cap
M[g_{2}]$ such that $\varphi(y_{1})=\varphi(y_{2})$, and
$\varphi^{M[g_{1}]}(y_{1})=\varphi^{M[g_{2}]}(y_{2})$.

Furthermore $\sigma^{M[g_{1}]}=\sigma^{M[g_{2}]}$ where

$\sigma^{M[g_{i}]}:(\boldsymbol{\delta}^{1}_{\omega})^{M[g_{i}]}\rightarrow\boldsymbol{\delta}^{1}_{\omega}$

is given by Definition 4.6.

Let $h$ be a $\Pi^{1}_{\omega}$ code for a subset of $\mathbb{R}^{2}$. Then
$(T^{h}(x))^{M[g_{1}]}=(T^{h}(x))^{M[g_{2}]}$ and
$\sigma^{M[g_{1}],h,x}=\sigma^{M[g_{2}],h,x}$ where

$\sigma^{M[g_{i}],h,x}:(T^{h}(x))^{M[g_{i}]}\rightarrow(T^{h}(x))^{V}$

is the tree embedding from Definition 4.6.

###### Proof.

All of this follows easily from Corollary 5.3 and its proof.

Let $y_{1}\in\mathbb{R}\cap M[g_{1}]$ be as in the statement of the Lemma. The
proof of Corollary 5.3 gives us a $y_{2}\in\mathbb{R}\cap M[g_{2}]$ such that
$\varphi(y_{1})=\varphi(y_{2})$.

Since this is true for all projective norms $\varphi^{\prime}$ we have that
$\operatorname{ran}(\sigma^{M[g_{1}]})=\operatorname{ran}(\sigma^{M[g_{2}]})$
and since $\sigma^{M[g_{i}]}$ is the inverse of the transitive collapse of its
range, we have that $\sigma^{M[g_{1}]}=\sigma^{M[g_{2}]}$ and so
$\varphi^{M[g_{1}]}(y_{1})=\varphi^{M[g_{2}]}(y_{2})$.

Suppose $(s,u)\in(T^{h}(x))^{M[g_{1}]}$. Let
$n=\operatorname{lh}(s)=\operatorname{lh}(u)$. Then there is
$y_{1}\in\mathbb{R}\cap M[g_{1}]$ extending $s$ such that $(\forall
i<n)\,(y_{1},x)\in G^{i}_{h(i)}$ and
$u(i)=(\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{1},x))^{M[g_{1}]}$.

By Corollary 5.3, relativized to $x$, there is a $y_{2}\in\mathbb{R}\cap
M[g_{2}]$ such that

  * •

$y_{2}\\!\\!\upharpoonright\\!n=y_{1}\\!\\!\upharpoonright\\!n$,

  * •

$(y_{2},x)\in\bigcap\limits_{i<n}G^{i}_{h(i)}$, and

  * •

for $i<n$,
$\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{2},x)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{1},x)$.

So then for $i<n$,
$(\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{2},x))^{M[g_{2}]}=(\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{1},x))^{M[g_{1}]}$.
So $(s,u)\in(T^{h}(x))^{M[g_{2}]}$. We have shown that
$(T^{h}(x))^{M[g_{1}]}\subseteq(T^{h}(x))^{M[g_{2}]}$ and the same argument
shows that $(T^{h}(x))^{M[g_{2}]}\subseteq(T^{h}(x))^{M[g_{1}]}$.

That $\sigma^{M[g_{1}],h,x}=\sigma^{M[g_{2}],h,x}$ now follows immediately. ∎

######  Definition 7.15.

Let $M$ be a countable transitive model of

$\operatorname{ZFC}-\text{ Replacement }+\text{ ``}\exists\text{ cofinally
many inaccessible cardinals.'' }$

Suppose that every set-generic extension of $M$ is projectively correct.

Let $h$ be a $\Pi^{1}_{\omega}$ code for a subset of $\mathbb{R}^{2}$ and let
$x\in\mathbb{R}\cap M$.

Let $\delta$ be a cardinal of $M$. Then we define

  * •

$X^{M,\delta}=\operatorname{ran}(\sigma^{M[g]})$

  * •

$\bar{T}^{h,x,M,\delta}=\operatorname{ran}(\sigma^{M[g],h,x})$

where $g$ is any $\operatorname{Coll}(\delta,\omega)$-generic over $M$.

The definitions makes sense because by Lemma 7.14, applied to a rank initial
segment of $M$, $\sigma^{M[g]}$ and $\sigma^{M[g],h,x}$ don’t depend on the
choice of $g$.

$X^{M,\delta}\subset\boldsymbol{\delta}^{1}_{\omega}$, namely
$X^{M,\delta}=\big{\\{}\varphi(y)\,\mid\,\varphi$ is a projective norm and
$y\in\mathbb{R}\cap M[g]$ for some
$g\subset\operatorname{Coll}(\delta,\omega)$ generic over $M\big{\\}}$.

$\bar{T}^{h,x,M,\delta}\subset(T^{h}(x))^{V}$, namely
$\bar{T}^{h,x,M,\delta}=\big{\\{}(s,u)\in(T^{h}(x))^{V}\,\mid\,\exists
g\,\operatorname{Coll}(\delta,\omega)$-generic over $M$ and $\exists
y\in\mathbb{R}\cap M[g]$ s.t. $y$ extends $s$ and $(\forall
i<\operatorname{lh}(u))\,(y,x)\in G^{i}_{h(i)}$ and
$u(i)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y,x)\big{\\}}$.

If $\delta_{1}<\delta_{2}$ then $X^{M,\delta_{1}}\subset X^{M,\delta_{2}}$ and
$\bar{T}^{h,x,M,\delta_{1}}\subset\bar{T}^{h,x,M,\delta_{2}}$.

Let

$X^{M,\infty}=\bigcup\left\\{\,X^{M,\delta}\,\left|\,\delta\in
M\,\right.\right\\}$
$\bar{T}^{h,x,M,\infty}=\bigcup\left\\{\,\bar{T}^{h,x,M,\delta}\,\left|\,\delta\in
M\,\right.\right\\}.$

Let $\sigma^{M,\infty}$ be the inverse of the transitive collapse of
$X^{M,\infty}$.

Let

$T^{h,x,M,\infty}=\left\\{\,(s,u)\,\left|\,(s,\sigma^{M,\infty}\circ
u)\in\bar{T}^{h,x,M,\infty}\,\right.\right\\}.$

Very roughly speaking $T^{h,x,M,\infty}$ is “the version of $T^{h}(x)$ in the
symmetric collapse over $M$ of the ordinals of $M$.”

One example of a model $M$ that satisfies the hypotheses of Definition 7.15 is
$M^{\text{ld}}[g]$ for some generic $g$. The next lemma states that in this
case a $\Pi^{1}_{\omega}(x)$-tree in the stationary tower ultrapower is equal
to $T^{h,x,M^{\text{ld}}[g],\infty}$.

######  Lemma 7.16.

Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M^{\text{ld}},G)$ be a
ladder stationary tower embedding. Let $g\in M^{*}$ be
$\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}$ for some
cardinal $\delta\in M^{\text{ld}}$. Let $h$ be a $\Pi^{1}_{\omega}$ code for a
subset of $\mathbb{R}^{2}$ and let $x\in\mathbb{R}\cap M^{\text{ld}}[g]$. Then
$(T^{h}(x))^{M^{*}}=T^{h,x,M^{\text{ld}}[g],\infty}$.

###### Proof.

Let
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$
be ladder over $M^{\text{ld}}$ such that $\pi:M^{\text{ld}}\to
M^{*}=\operatorname{Ult}(M^{\text{ld}},G)$ is a ladder stationary tower
embedding for
$\left\langle\,\delta_{n},\gamma_{n}\,\left|\,n\in\omega\,\right.\right\rangle$.

Fix $m\in\omega$ and let $y\in\mathbb{R}\cap M^{*}$ be such that
$(y,x)\in\bigcap\limits_{i<m}G^{i}_{h(i)}$.

Let $n$ be even and large enough that $2m+1<n$, $\delta<\delta_{n}$ and
$g,y,x\in\operatorname{Ult}(J^{M^{\text{ld}}}_{\gamma_{n}},G\cap
J^{M^{\text{ld}}}_{\delta_{n}})$. There is such an $n$ by the proof of Claim 2
in the proof of Lemma 5.4 where we showed that $M^{*}$ is projectively
correct.

There is a $k$, $\operatorname{Coll}(\delta_{n},\omega)$-generic over
$J^{M^{\text{ld}}}_{\gamma_{n}}$ such that $g,y,x\in
J^{M^{\text{ld}}}_{\gamma_{n}}[k]$. By general forcing theory there is a
$k_{1}$ such that
$J^{M^{\text{ld}}}_{\gamma_{n}}[k]=J^{M^{\text{ld}}}_{\gamma_{n}}[g][k_{1}]$
with $k_{1}$ $\operatorname{Coll}(\delta_{n},\omega)$-generic over
$J^{M^{\text{ld}}}_{\gamma_{n}}[g]$.

Let $k_{2}$ be $\operatorname{Coll}(\delta_{n},\omega)$-generic over
$M^{\text{ld}}[g]$. In particular $k_{2}$ is
$\operatorname{Coll}(\delta_{n},\omega)$-generic over
$J^{M^{\text{ld}}}_{\gamma_{n}}[g]$.

By Corollary 5.3, applied to $J^{M^{\text{ld}}}_{\gamma_{n}}[g]$, and
relativized to $x$, there is a $y_{2}\in
J^{M^{\text{ld}}}_{\gamma_{n}}[g][k_{2}]$ such that

  * •

$y_{2}\\!\\!\upharpoonright\\!m=y\\!\\!\upharpoonright\\!m$,

  * •

$(y_{2},x)\in\bigcap\limits_{i<m}G^{i}_{h(i)}$, and

  * •

for $i<m$,
$\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{2},x)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y,x)$.

Since $y_{2}\in M^{\text{ld}}[g][k_{2}]$, we have shown that
$\operatorname{ran}(\sigma^{M^{*},h,x})\subseteq\bar{T}^{h,x,M^{\text{ld}}[g],\infty}$.

Conversely, suppose that $k$ is $\operatorname{Coll}(\delta,\omega)$-generic
over $M^{\text{ld}}[g]$ for some cardinal $\delta$ of $M^{\text{ld}}[g]$ and
$y\in\mathbb{R}\cap M^{\text{ld}}[g][k]$ with
$(y,x)\in\bigcap\limits_{i<m}G^{i}_{h(i)}$. Let $k_{2}\in M^{*}$ be
$\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}[g]$. Then
there is a $y_{2}\in M^{\text{ld}}[g][k_{2}]$ such that

  * •

$y_{2}\\!\\!\upharpoonright\\!m=y\\!\\!\upharpoonright\\!m$,

  * •

$(y_{2},x)\in\bigcap\limits_{i<m}G^{i}_{h(i)}$, and

  * •

for $i<m$,
$\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y_{2},x)=\varphi^{i_{0}}_{h(i_{0}),i_{1}}(y,x)$.

Since $y_{2}\in M^{*}$, we have shown that
$\operatorname{ran}(\sigma^{M^{*},h,x})=\bar{T}^{h,x,M^{\text{ld}}[g],\infty}$.

Since the previous argument may be applied to any projective norm $\varphi$ we
also have that
$\operatorname{ran}(\sigma^{M^{*}})=X^{M^{\text{ld}}[g],\infty}$.

So then we get that $(T^{h}(x))^{M^{*}}=T^{h,x,M^{\text{ld}}[g],\infty}$.

∎

######  Corollary 7.17.

Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M^{\text{ld}},G)$ be a
ladder stationary tower embedding. Let $g\in M^{*}$ be
$\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}$ for some
cardinal $\delta\in M^{\text{ld}}$. Let $h$ be a $\Pi^{1}_{\omega}$ code for a
subset of $\mathbb{R}^{2}$ and let $x\in\mathbb{R}\cap M^{\text{ld}}[g]$.

Let $\psi:M^{\text{ld}}[g]\to N^{*}=\operatorname{Ult}(M^{\text{ld}}[g],H)$ be
a ladder stationary tower embedding above $g$.

Then $(T^{h}(x))^{M^{*}}=(T^{h}(x))^{N^{*}}$.

###### Proof.

We can apply Lemma 7.16 twice to conclude that
$(T^{h}(x))^{M^{*}}=T^{h,x,M^{\text{ld}}[g],\infty}=(T^{h}(x))^{N^{*}}$. ∎

As we suggested above, we can use Corollary 7.17 to show that the rank of a
correctly wellfounded tree is less than the rank of an incorrectly wellfounded
tree in $M^{*}$.

###### Proof of Lemma 7.13.

Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M^{\text{ld}},G)$ be a
ladder stationary tower embedding. Let $g\in M^{*}$ be
$\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}$ for some
cardinal $\delta\in M^{\text{ld}}$. Let $h_{1}$ and $h_{2}$ be
$\Pi^{1}_{\omega}$ codes for a subsets of $\mathbb{R}^{2}$ and let
$x,y\in\mathbb{R}\cap M^{\text{ld}}[g]$. Suppose that $(T^{h_{1}})^{M^{*}}(x)$
is correctly wellfounded and $(T^{h_{2}})^{M^{*}}(y)$ is incorrectly
wellfounded. We must show that
$\operatorname{rank}((T^{h_{1}})^{M^{*}}(x))<\operatorname{rank}((T^{h_{2}})^{M^{*}}(y))$.

$(T^{h_{1}}(x))^{M^{\text{ld}}[g]}$ is correctly wellfounded and
$(T^{h_{2}}(y))^{M^{\text{ld}}[g]}$ is incorrectly wellfounded, or illfounded.
(It will follow from our proof that it is not illfounded.) Because
$M^{\text{ld}}[g]$ can be rearranged as $M^{\text{ld}}_{z}$ for some
$z\in\mathbb{R}$, we can apply Theorem 4.11 to conclude that
$M^{\text{ld}}[g]\models\text{``}\operatorname{rank}(T^{h_{1}}(x))<\operatorname{rank}(T^{h_{2}}(y))$”.
(For convenience we take the previous sentence to include the possibility that
$\operatorname{rank}(T^{h_{2}}(y)=\infty$.) Let
$\psi:M^{\text{ld}}[g]\rightarrow
N^{*}=\operatorname{Ult}(M^{\text{ld}}[g],H)$ be a ladder stationary tower
embeddding above $g$. Then
$N^{*}\models\text{``}\operatorname{rank}(T^{h_{1}}(x))<\operatorname{rank}(T^{h_{2}}(y))$”
and so
$\operatorname{rank}((T^{h_{1}})^{N^{*}}(x))<\operatorname{rank}((T^{h_{2}})^{N^{*}}(y))$.
By Corollary 7.17,$(T^{h_{1}})^{N^{*}}(x)=(T^{h_{1}})^{M^{*}}(x)$ and
$(T^{h_{2}})^{N^{*}}(y)=(T^{h_{2}})^{M^{*}}(y)$. So
$\operatorname{rank}((T^{h_{1}})^{M^{*}}(x))<\operatorname{rank}((T^{h_{2}})^{M^{*}}(y))$.
∎

What remains in this section is to give the

###### Proof of Lemma 7.8.

The proof was essentially given already in the discussion following Definition
7.9.

Let $T\in M^{\text{ld}}$ be a $\boldsymbol{\Pi}^{1}_{\omega}$-tree that is
correctly wellfounded. We need to find an
$(n,w)\in\operatorname{NSat}^{M^{\text{ld}}}$ such that
$\operatorname{rank}(T)<\operatorname{rank}(T^{M^{\text{ld}}}_{n,w})$.

Let $\pi:M^{\text{ld}}\to M^{*}=\operatorname{Ult}(M,G)$ be a ladder
stationary tower embedding. Let $T^{*}=\pi(T)$. It suffices to show that
$M^{*}\models\text{``}\operatorname{rank}(T^{*})<\operatorname{rank}(T_{n,w})$
for some $(n,w)\in\operatorname{NSat}^{M^{*}}$”.

By Lemma 7.13 it suffices to find
$(n,w)\in\operatorname{NSat}^{M^{\text{ld}}}$ such that $(T_{n,w})^{M^{*}}$ is
incorrectly wellfounded.

Let $P^{*}$ be the least initial segment of $M^{*}$ such that
$\omega_{1}^{M^{\text{ld}}}\in P^{*}$ and and $P^{*}$ projects to $\omega$.
Let $w\in\operatorname{WO}\cap M^{*}$ with $|w|=\operatorname{ord}(P^{*})$ and
with $w\in M^{\text{ld}}[g]$ for some $g$ that is
$\operatorname{Coll}(\delta,\omega)$-generic over $M^{\text{ld}}$ for some
cardinal $\delta$ of $M^{\text{ld}}$. Let $n$ be given by Lemma 7.12. Then
$(n,w)\in\operatorname{NSat}^{M^{*}}$ and $(T_{n,w})^{M^{*}}$ is incorrectly
wellfounded. ∎

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