# The consistency strength of determinacy when all sets are universally Baire

Sandra Müller  Sandra Müller, Institut für Diskrete Mathematik und Geometrie,
TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria, and Institut für
Mathematik, Universität Wien, Kolingasse 14-16, 1090 Wien, Austria.
<EMAIL_ADDRESS>
###### Abstract.

It is known that the large cardinal strength of the Axiom of Determinacy when
enhanced with the hypothesis that all sets of reals are universally Baire is
much stronger than the Axiom of Determinacy itself. Sargsyan conjectured it to
be as strong as the existence of a cardinal that is both a limit of Woodin
cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson used a
generalization of Woodin’s derived model construction to show that this
conjectured result would be optimal. In this paper we introduce a new
translation procedure for hybrid mice extending work of Steel, Zhu and
Sargsyan and apply it to prove Sargsyan’s conjecture.

###### Key words and phrases:

Determinacy, Inner Model Theory, Universally Baire, Woodin Cardinal, Strong
Cardinal, Hod Mouse

###### 2010 Mathematics Subject Classification:

03E45, 03E60, 03E55, 03E15

The author gratefully acknowledges funding from L’ORÉAL Austria, in
collaboration with the Austrian UNESCO Commission and in cooperation with the
Austrian Academy of Sciences - Fellowship _Determinacy and Large Cardinals_
and the support of the Elise Richter grant number V844 of the Austrian Science
Fund (FWF). Finally, the author would like to thank Grigor Sargsyan for
helpful discussions at the beginning of this project and the support through
his NSF Career Award DMS-1352034 during a research visit at Rutgers University
in the fall of 2019.

##  1\. Introduction

A central theme in inner model theory is the deep connection between
determinacy principles and inner models with large cardinals. One hierarchy of
determinacy principles is given by imposing additional structural properties
on a model of the Axiom of Determinacy ($\operatorname{AD}$). Examples of such
structural properties are “$\theta_{0}<\Theta$”, “all sets of reals are
Suslin”, “$\Theta$ is regular”, or the Largest Suslin Axiom, see for example
[24, 21, 10, 15]. Woodin used his famous derived model construction to show
that if there is a cardinal $\lambda$ that is a limit of Woodin cardinals and
a limit of ${<}\lambda$-strong cardinals, then there is a model of
$\text{``}\operatorname{AD}^{+}+\text{ all sets of reals are Suslin''},$ see
[24]. Steel showed that this is optimal [21]. By results of Martin and Woodin,
see [24, Theorems 9.1 and 9.2], assuming $\operatorname{AD}$, the statement
“all sets of reals are Suslin” is equivalent to the Axiom of Determinacy for
games on reals ($\operatorname{AD}_{\mathbb{R}}$).

Being universally Baire as defined by Feng, Magidor and Woodin [1] is a non-
trivial strengthening of being Suslin. Larson, Sargsyan and Wilson showed in
[4] via an extension of Woodin’s derived model construction that if there is a
cardinal $\lambda$ that is a limit of Woodin cardinals and a limit of strong
cardinals, then there is a model of $\text{``}\operatorname{AD}+$ all sets of
reals are universally Baire”. One challenge in their argument is that no model
of the form $L(\mathcal{P}(\mathbb{R}))$ is a model of
$\text{``}\operatorname{AD}+$ all sets of reals are universally Baire”.
Sargsyan conjectured that a cardinal $\lambda$ that is a limit of Woodin
cardinals and a limit of strong cardinals is the optimal hypothesis to
construct a model of $\text{``}\operatorname{AD}+$ all sets of reals are
universally Baire”. We prove here that this conjecture is true. More
precisely, we extend Sargsyan’s translation procedure from [11] to prove the
following theorem.

######  Theorem 1.1.

Suppose there is a proper class model of the theory “$\operatorname{AD}+$ all
sets of reals are universally Baire”. Then there is a transitive model
$\mathcal{M}$ of $\operatorname{ZFC}$ containing all ordinals such that
$\mathcal{M}$ has a cardinal $\lambda$ that is a limit of Woodin cardinals and
a limit of strong cardinals.

Translation procedures can be used to transform a hybrid strategy mouse into
an ordinary mouse while generating additional large cardinals from the
strategy predicate. They go back to Steel and were first considered in [22]
and later extended by Zhu in [29]. Steel’s translation procedure can be used
to prove Woodin’s result that
“$\operatorname{ZF}+\operatorname{AD}^{+}+\;\theta_{0}<\theta$” is
equiconsistent with “$\operatorname{ZFC}$ \+ there is a pair of cardinals
$\kappa,\lambda$ such that $\lambda$ is a limit of Woodin cardinals and
$\kappa$ is ${<}\lambda$-strong”. Building on this, Sargsyan developed in [11]
a translation procedure that works locally and constructs a model with
infinitely many Woodin cardinals and one strong cardinal. We extend Sargsyan’s
work and develop a new translation procedure that allows us to construct a
model with a cardinal that is both a limit of Woodin cardinals and a limit of
strong cardinals, cf. Definition 4.3. The model $\mathcal{M}$ we construct
will in fact be a countably iterable premouse. Note that the premouse
constructed in [11] was not known to be iterable.

######  Theorem 1.2.

Let $\mathcal{W}$ be a translatable structure as in Definition 3.14 and let
$\mathcal{M}$ be the result of an $\mathcal{M}$-construction in $\mathcal{W}$
as in Definition 4.3. Then $\mathcal{M}$ is a countably iterable proper class
premouse with a cardinal $\lambda$ that is a limit of Woodin cardinals and a
limit of strong cardinals.

By combining our work here with the results in [4] we obtain the following
corollary.

######  Corollary 1.3.

The following theories are equiconsistent.

$T_{1}$:

$\operatorname{ZF}+\operatorname{AD}+$ all sets of reals are universally
Baire.

$T_{2}$:

$\operatorname{ZFC}+$ there is a limit of Woodin cardinals that is a limit of
strong cardinals.

##  2\. Preliminaries and notation

We start with the definition of universally Baire sets of reals, see for
example [2, Definition 32.21] or [17, Definition 8.6].

######  Definition 2.1.

Let $(S,T)$ be trees on $\omega\times\kappa$ for some ordinal $\kappa$ and let
$Z$ be any set. We say _$(S,T)$ is $Z$-absolutely complementing_ iff

$p[S]={}^{\omega}\omega\setminus p[T]$

in every $\operatorname{Col}(\omega,Z)$-generic extension of $V$.

######  Definition 2.2 (Feng-Magidor-Woodin, [1]).

A set of reals $A$ is _universally Baire (uB)_ if for every $Z$, there are
$Z$-absolutely complementing trees $(S,T)$ with $p[S]=A$.

We assume that the reader is familiar with the basic concepts of inner model
theory and refer to [25] for an overview. But in contrast to [25] we will not
use Mitchell-Steel indexing for the extenders on the sequence of our premice.
We use Jensen-indexing for the extenders on our sequences, cf. [3] or for
example [26, Section 1], with the following exception. We say an ordinal $\xi$
is a _$0$ -weak cutpoint_ in a premouse $\mathcal{M}$ iff $\xi$ is a cutpoint
in $\mathcal{M}$. Furthermore, we recursively say $\xi$ is an _$i+1$ -weak
cutpoint_ iff the only extenders overlapping $\xi$ have critical points
$\kappa_{j}$, $j\leq i$, that are limits of Woodin cardinals and $j$-weak
cutpoints in $\mathcal{M}$. More precisely, $\xi$ is an $i+1$-weak cutpoint
iff for every extender $E_{\alpha}^{\mathcal{M}}$ on the sequence of
$\mathcal{M}$, if $\operatorname{crit}(E_{\alpha}^{\mathcal{M}})<\xi<\alpha$,
then $\operatorname{crit}(E_{\alpha}^{\mathcal{M}})$ is a limit of Woodin
cardinals and a $j$-weak cutpoint in $\mathcal{M}$ for some $j\leq i$.

Analogous to the indexing in [11], the following extenders are indexed
differently than in Jensen indexing: Suppose $E$ is an extender with critical
point $\kappa$ that is a limit of Woodin cardinals and a $j$-weak cutpoint for
some $j<\omega$. Then $E$ is indexed at the successor of the least $j$-weak
cutpoint above $\kappa$ of its own ultrapower. More precisely, if
$E_{\alpha}^{\mathcal{M}}$ is an extender on the sequence of $\mathcal{M}$
such that $\operatorname{crit}(E_{\alpha}^{\mathcal{M}})$ is a limit of Woodin
cardinals in $\mathcal{M}$ and a $j$-weak cutpoint for some $j<\omega$, then
$\alpha$, the index of $E_{\alpha}^{\mathcal{M}}$ on the
$\mathcal{M}$-sequence, is equal to
$(\xi^{+})^{\operatorname{Ult}(\mathcal{M},E_{\alpha}^{\mathcal{M}})}$, where
$\xi$ is the least $j$-weak cutpoint of
$\operatorname{Ult}(\mathcal{M},E_{\alpha}^{\mathcal{M}})$ above
$\operatorname{crit}(E_{\alpha}^{\mathcal{M}})^{+}$. The reason why this
indexing is much more convenient for our construction than the usual Jensen
indexing becomes apparent in the definition of the operators, cf. Section 5.6,
and the proof of Lemma 6.6.

We will work below a model of “$\operatorname{AD}_{\mathbb{R}}+\;\Theta$ is
regular” and refer to hybrid strategy mice as in [10] but we will also use the
comparison theorem from [26], cf. for example the proof of Lemma 5.11 or the
proof of Lemma 6.4. Note that in our context this comparison theorem can be
applied to hybrid strategy mice in the sense of [10] as well.

##  3\. Getting a translatable structure

In this section, we work in a model of “$\operatorname{AD}_{\mathbb{R}}+\text{
all sets of reals are uB}$” in order to define a hybrid structure that we are
going to translate into a model with a cardinal $\lambda$ that is both a limit
of Woodin cardinals and a limit of strong cardinals in the rest of this paper.
Recall that by a result of Martin and Woodin, cf. [24, Theorem 9.1],
$\operatorname{AD}_{\mathbb{R}}$ holds in every model of
“$\operatorname{AD}+\text{ all sets of reals are uB}$” as clearly every
universally Baire set of reals is Suslin. The arguments in this section will
come from descriptive inner model theory and will look familiar to readers who
have seen _$\operatorname{HOD}$ computations_ (cf., e.g. [27, 8, 10, 28]).

###  3.1. Hod premice

We consider _hod premice_ as in [10, Definition 1.34] and just briefly recall
some important notions here. Let $\mathcal{P}$ be a hod premouse. Then, as in
[10] we let $\lambda^{\mathcal{P}}=o.t.(Y^{\mathcal{P}})$ and
$(\delta_{\alpha}^{\mathcal{P}}\mid\alpha\leq\lambda^{\mathcal{P}})$ be the
sequence of Woodin cardinals of $\mathcal{P}$ and their limits enumerated in
increasing order. All $\delta_{\alpha}^{\mathcal{P}}$ will be strong cutpoints
of $\mathcal{P}$ and our hod premice will always satisfy

$\lambda^{\mathcal{P}}\leq\omega,$

so they will for example look like [10, Figure 1.7.2] (an ordinary premouse,
$\lambda^{\mathcal{P}}=0$), [10, Figure 1.7.3] ($\lambda^{\mathcal{P}}=1$), or
[10, Figure 1.7.4] ($\lambda^{\mathcal{P}}=\omega$). As in [10, Definition
1.34], we use

  1. (1)

$\mathcal{P}(i)$ to denote $\mathcal{P}$ up to its $i$’s level for
$i\leq\lambda^{\mathcal{P}}$ and

  2. (2)

let $\mathcal{P}(\lambda^{\mathcal{P}})=\mathcal{P}$.

Moreover, we let

  1. (1)

$\Sigma^{\mathcal{P}}_{i}$ denote the internal strategy of $\mathcal{P}(i)$
for $i<\lambda^{\mathcal{P}}$ and

  2. (2)

if $\lambda^{\mathcal{P}}=\omega$, we let
$\Sigma^{\mathcal{P}}=\Sigma^{\mathcal{P}}_{{<}\omega}=\bigoplus_{i<\omega}\Sigma_{i}^{\mathcal{P}}.$

Whenever it does not lead to confusion, we drop the superscript $\mathcal{P}$
and just write $\Sigma_{i}$ and $\Sigma$. For a hod premouse $\mathcal{P}$
such that $\lambda^{\mathcal{P}}=i<\omega$, we write

$\mathcal{P}^{-}=\mathcal{P}|(\delta_{i-1}^{+\omega})^{\mathcal{P}}.$

If $\lambda^{\mathcal{P}}=\omega$, we write

$\mathcal{P}^{-}=\mathcal{P}|\delta^{\mathcal{P}}_{\omega}.$

We will consider strategies for hod premice that satisfy hull condensation
(cf., [10, Definition 1.31]).

######  Definition 3.1.

Let $\mathcal{P}$ be a hod premouse and let $\Sigma$ be an
$(\omega_{1},\omega_{1}+1)$-iteration strategy for $\mathcal{P}$. Then we say
$\Sigma$ has _hull condensation_ if for any two stacks $\vec{\mathcal{T}}$ and
$\vec{\mathcal{U}}$ on $\mathcal{P}$ such that $\vec{\mathcal{T}}$ is
according to $\Sigma$ and $(\mathcal{P},\vec{\mathcal{U}})$ is a hull of
$(\mathcal{P},\vec{\mathcal{T}})$ in the sense of [10, Definition 1.30],
$\vec{\mathcal{U}}$ is according to $\Sigma$ as well.

Recall the definition of a hod pair (cf., [10, Definition 1.36]).

######  Definition 3.2.

We say $(\mathcal{P},\Sigma)$ is a _hod pair_ iff

  1. (1)

$(\mathcal{P},\Sigma)$ is an lsm pair (cf., [10, Definition 1.23]),

  2. (2)

$\mathcal{P}$ is a hod premouse, and

  3. (3)

$\Sigma$ is an $(\omega_{1},\omega_{1}+1)$-iteration strategy for
$\mathcal{P}$ with hull condensation.

As in [10, Definition 2.26] we define the lower part closure relative to
$\Sigma$ as follows.

######  Definition 3.3.

Let $\Sigma$ be an iteration strategy with hull condensation for some premouse
$\mathcal{M}_{\Sigma}$ and $a$ a transitive self well-ordered111A transitive
set $a$ is called _self well-ordered_ iff $a\cup\\{a\\}$ is well-ordered in
$L_{1}(a)$. set such that $\mathcal{M}_{\Sigma}\in a$. Then we let

  1. (1)

$Lp_{0}^{\Sigma}(a)=a\cup\\{a\\}$,

  2. (2)

$Lp^{\Sigma}(a)=Lp_{1}^{\Sigma}(a)=\bigcup\\{\mathcal{M}\mid\mathcal{M}$ is a
countably iterable222That means, every countable substructure of $\mathcal{M}$
is $(\omega_{1},\omega_{1}+1)$-iterable. sound $\Sigma$-mouse over $a$ such
that $\rho_{\omega}(\mathcal{M})=a\\}$,

  3. (3)

$Lp_{\alpha+1}^{\Sigma}(a)=Lp^{\Sigma}(Lp_{\alpha}^{\Sigma}(a))$, and

  4. (4)

$Lp_{\lambda}^{\Sigma}(a)=\bigcup_{\alpha<\lambda}Lp_{\alpha}^{\Sigma}(a)$,
for limit ordinals $\lambda$.

The following is a special case of $Lp^{\Sigma}$ for ordinary mice. In case
$a$ is a premouse we do not require that $a\cap\operatorname{Ord}$ is a
cutpoint of $\mathcal{M}$ in the definition of $Lp(a)$.

######  Definition 3.4.

Let $a$ be a transitive self well-ordered set. Then we let

  1. (1)

$Lp_{0}(a)=a\cup\\{a\\}$,

  2. (2)

$Lp(a)=Lp_{1}(a)=\bigcup\\{\mathcal{M}\mid\mathcal{M}$ is a countably iterable
sound mouse over $a$ such that $\rho_{\omega}(\mathcal{M})=a\\}$,

  3. (3)

$Lp_{\alpha+1}(a)=Lp(Lp_{\alpha}(a))$, and

  4. (4)

$Lp_{\lambda}(a)=\bigcup_{\alpha<\lambda}Lp_{\alpha}(a)$, for limit ordinals
$\lambda$.

We define fullness preservation analogous to [10, Definition 2.27].

######  Definition 3.5.

Let $(\mathcal{P},\Sigma)$ be a hod pair. Then $\Sigma$ is _fullness
preserving_ if whenever $\vec{\mathcal{T}}$ is a stack on $\mathcal{P}$ via
$\Sigma$ with last model $\mathcal{Q}$ such that $\pi^{\vec{\mathcal{T}}}$
exists, $i+1\leq\lambda^{\mathcal{Q}}$ and $\eta\geq\delta_{i}^{\mathcal{Q}}$
is a strong cutpoint of $\mathcal{Q}(i+1)$, then

$\mathcal{Q}|(\eta^{+})^{\mathcal{Q}(i+1)}=Lp^{\Sigma_{\mathcal{Q}(i),\vec{\mathcal{T}}}}(\mathcal{Q}|\eta).$

Following [10, Section 4.3], we can now define direct limits of hod pairs as
follows. For a hod pair $(\mathcal{P},\Sigma)$ such that $\Sigma$ has branch
condensation (cf., [10, Definition 2.14]) and is fullness preserving, let

$\displaystyle pB(\mathcal{P},\Sigma)=\\{\mathcal{Q}$
$\displaystyle\mid\exists\,\vec{\mathcal{T}}\text{ stack on }\mathcal{P}\text{
via }\Sigma\text{ with last model $\mathcal{R}$ such that}$
$\mathcal{P}$-to-$\mathcal{R}$ does not drop and $\mathcal{Q}$ is a hod
initial segment of $\mathcal{R}$, $\displaystyle\text{i.e.,
$\mathcal{Q}=\mathcal{R}(\alpha)$ for some $\alpha<\lambda^{\mathcal{R}}$}\\}$

and

$\displaystyle pI(\mathcal{P},\Sigma)=\\{\mathcal{Q}$
$\displaystyle\mid\exists\,\vec{\mathcal{T}}\text{ stack on }\mathcal{P}\text{
via }\Sigma\text{ with last model $\mathcal{Q}$}$ $\displaystyle\text{such
that $\mathcal{P}$-to-$\mathcal{Q}$ does not drop}\\}.$

We let $\mathcal{M}_{\infty}(\mathcal{P},\Sigma)$ be the direct limit of all
$\Sigma$-iterates of $\mathcal{P}$ under the iteration maps. That means

$\displaystyle\mathcal{M}_{\infty}(\mathcal{P},\Sigma)=\operatorname{dirlim}((\mathcal{Q},$
$\displaystyle\Sigma_{\mathcal{Q}}),\pi^{\Sigma}_{\mathcal{Q},\mathcal{R}}\mid\mathcal{Q},\mathcal{R}\in
pB(\mathcal{P},\Sigma)\;\wedge$
$\displaystyle\exists\alpha\leq\lambda^{\mathcal{R}}(\mathcal{R}(\alpha)\in
pI(\mathcal{Q},\Sigma_{\mathcal{Q}})).$

###  3.2. Suitable premice

We define suitable premice and related concepts as in [10, Section 4.1] and
only recall the most relevant notions for the reader’s convenience. The reader
can consult [14, 28] for similar notions.

We first define suitability at $\kappa_{0}$. The role of $\kappa_{0}$ will
become clear later in the definition of the translation procedure. For now we
could also call this notion $0$-suitability333This is different from
$0$-suitability as defined in [7] or [8], so to avoid confusion we decided to
call it $\kappa_{0}$-suitability here. and it suffices to notice that these
mice are relevant for the first level of our construction. Note that
$\kappa_{0}$-suitable mice are ordinary and not hybrid.

######  Definition 3.6.

We say $\mathcal{P}$ is _suitable at $\kappa_{0}$_ (or $\kappa_{0}$-suitable)
iff the following holds:

  1. (1)

There is some ordinal $\delta$ such that

$\mathcal{P}\vDash\text{``}\delta\text{ is a Woodin cardinal''}$

and
$\mathcal{P}\cap\operatorname{Ord}=\sup_{n<\omega}(\delta^{+n})^{\mathcal{P}}$.

  2. (2)

$\mathcal{P}$ is a hod premouse with $\lambda^{\mathcal{P}}=0$, so
$\mathcal{P}$ is an ordinary premouse.

  3. (3)

For any $\mathcal{P}$-cardinal $\eta$, if $\eta$ is a strong cutpoint, then

$\mathcal{P}|(\eta^{+})^{\mathcal{P}}=Lp(\mathcal{P}|\eta).$

In general, our suitable premice are hybrid and defined as follows. Again, the
role of $\kappa_{i}$ will become clear later and we could have called these
premice $i$-suitable here.

######  Definition 3.7.

Let $i>0$. We say $\mathcal{P}$ is _$\Sigma$ -suitable at $\kappa_{i}$_ (or
$(\kappa_{i},\Sigma)$-suitable) iff the following holds:

  1. (1)

There is some ordinal $\delta$ such that

$\mathcal{P}\vDash\text{``}\delta\text{ is a Woodin cardinal''}$

and
$\mathcal{P}\cap\operatorname{Ord}=\sup_{n<\omega}(\delta^{+n})^{\mathcal{P}}$.

  2. (2)

$\mathcal{P}$ is a hod premouse with $\lambda^{\mathcal{P}}=i$.

  3. (3)

$(\mathcal{P}^{-},\Sigma)$ is a hod pair such that $\Sigma$ has branch
condensation and is fullness preserving.

  4. (4)

$\mathcal{P}$ is a $\Sigma$-mouse above $\mathcal{P}(i-1)$.

  5. (5)

For any $\mathcal{P}$-cardinal $\eta>\delta^{\mathcal{P}}_{i-1}$, if $\eta$ is
a strong cutpoint, then

$\mathcal{P}|(\eta^{+})^{\mathcal{P}}=Lp^{\Sigma}(\mathcal{P}|\eta).$

If it is clear which strategy $\Sigma$ we consider, we will also just call
$\mathcal{P}$ suitable at $\kappa_{i}$ or $\kappa_{i}$-suitable.

To unify the notation we sometimes call a premouse $\Sigma$-suitable at
$\kappa_{0}$ as well and mean suitable at $\kappa_{0}$. In this case $\Sigma$
could be empty or undefined.

We define $\Sigma$-short tree, $\Sigma$-maximal tree, $\Sigma$-correctly
guided finite stack, the last model of a $\Sigma$-correctly guided finite
stack, $S(\Sigma)$, $F(\Sigma)$, and $f$-iterability as in [10, Definitions
4.3 - 4.7], adapted to our definition of suitability. We again omit the
reference to $\Sigma$ if $\Sigma=\emptyset$ or it is clear from the context to
which strategy $\Sigma$ we are referring. As in [10, Section 4.2] we can
define when a $\kappa_{i}$-suitable premouse is $\vec{A}$-iterable444Sargsyan
calls this $B$-iterability. for a sequence $\vec{A}$ of $\operatorname{OD}$
sets of reals. This also leads to a definition of $\vec{A}$-correct branch,
cf., [18, Definition 7.7.3].

######  Definition 3.8.

Let $i\geq 0$. We say
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\gamma)$ is a _pre-
correctly guided countable stack_ on some premouse
$\mathcal{R}=\mathcal{R}_{0}$ that is $\Sigma$-suitable at $\kappa_{i}$ iff
for every limit ordinal $\nu<\gamma$ and every $m<\omega$,
$(\mathcal{R}_{\nu+k},\mathcal{T}_{\nu+k}\mid k<m)$ is a $\Sigma$-correctly
guided finite stack on $\mathcal{R}_{\nu}$.

###  3.3. Properties of iteration strategies

We will need the following standard properties of iteration strategies. The
first one is super fullness preservation. This notion goes back to [10,
Definition 3.33] and is also used in [11, Definition 1.3]. We simply adapt the
definition to our context.

######  Definition 3.9.

Let $(\mathcal{P},\Sigma)$ be a hod pair such that $\Sigma$ has branch
condensation and is fullness preserving. Then we say $\Sigma$ is _super
fullness preserving_ iff for every $\mathcal{Q}\in pI(\mathcal{P},\Sigma)$ and
every $i<\lambda^{\mathcal{Q}}$, there are a
$\operatorname{Col}(\omega,\mathcal{Q}(i))$-term relations $\tau_{i}$ and
$w_{i}$ such that

  1. (1)

$\tau_{i}$ captures the set

$\displaystyle U_{i}=\\{(x,y)\in\mathbb{R}^{2}\mid x$ $\displaystyle\text{
codes a transitive set }a\in\operatorname{HC}\text{ and $y$ codes
$\mathcal{N}$}$ $\displaystyle\text{such that }\mathcal{N}\unlhd
Lp^{\Sigma_{\mathcal{Q}(i)}}(a)\text{ and }\rho_{\omega}(\mathcal{N})=a\\},$

  2. (2)

$w_{i}$ captures the set

$\displaystyle W_{i}=\\{(x,y,$ $\displaystyle z)\in\mathbb{R}^{3}\mid(x,y)\in
U_{i}\text{ and if $\mathcal{N}$ is the $\Sigma_{\mathcal{Q}(i)}$-premouse
coded by $y$}$ $\displaystyle\text{then $z$ codes a tree according to the
unique strategy of }\mathcal{N}\\}.$

Now, following [11, Definition 1.4], we can define that a model $M$ is
generically closed under $\Sigma$ if $\Sigma$ is super fullness preserving in
all generic extensions of $M$.

######  Definition 3.10.

Let $\Sigma$ be a super fullness preserving strategy in some model $M$. Then
$M$ is _generically closed under $\Sigma$_ iff for all $M$-cardinals $\eta$
and all ${\leq}\eta$-generics $g$ over $M$, there is a unique
$((\eta^{+})^{M},(\eta^{+})^{M})$-strategy $\Sigma^{g,\eta}\in M[g]$ such that
$\Sigma^{g,\eta}\upharpoonright M$ is the
$((\eta^{+})^{M},(\eta^{+})^{M})$-fragment of $\Sigma$ and

$M[g]\vDash\text{``}\Sigma^{g,\eta}\text{ is super fullness preserving''}.$

In this case we write
$\Sigma^{g}=\bigcup_{\eta\in\operatorname{Ord}}\Sigma^{g,\eta}$.

######  Definition 3.11.

Let $(\mathcal{P},\Sigma)$ be a hod pair and suppose $\Sigma$ is super
fullness preserving in some model $M$ that is generically closed under
$\Sigma$. Let $i<\omega$ and let $\tau$ be a
$\operatorname{Col}(\omega,\mathcal{P}(i))$-term relation for a set of reals.
Then _$\Sigma$ guides $\tau$ correctly_ iff for any $g$ generic over $M$ and
any $\Sigma^{g}$-iterates $\mathcal{Q}$ and $\mathcal{R}$ of $\mathcal{P}$
such that $\mathcal{P}$-to-$\mathcal{Q}$ and $\mathcal{P}$-to-$\mathcal{R}$ do
not drop, whenever $x\in\mathbb{R}^{M[g]}$, $g_{\mathcal{Q}}$ is
$\operatorname{Col}(\omega,\mathcal{Q}(i))$-generic over $\mathcal{Q}$,
$g_{\mathcal{R}}$ is $\operatorname{Col}(\omega,\mathcal{R}(i))$-generic over
$\mathcal{R}$, and
$x\in\mathcal{Q}[g_{\mathcal{Q}}]\cap\mathcal{R}[g_{\mathcal{R}}]$, then

$x\in i(\tau)_{g_{\mathcal{Q}}}\text{ iff }x\in j(\tau)_{g_{\mathcal{R}}},$

where $i\colon\mathcal{P}\rightarrow\mathcal{Q}$ and
$j\colon\mathcal{P}\rightarrow\mathcal{R}$ denote the iteration embeddings.

In this case, we write $A^{g}_{\Sigma,\tau}$ for the set of reals in $M[g]$
determined by $\Sigma^{g}$ and $\tau$.

Let $\mathcal{P}$ be $\Sigma$-suitable at $\kappa_{i}$ for some $i<\omega$ and
let $\tau$ be a
$\operatorname{Col}(\omega,((\delta^{\mathcal{P}})^{+n})^{\mathcal{P}})$-term
relation for a set of reals for some $n<\omega$. Let

$\gamma_{\tau}^{\mathcal{P}}=\sup(Hull_{1}^{\mathcal{P}}(\mathcal{P}^{-}\cup\\{\tau\\})\cap\delta^{\mathcal{P}})$

and

$H_{\tau}^{\mathcal{P}}=Hull_{1}^{\mathcal{P}}(\gamma_{\tau}^{\mathcal{P}}\cup\\{\tau\\}).$

Now, the following definition adapts [11, Definition 1.6] to our context (cf.,
[10, Definition 4.14]).

######  Definition 3.12.

Let $\mathcal{P}$ be $\Sigma$-suitable at $\kappa_{i}$ for some $i<\omega$ and
suppose $(\mathcal{P},\Lambda)$ is a hod pair such that $\Lambda$ is super
fullness preserving in some model $M$ that is generically closed under
$\Lambda$. Then we say _$\Lambda$ is guided by the sequence
$\vec{\tau}=(\tau_{k}\mid k<\omega)$_ iff for any $g$ generic over $M$ the
following holds.

  1. (1)

For every $k<\omega$ there is some $n<\omega$ such that $\tau_{k}$ is a
$\operatorname{Col}(\omega,((\delta^{\mathcal{P}})^{+n})^{\mathcal{P}})$-term
relation for a set of reals and $\Lambda^{g}$ guides $\tau_{k}$ correctly.

  2. (2)

For each $\Lambda^{g}$-iterate $\mathcal{Q}$ of $\mathcal{P}$ such that
$\mathcal{P}$-to-$\mathcal{Q}$ does not drop with iteration embedding
$\pi\colon\mathcal{P}\rightarrow\mathcal{Q}$,

$\mathcal{Q}=\bigcup_{k<\omega}H_{\pi(\tau_{k})}^{\mathcal{Q}}.$

  3. (3)

For each $\Lambda^{g}$-iterate $\mathcal{Q}$ of $\mathcal{P}$ such that
$\mathcal{P}$-to-$\mathcal{Q}$ does not drop with iteration embedding
$\pi\colon\mathcal{P}\rightarrow\mathcal{Q}$ and each normal tree
$\mathcal{T}$ in the domain of $\Lambda^{g}_{\mathcal{Q}}$, letting
$b=\Lambda^{g}_{\mathcal{Q}}(\mathcal{T})$, if $\pi_{b}^{\mathcal{T}}$ exists,
then $b$ is the unique branch through $\mathcal{T}$ such that for every
$k<\omega$,

$\pi_{b}^{\mathcal{T}}(\pi(\tau_{k}))\text{ locally term captures
}A^{g}_{\Lambda,\tau_{k}}.$

  4. (4)

For each $\eta$ that is a limit of Woodin cardinals in $M[g]$ such that
$\mathcal{P}\in V_{\eta}^{M[g]}$, if $h$ is
$\operatorname{Col}(\omega,{<}\eta)$-generic over $M[g]$, then for every
$k<\omega$, $A_{\Lambda,\tau_{k}}^{g*h}$ is an $\operatorname{OD}$ set of
reals in the derived model of $M[g]$ as computed by $h$.

Moreover, we say _$\Lambda$ is strongly guided by the sequence
$\vec{\tau}=(\tau_{k}\mid k<\omega)$_ iff for any $\Lambda^{g}$-iterate
$\mathcal{Q}$ of $\mathcal{P}$ such that $\mathcal{P}$-to-$\mathcal{Q}$ does
not drop with iteration embedding $\pi\colon\mathcal{P}\rightarrow\mathcal{Q}$
and any premouse $\mathcal{R}$ such that there are embeddings
$\sigma\colon\mathcal{P}\rightarrow\mathcal{R}$ and
$l\colon\mathcal{R}\rightarrow\mathcal{Q}$ with $\pi=l\circ\sigma$, it follows
that $\mathcal{R}$ is $\Sigma$-suitable at $\kappa_{i}$ and for every
$k<\omega$, $l^{-1}(\pi(\tau_{k}))$ locally term captures
$A^{g}_{\Lambda,\tau_{k}}$.

###  3.4. Translatable structures

Work in a model of

$\operatorname{AD}_{\mathbb{R}}+\text{ all sets of reals are universally
Baire.}$

By [12, Corollary 5.4], we may and will assume $\Theta=\theta_{\omega}$. Let

$\displaystyle\mathcal{M}_{\infty}=\bigcup\\{\mathcal{M}_{\infty}$
$\displaystyle(\mathcal{P},\Sigma)|\theta_{i}\mid(\mathcal{P},\Sigma)\text{ is
a countable hod pair such }$ $\displaystyle\text{that }\Sigma\text{ has branch
condensation and is fullness}$ $\displaystyle\;\;\;\text{preserving and
}w(Code(\Sigma))=\theta_{i}\\}.$

Here $w(Code(\Sigma))$ denotes the Wadge-rank of the canonical code of
$\Sigma$ as a set of reals. Note that $\mathcal{M}_{\infty}$ is well-defined
as the direct limit models $\mathcal{M}_{\infty}(\mathcal{P},\Sigma)$ line up
(cf. [10, Lemma 0.23 and Theorem 4.24] or, in a more general context but for a
different notion of hod pair, [26]). Let $\Lambda$ be the canonical iteration
strategy for $\mathcal{M}_{\infty}$ given by the join of the strategies for
its hod initial segments. As usual, each $\theta_{i}$ in our background
universe is a Woodin cardinal in $\mathcal{M}_{\infty}$ and we write
$(\delta_{i}\mid i<\omega)$ for the sequence of Woodin cardinals in
$\mathcal{M}_{\infty}$.

Note that the Woodin cardinals $\delta_{i}$, $i<\omega$, in
$\mathcal{M}_{\infty}$ are cutpoints, so the strategy $\Lambda$ of
$\mathcal{M}_{\infty}$ naturally splits into strategies $\Sigma_{i}$ between
consecutive Woodin cardinals $\delta_{i-1}$ and $\delta_{i}$, letting
$\delta_{-1}=0$. In particular, $\mathcal{M}_{\infty}$ can see its own
iteration strategy.

Universal Baireness ensures that we can canonically extend $\Lambda$ to an
iteration strategy acting on stacks of normal iteration trees of length
$\lambda$ for any ordinal $\lambda$. See [9, Lemma 1.21] for this argument and
in particular for how iteration strategies need to be coded as sets of reals
to make the coding sufficiently absolute. We also write $\Lambda$ for this
canonical extension of the original strategy $\Lambda$ of
$\mathcal{M}_{\infty}$.

The following lemma now easily follows from the fact that we are working in a
model of “all sets of reals are universally Baire”.

######  Lemma 3.13.

$\Lambda$ is an $(\operatorname{Ord},\operatorname{Ord})$-iteration strategy
for $\mathcal{M}_{\infty}$ and $\mathcal{M}_{\infty}$ is generically closed
under $\Lambda$.

Now we are ready to define translatable structures. This is crucial for the
rest of this article as a translatable structure is what we want to translate
into a model with a cardinal that is both a limit of Woodin cardinals and a
limit of strong cardinals.

######  Definition 3.14.

Let $\mathcal{W}$ be a proper class hybrid strategy premouse in the sense of
[10, Chapter 1]. Then $\mathcal{W}$ is a _translatable structure_ iff there is
a sequence of ordinals $(\delta_{i}\mid i<\omega)$ such that $\mathcal{W}$
satisfies the following conditions.

  1. (1)

$\mathcal{W}\vDash\text{``}\delta_{i}$ is a Woodin cardinal and a cutpoint for
every $i<\omega$ and these are the only Woodin cardinals”.

  2. (2)

In $\mathcal{W}$, let
$\mathcal{P}_{0}=\mathcal{W}_{\omega}(\mathcal{W}|\delta_{0})$ and
$\mathcal{P}_{i}=Lp_{\omega}^{\Sigma_{i-1}}(\mathcal{W}|\delta_{i-1})$ for
$i>0$, where $\Sigma_{i}$ denotes the strategy for $\mathcal{P}_{i}$. Then
$\mathcal{P}_{0}$ is suitable at $\kappa_{0}$ and $\mathcal{P}_{i}$ is
$\Sigma_{i-1}$-suitable at $\kappa_{i}$ for every $i>0$.

  3. (3)

$\Sigma_{i}$ is a super fullness preserving iteration strategy for
$\mathcal{P}_{i}$ with hull condensation such that $\mathcal{W}$ is
generically closed under $\Sigma_{i}$ and there is a set of term relations
$\vec{\tau}\subseteq\mathcal{P}_{i}$ such that whenever $g$ is
$\operatorname{Col}(\omega,\mathcal{P}_{i})$-generic over $\mathcal{W}$ and
$(\tau_{k}\mid k<\omega)$ is a generic enumeration of $\vec{\tau}$ in
$\mathcal{W}[g]$, then $(\Sigma_{i})^{g}$ is strongly guided by $(\tau_{k}\mid
k<\omega)$.

  4. (4)

$\mathcal{W}$ is internally
$(\operatorname{Ord},\operatorname{Ord})$-iterable.

  5. (5)

$\mathcal{W}\vDash$ “there is no inner model with a superstrong cardinal”.

  6. (6)

Let $\delta_{\omega}$ be the limit of $\delta_{i}$, $i<\omega$. Let $G$ be
$\operatorname{Col}(\omega,{<}\delta_{\omega})$-generic over $\mathcal{W}$ and
let $M$ be the derived model computed in $\mathcal{W}(\mathbb{R}^{*})$. Let
$\Phi=(\Sigma^{\mathcal{W}})^{G}\upharpoonright\operatorname{HC}^{M}$. Then
$\Phi\in M$ and in $M$, $\mathcal{W}|(\delta_{i}^{+\omega})^{\mathcal{W}}$ is
a $\kappa_{i}$-suitable $\emptyset$-iterable $\Phi$-premouse (cf. [11,
Definition 1.7] for the definition of $\emptyset$-iterability).

######  Remark 3.15.

We analogously define set-sized translatable structures $\mathcal{W}$ that
satisfy a sufficiently large fragment of $\operatorname{ZFC}$.

Recall that $(\mathcal{M}_{\infty},\Lambda)$ is a hod pair. It easily follows
that constructing relative to the strategy $\Lambda$ above
$\mathcal{M}_{\infty}$ does not project across
$\mathcal{M}_{\infty}\cap\operatorname{Ord}$ and hence
$L^{\Lambda}[\mathcal{M}_{\infty}]$ is a translatable structure.

######  Lemma 3.16.

$L^{\Lambda}[\mathcal{M}_{\infty}]$ is a translatable structure.

##  4\. The translation procedure

For the rest of this article, fix a translatable structure $\mathcal{W}$ and
let $\Sigma=\bigoplus_{i<\omega}\Sigma_{i}$ be the associated iteration
strategy. The extenders we want to use to witness that certain cardinals are
strong in the translated structure are of the following form.

######  Definition 4.1.

We say a $(\kappa,\lambda)$-extender $E$ is _generically countably complete_
if it is countably complete in all ${<}\kappa$-generic extensions, i.e.,
whenever $\mathbb{P}$ is a partial order with $|\mathbb{P}|<\kappa$ and $G$ is
$\mathbb{P}$-generic over $V$, then in $V[G]$, for every sequence $(a_{i}\mid
i<\omega)$ of sets in $[\lambda]^{<\omega}$ and every sequence $(A_{i}\mid
i<\omega)$ of sets $A_{i}\in E_{a_{i}}$ there is a function
$\tau\colon\bigcup_{i}a_{i}\rightarrow\kappa$ such that $\tau\,"a_{i}\in
A_{i}$ for each $i<\omega$.

It is a well-known fact for countably complete extenders that they provide
realizations, see for example [17, Lemma 10.64]. We will crucially use (a
generalization of) this in the proof of Lemma 6.3 to obtain iterability for
our translated structures.

######  Fact 4.2.

Let $\mathcal{M}$ be a premouse and $\bar{\mathcal{M}}$ the transitive
collapse of a countable $\Sigma_{n}$-elementary substructure of $\mathcal{M}$
for some $n<\omega$. Let
$\sigma\colon\bar{\mathcal{M}}\rightarrow_{\Sigma_{n}}\mathcal{M}$ be the
uncollapse map. Then for every extender $E\in\bar{\mathcal{M}}$ such that
$\sigma(E)$ is countably complete, if
$\pi_{E}\colon\bar{\mathcal{M}}\rightarrow\operatorname{Ult}(\bar{\mathcal{M}},E)$
denotes the ultrapower embedding, there is a realization map
$\sigma^{*}\colon\operatorname{Ult}(\bar{\mathcal{M}},E)\rightarrow\mathcal{M}$,
cf. Figure 1.

$\sigma$$\mathcal{M}$$\bar{\mathcal{M}}$$\pi_{E}$$\operatorname{Ult}(\bar{\mathcal{M}},E)$$\sigma^{*}$
Figure 1. Realizing a countably complete extender.

Now we can define our translation procedure. We aim to construct a premouse
with strong cardinals $\kappa_{i}$, $i<\omega$ inside $\mathcal{W}$. The
cardinals $\kappa_{i}$ are going to be the least ${<}\delta_{i+1}$-strong
cardinals in $\mathcal{W}$. The idea is that extenders with critical point
$\kappa\neq\kappa_{i}$ for all $i<\omega$ come from a standard fully
backgrounded extender construction as in [6] (without the smallness
assumption). In addition, we add extra extenders witnessing that $\kappa_{i}$
is strong for each $i<\omega$. These extenders with critical point
$\kappa_{i}$ are generically countably complete. This will later allow us to
prove that certain elementary substructures of the translated premouse are in
fact sufficiently iterable (cf. Section 6.1) to show that the construction
converges and the cardinals $\kappa_{i}$ are indeed fully strong (cf. Section
6.3).

######  Definition 4.3.

Let $\mathcal{W}$ be a translatable structure with Woodin cardinals
$(\delta_{i}\mid i<\omega)$. Let $(\kappa_{i}\mid i<\omega)$ be a sequence of
cardinals such that for each $i<\omega$, $\kappa_{i}$ is the least
${<}\delta_{i+1}$-strong cardinal above $\delta_{i}$ in $\mathcal{W}$. We say
$(\mathcal{M}_{\xi},\mathcal{N}_{\xi}\mid\xi\leq\Omega)$ is the _mouse
construction_ (or short _$\mathcal{M}$ -construction_) in $\mathcal{W}$ for
$\Omega\leq\operatorname{Ord}$ if it is given inductively as follows.

  1. (1)

$\mathcal{N}_{0}$ is the result of a fully backgrounded extender construction
in $\mathcal{W}|\delta_{1}$.

  2. (2)

For $\xi+1<\Omega$, if
$\mathcal{M}_{\xi}=(J_{\alpha}^{\vec{E}},\in,\vec{E},\emptyset)$ is a passive
premouse, we define $\mathcal{N}_{\xi+1}$ as follows.

    1. (a)

If there is an extender $F^{*}$ on the sequence of $\mathcal{W}$ and an
extender $F$ over $\mathcal{M}_{\xi}$, both with critical point
$\kappa\notin\\{\kappa_{i}\mid i<\omega\\}$ or $\kappa=\kappa_{i}$ for some
$i<\omega$ with $\alpha<\delta_{i+1}$, such that
$(J_{\alpha}^{\vec{E}},\in,\vec{E},F)$ is a premouse and for some
$\nu<\alpha$,
$V_{\nu+\omega}^{\mathcal{W}}\subseteq\operatorname{Ult}(\mathcal{W},F^{*})$
and

$F\upharpoonright\nu=F^{*}\cap([\nu]^{<\omega}\times J_{\alpha}^{\vec{E}}),$

we let

$\mathcal{N}_{\xi+1}=(J_{\alpha}^{\vec{E}},\in,\vec{E},F).$

    2. (b)

If (2a) fails and there is a generically countably complete extender $F$ in
$\mathcal{W}$ with critical point $\kappa_{i}$ for some $i<\omega$ such that
$\delta_{i+1}\leq\alpha$ and $(J_{\alpha}^{\vec{E}},\in,\vec{E},F)$ is a
premouse, we let

$\mathcal{N}_{\xi+1}=(J_{\alpha}^{\vec{E}},\in,\vec{E},F).$

    3. (c)

If both (2a) and (2b) fail, we let

$\mathcal{N}_{\xi+1}=(J_{\alpha+1}^{\vec{E}},\in,\vec{E},\emptyset).$

In all cases, we let
$\mathcal{M}_{\xi+1}=\mathbb{C}_{\omega}(\mathcal{N}_{\xi+1})$, if it is
defined.

  3. (3)

For $\xi+1<\Omega$, if
$\mathcal{M}_{\xi}=(J_{\alpha}^{\vec{E}},\in,\vec{E},F)$ is active, we let

$\mathcal{N}_{\xi+1}=(J_{\alpha+1}^{\vec{E}^{\prime}},\in,\vec{E}^{\prime},\emptyset)$

for $\vec{E}^{\prime}=\vec{E}^{\frown}F$. Moreover, we let
$\mathcal{M}_{\xi+1}=\mathbb{C}_{\omega}(\mathcal{N}_{\xi+1})$, if it is
defined.

  4. (4)

If $\lambda\leq\Omega$ is a limit ordinal or $\lambda=\operatorname{Ord}$, we
let $\mathcal{N}_{\lambda}=\mathcal{M}_{\lambda}$ be the unique passive
premouse such that for all ordinals $\beta$,
$\omega\beta\in\mathcal{N}_{\lambda}\cap\operatorname{Ord}$ iff
$J_{\beta}^{\mathcal{N}_{\alpha}}$ is defined and eventually constant as
$\alpha$ converges to $\lambda$, and for all such
$\omega\beta\in\mathcal{N}_{\lambda}\cap\operatorname{Ord}$,
$J_{\beta}^{\mathcal{N}_{\lambda}}$ is given by the eventual value of
$J_{\beta}^{\mathcal{N}_{\alpha}}$ as $\alpha$ converges to $\lambda$.

We say the $\mathcal{M}$-construction breaks down and stop the construction if
$\mathbb{C}_{\omega}(\mathcal{N}_{\xi})$ is not defined for some
$\xi\in\operatorname{Ord}$. Otherwise we say the $\mathcal{M}$-construction
converges and $\mathcal{M}=\mathcal{M}_{\operatorname{Ord}}$ is the result of
the construction.

######  Remark 4.4.

We can similarly perform $\mathcal{M}$-constructions in set-sized translatable
structures that satisfy a sufficiently large fragment of $\operatorname{ZFC}$.

For the rest of this article, we let $\mathcal{M}$ be the result of an
$\mathcal{M}$-construction up to $\Omega$ in the translatable structure
$\mathcal{W}$ and argue that it converges, i.e. $\Omega=\operatorname{Ord}$,
and has the desired properties.

##  5\. Characterizing extenders

To characterize the extenders at the different “future strong cardinals”
$\kappa_{i}$, $i<\omega$, we first need to define direct limit systems.

###  5.1. Direct limit systems at $\kappa_{0}$

For simplicity, we start with a direct limit system at $\kappa_{0}$, the first
cardinal that is strong up to $\delta_{1}$ in $\mathcal{W}$. The general case
for $\kappa_{i}$, $i>0$, will be handled in the next section. We write
$\Sigma=\Sigma_{0}$ and let $\mathcal{M}^{0}_{\infty}$ be the direct limit of
all $\Sigma$-iterates of $\mathcal{P}=Lp_{\omega}(\mathcal{M}|\delta_{0})$
that are countable in $\mathcal{W}[g]$ for some generic
$g\subseteq\operatorname{Col}(\omega,{<}\kappa_{0})$. It will be useful for us
to consider $\mathcal{M}^{0}_{\infty}$ as a direct limit of a different direct
limit system as follows.

Write $\mathcal{Q}=\mathcal{M}|\delta_{1}$ and note that $\mathcal{Q}$ is
equal to the result of a fully backgrounded extender construction in
$\mathcal{W}|\delta_{1}$ (cf. [7, Lemma 2.21]). Let $G$ be
$\operatorname{Col}(\omega,{<}\kappa_{0})$-generic over $\mathcal{Q}$ and let
$M$ be the derived model of $\mathcal{Q}[G]$. Working in $M$, let

$\mathcal{F}^{0}=\\{(\mathcal{N},\vec{A})\mid\mathcal{N}\text{ is countable,
suitable at $\kappa_{0}$, and }\vec{A}\text{-iterable}\\}.$

As in the standard $\operatorname{HOD}$-computation arguments we get that
$\mathcal{M}^{0}_{\infty}$ is the direct limit of $\mathcal{F}^{0}$ (cf. [10,
Theorem 4.24]).

######  Lemma 5.1.

$\mathcal{M}^{0}_{\infty}$ is the direct limit of $\mathcal{F}^{0}$.

We can identify $\mathcal{M}^{0}_{\infty}$ inside $\mathcal{M}$ as the quasi-
limit of certain fully backgrounded constructions as follows. We say an
ordinal $\delta^{*}$ is _$\kappa_{0}$ -good_ iff $\delta^{*}$ is the least
Woodin cardinal above $\nu$ for some cutpoint $\nu$ that is not a limit of
Woodin cardinals. For any $\kappa_{0}$-good ordinal $\delta^{*}$ in
$\mathcal{M}|\kappa_{0}$, let

$\mathcal{N}^{-}_{\delta^{*}}=(L[E]^{\geq\delta^{*}}(\emptyset))^{\mathcal{M}|\kappa_{0}}$

be the result of a fully backgrounded $L[E]$-construction in
$\mathcal{M}|\kappa_{0}$ using extenders with critical point above
$\delta^{*}$ up to the level where it reaches a Woodin cardinal, i.e., we
construct until

$L[\mathcal{N}^{-}_{\delta^{*}}]\vDash\text{``}\mathcal{N}^{-}_{\delta^{*}}\cap\operatorname{Ord}\text{
is Woodin''}.$

Let $\mathcal{N}_{\delta^{*}}=Lp_{\omega}(\mathcal{N}_{\delta^{*}}^{-})$. As
$\kappa_{0}$ is a limit of Woodin cardinals (cf. [11, Lemma 1.13]), there are
plenty of such “windows” of Woodin cardinals below $\kappa_{0}$. We aim to
take the quasi-limit of the models constructed in these “windows” in the
following sense (cf. [18, Section 7.7]555This definition is analogous to what
is called a suitable sequence $(\mathcal{N}_{\alpha}\mid\alpha<\beta)$ in [18,
Section 7.7]. or [27, Section 5.2]). We formulate the next definition in the
general case for an arbitrary $i<\omega$.

######  Definition 5.2.

Let $i<\omega$ and let
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\gamma)$ be a pre-
correctly guided countable stack on some $\kappa_{i}$-suitable premouse
$\mathcal{R}_{0}$. Let $\vec{A}=(A_{n}\mid n<\omega)$ be a sequence of
$\operatorname{OD}$ sets of reals and write $\vec{A}_{n}=(A_{k}\mid k\leq n)$
for its initial segments. Then we define inductively for
$2\leq\beta\leq\gamma$:

  1. (1)

when $(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ is
_correctly guided_ ,

  2. (2)

when $[\xi,\beta)$ is _$\vec{A}_{n}$ -guided_ for $\xi<\beta$ and $n<\omega$,

  3. (3)

the _$\vec{A}_{n}$ -guided embedding_

$\pi_{\xi,\eta}^{\vec{A}_{n}}\colon
H_{\vec{A}_{n}}^{\mathcal{R}_{\xi}}\rightarrow
H_{\vec{A}_{n}}^{\mathcal{R}_{\eta}}$

for $\xi\leq\eta<\beta$ such that $[\xi,\beta)$ is $\vec{A}_{n}$-guided, and

  4. (4)

the _quasi-limit $\operatorname{qlim}_{\alpha<\beta}\mathcal{R}_{\alpha}$_ for
limit ordinals $\beta$.

We say $(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ is
correctly guided if
$\mathcal{R}_{\lambda}=\operatorname{qlim}_{\alpha<\lambda}\mathcal{R}_{\alpha}$
for all limit ordinals $\lambda<\beta$ and assume inductively for the rest of
the definition that
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ is correctly
guided. In addition, we assume inductively that the $\vec{A}_{n}$-guided
embeddings commute, i.e., for $\xi<\eta<\nu<\beta$ such that $[\xi,\nu+1)$ is
$\vec{A}_{n}$-guided,

$\pi^{\vec{A}_{n}}_{\xi,\nu}=\pi^{\vec{A}_{n}}_{\eta,\nu}\circ\pi^{\vec{A}_{n}}_{\xi,\eta}.$

The inductive definition splits into three cases.

######  Case 1.

$\beta$ is a limit ordinal.

We start by letting $[\xi,\beta)$ be $\vec{A}_{n}$-guided iff $[\xi,\eta)$ is
$\vec{A}_{n}$-guided for all $\eta<\beta$. Suppose that for every $n<\omega$
there is some $\xi_{n}<\beta$ such that $[\xi_{n},\beta)$ is
$\vec{A}_{n}$-guided. Then we let

$\mathcal{I}=\\{(\xi,\vec{A}_{n})\mid[\xi,\beta)\text{ is
$\vec{A}_{n}$-guided}\\}$

and consider the direct limit system

$\mathcal{F}=\\{(H_{\vec{A}_{n}}^{\mathcal{R}_{\xi}},\pi^{\vec{A}_{n}}_{\xi,\eta})\mid(\xi,\vec{A}_{n}),(\eta,\vec{A}_{m})\in\mathcal{I}\text{
and }(\xi,\vec{A}_{n})\leq(\eta,\vec{A}_{m})\\},$

where $\pi^{\vec{A}_{n}}_{\xi,\eta}\colon
H_{\vec{A}_{n}}^{\mathcal{R}_{\xi}}\rightarrow
H_{\vec{A}_{n}}^{\mathcal{R}_{\eta}}\subseteq
H_{\vec{A}_{m}}^{\mathcal{R}_{\eta}}$ is the $\vec{A}_{n}$-guided embedding
and $(\xi,\vec{A}_{n})\leq(\eta,\vec{A}_{m})$ is defined as $\xi\leq\eta$ and
$n\leq m$. Now let

$\mathcal{R}_{\beta}=\operatorname{qlim}_{\alpha<\beta}\mathcal{R}_{\alpha}=\operatorname{dirlim}\mathcal{F}.$

Note that there is no need to define $\vec{A}_{n}$-guided embeddings in this
case.

######  Case 2.

$\beta=\lambda+1$ for some limit ordinal $\lambda$.

In this case we let $[\xi,\beta)$ be $\vec{A}_{n}$-guided iff $\xi=\lambda$ or
$[\xi,\lambda)$ is $\vec{A}_{n}$-guided. For $\xi\leq\lambda<\beta$ such that
$[\xi,\beta)$ is $\vec{A}_{n}$-guided we now define the $\vec{A}_{n}$-guided
embedding $\pi_{\xi,\lambda}^{\vec{A}_{n}}$. For $\xi=\lambda$ let
$\pi_{\xi,\lambda}^{\vec{A}_{n}}=\operatorname{id}$. For $\xi<\lambda$,
consider the direct limit embedding $\pi\colon
H_{\vec{A}_{n}}^{\mathcal{R}_{\xi}}\rightarrow
H_{\vec{A}_{n}}^{\mathcal{R}_{\lambda}}\subseteq\mathcal{R}_{\lambda}$ giving
rise to $\mathcal{R}_{\lambda}$ and let $\pi_{\xi,\lambda}^{\vec{A}_{n}}=\pi$.

######  Case 3.

$\beta=\eta+1$ for some successor ordinal $\eta=\nu+1$.

Here we let $[\xi,\beta)$ be $\vec{A}_{n}$-guided iff $\xi=\eta$, or
$[\xi,\eta)$ is $\vec{A}_{n}$-guided and either

  1. (i)

$\mathcal{T}_{\nu}$ is a short tree and the corresponding embedding
$i^{\mathcal{T}_{\nu}}\colon\mathcal{R}_{\nu}\rightarrow\mathcal{R}_{\eta}$ is
$\vec{A}_{n}$-correct, or

  2. (ii)

$\mathcal{T}_{\nu}$ is a maximal tree and has a cofinal $\vec{A}_{n}$-correct
branch.

Then there is a canonical $\vec{A}_{n}$-guided embedding
$\pi^{\vec{A}_{n}}_{\nu,\eta}\colon
H_{\vec{A}_{n}}^{\mathcal{R}_{\nu}}\rightarrow
H_{\vec{A}_{n}}^{\mathcal{R}_{\eta}}$ and we can define the
$\vec{A}_{n}$-guided embedding

$\pi^{\vec{A}_{n}}_{\xi,\eta}=\pi^{\vec{A}_{n}}_{\nu,\eta}\circ\pi^{\vec{A}_{n}}_{\xi,\nu}$

for $\xi<\nu$.

This finishes the definition of correctly guided stacks
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ and their quasi-
limits. We can similarly define quasi-limits of directed systems of mice
instead of just stacks.

Now we can define quasi-iterability.

######  Definition 5.3.

Let $\mathcal{R}$ be a $\kappa_{0}$-suitable premouse.

  1. (1)

We say $\mathcal{R}$ is _quasi-iterable_ iff for every limit ordinal $\beta$,
every correctly guided countable stack
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ with
$\mathcal{R}=\mathcal{R}_{0}$ has a $\kappa_{0}$-suitable quasi-limit
$\mathcal{R}_{\beta}$. In this case, we say that $\mathcal{R}_{\alpha}$ for
$\alpha\leq\beta$ is a _quasi-iterate_ of $\mathcal{R}$.

  2. (2)

For any finite sequence $\vec{A}$ of $\operatorname{OD}$ sets of reals we say
$\mathcal{R}$ is _$\vec{A}$ -quasi-iterable_ iff $\mathcal{R}$ is quasi-
iterable and for every correctly guided countable stack
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ with
$\mathcal{R}=\mathcal{R}_{0}$, $[0,\beta)$ is $\vec{A}$-guided. In this case,
we let $\mathcal{R}_{\beta}$ be the quasi-limit of
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ and say
$\mathcal{R}_{\alpha}$ for $\alpha\leq\beta$ is an _$\vec{A}$ -quasi-iterate_
of $\mathcal{R}$.

  3. (3)

For any sequence $\vec{A}=(A_{n}\mid n<\omega)$ of $\operatorname{OD}$ sets of
reals we say $\mathcal{R}$ is _$\vec{A}$ -quasi-iterable_ iff $\mathcal{R}$ is
quasi-iterable and for every correctly guided countable stack
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ with
$\mathcal{R}_{0}=\mathcal{R}$, $[0,\beta)$ is $\vec{A}_{n}$-guided for all
$n<\omega$, where $\vec{A}_{n}=(A_{k}\mid k\leq n)$. In this case, we let
$\mathcal{R}_{\beta}$ be the quasi-limit of
$(\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\mid\alpha<\beta)$ and say
$\mathcal{R}_{\alpha}$ for $\alpha\leq\beta$ is an _$\vec{A}$ -quasi-iterate_
of $\mathcal{R}$.

Next, we argue that we can consider the quasi-limit of the premice
$\mathcal{N}_{\delta^{*}}$ for all $\kappa_{0}$-good ordinals $\delta^{*}$ in
$\mathcal{M}|\kappa_{0}$. In order to make sense of the quasi-limit, we need
to show that the $\mathcal{N}_{\delta^{*}}$’s form a correctly guided stack of
$\kappa_{0}$-suitable premice. Let $\vec{\tau}=(\tau_{k}\mid k<\omega)$ be a
sequence of term relations such that $\Sigma^{g}$ is strongly guided by
$\vec{\tau}$. Let $\vec{B}$ be such that whenever $\vec{B}=(B_{k}\mid
k<\omega)$ is a generic enumeration of $\vec{B}$ in some $\mathcal{W}[g]$,
then $B_{k}=A_{\Sigma,\tau_{k}}^{g}$. Then every iteration according to
$\Sigma^{g}$ is $\vec{B}$-correct and
$\mathcal{P}=Lp_{\omega}(\mathcal{W}|\delta_{0})$ is $\vec{B}$-quasi-iterable.
We summarize this observation in the next lemma.

######  Lemma 5.4.

$\mathcal{P}=Lp_{\omega}(\mathcal{W}|\delta_{0})$ is suitable at $\kappa_{0}$
and $\vec{B}$-quasi-iterable.

######  Lemma 5.5.

Let $\delta^{*}$ be a $\kappa_{0}$-good ordinal. Then
$\mathcal{N}_{\delta^{*}}$ is a $\Sigma$-iterate of $\mathcal{P}$ and hence a
$\kappa_{0}$-suitable $\vec{B}$-quasi-iterable premouse.

###### Proof.

For this proof we work in $\mathcal{W}$ and compare $\mathcal{P}$ against the
construction of $\mathcal{N}_{\delta^{*}}$. By stationarity and universality
of fully backgrounded constructions666Cf. [10, Lemmas 2.11 and 2.13], note
that these results also hold in our setting where $\mathcal{N}_{\delta^{*}}$
is the result of a fully backgrounded construction in
$\mathcal{M}|\kappa_{0}$, that itself is an initial segment of the result of a
fully backgrounded construction in $\mathcal{W}$. Recall that we stop the
construction of $\mathcal{N}_{\delta^{*}}$ when it reaches a Woodin cardinal.,
$\mathcal{P}$ iterates to some level of the construction of
$\mathcal{N}_{\delta^{*}}$. As we stop the construction of
$\mathcal{N}_{\delta^{*}}^{-}$ when
$\mathcal{N}_{\delta^{*}}^{-}\cap\operatorname{Ord}$ is a Woodin cardinal and
let $\mathcal{N}_{\delta^{*}}=Lp_{\omega}(\mathcal{N}_{\delta^{*}}^{-})$, it
follows that $\mathcal{P}$ in fact iterates to $\mathcal{N}_{\delta^{*}}$.
This yields that each $\mathcal{N}_{\delta^{*}}$ is a $\kappa_{0}$-suitable
$\vec{B}$-quasi-iterable premouse. ∎

The same idea yields the following lemma.

######  Lemma 5.6.

The premice $\mathcal{N}_{\delta^{*}}$ for all $\kappa_{0}$-good ordinals
$\delta^{*}$ in $\mathcal{M}|\kappa_{0}$ have well-defined quasi-limits and,
together with these quasi-limits and the canonical iteration trees, form a
correctly guided countable stack.

###### Proof.

We start working in $\mathcal{W}$. For $\kappa_{0}$-good ordinals
$\delta^{*}<\delta^{**}$, by comparing $\mathcal{N}_{\delta^{*}}$ against the
construction of $\mathcal{N}_{\delta^{**}}$, the same argument as in the proof
of Lemma 5.5 gives that $\mathcal{N}_{\delta^{*}}$ iterates to (an initial
segment of) $\mathcal{N}_{\delta^{**}}$. By $\kappa_{0}$-suitability of both
$\mathcal{N}_{\delta^{*}}$ and $\mathcal{N}_{\delta^{**}}$ it cannot be a
proper initial segment, so $\mathcal{N}_{\delta^{*}}$ iterates to
$\mathcal{N}_{\delta^{**}}$. In $M$, all $\mathcal{N}_{\delta^{*}}$ are
$\vec{B}$-quasi-iterable. Therefore, it follows that quasi-limits of premice
of the form $\mathcal{N}_{\delta^{*}}$ are well-defined. Hence, the
$\mathcal{N}_{\delta^{*}}$’s together with their quasi-limits and the
canonical iteration trees form a correctly guided countable stack. ∎

Let $\mathcal{M}_{\infty}^{*}$ be the quasi-limit of the premice
$\mathcal{N}_{\delta^{*}}$ for all good ordinals $\delta^{*}$ in
$\mathcal{M}|\kappa_{0}$.

######  Lemma 5.7.

$\mathcal{M}^{0}_{\infty}=\mathcal{M}_{\infty}^{*}$.

###### Proof.

Recall that $\mathcal{M}^{0}_{\infty}$ is the direct limit of all
$\Sigma$-iterates of $\mathcal{P}$ that are countable in $\mathcal{W}[G]$,
where $G$ is $\operatorname{Col}(\omega,{<}\kappa_{0})$-generic over
$\mathcal{W}$. We argue that the models $\mathcal{N}_{\delta^{*}}$ are cofinal
in these $\Sigma$-iterates of $\mathcal{P}$. As in Lemma 5.1 this implies
$\mathcal{M}^{0}_{\infty}=\mathcal{M}_{\infty}^{*}$, as desired.

Let $\mathcal{R}$ be an arbitrary $\Sigma$-iterate of $\mathcal{P}$ that is
countable in $\mathcal{W}[G]$. In particular,
$\mathcal{R}\cap\operatorname{Ord}<\kappa_{0}$. As $\kappa_{0}$ is a limit of
Woodin cardinals in $\mathcal{M}|\kappa_{0}$ (cf. [11, Lemma 1.13]), we can
pick some $\kappa_{0}$-good ordinal
$\delta^{*}>\mathcal{R}\cap\operatorname{Ord}$. Consider
$\mathcal{N}_{\delta^{*}}$ and compare $\mathcal{R}$ against the construction
of $\mathcal{N}_{\delta^{*}}$. As in the proof of Lemma 5.5, $\mathcal{R}$
iterates to $\mathcal{N}_{\delta^{*}}$. This shows that the
$\mathcal{N}_{\delta^{*}}$’s are cofinal in the direct limit system giving
rise to $\mathcal{M}^{0}_{\infty}$. ∎

Recall that $\mathcal{M}_{\infty}^{0}$ is a $\Sigma$-iterate of $\mathcal{P}$
and that $\mathcal{Q}$ is obtained from a fully backgrounded construction in
$\mathcal{W}|\delta_{1}$. The proof of the successor case in the proof of [10,
Theorem 6.5] (cf. [11, Lemma 1.14]) shows that
$\Sigma_{\mathcal{M}_{\infty}^{0}}$ is in $\mathcal{M}$.

######  Lemma 5.8.

$\Sigma_{\mathcal{M}_{\infty}^{0}}\upharpoonright\mathcal{M}\in\mathcal{M}.$

###  5.2. Direct limit systems at $\kappa_{i}$

We now define more general direct limit systems that will be used to
characterize extenders with critical point $\kappa_{i}$ for $i>0$ and argue
that their direct limits $\mathcal{M}_{\infty}^{i}$ are in $\mathcal{M}$.
Working in $\mathcal{W}$, let $\mathcal{M}_{\infty}^{i}$ be the direct limit
of all $\Sigma$-iterates of
$\mathcal{P}^{i}=Lp_{\omega}(\mathcal{W}|\delta_{i})$ that are countable in
$\mathcal{W}[G]$, where $G$ is
$\operatorname{Col}(\omega,{<}\kappa_{i})$-generic over $\mathcal{W}$.

Write $\mathcal{Q}_{i}=\mathcal{M}|\delta_{i+1}$, let $G$ be
$\operatorname{Col}(\omega,{<}\kappa_{i})$-generic over $\mathcal{Q}_{i}$ and
let $M_{i}$ be the derived model of $\mathcal{Q}_{i}[G]$. Working in $M_{i}$,
let

$\mathcal{F}^{i}=\\{(\mathcal{N},\vec{A})\mid\mathcal{N}\text{ is countable,
suitable at $\kappa_{i}$, and }\vec{A}\text{-iterable}\\}.$

As in the standard $\operatorname{HOD}$-computation arguments we get that
$\mathcal{M}^{i}_{\infty}$ is the direct limit of $\mathcal{F}^{i}$ (cf. [10,
Theorem 4.24]).

######  Lemma 5.9.

$\mathcal{M}^{i}_{\infty}$ is the direct limit of $\mathcal{F}^{i}$.

Let $i>0$. We say an ordinal $\delta^{*}<\kappa_{i}$ is _$\kappa_{i}$ -good_
iff $\delta^{*}$ is the least Woodin cardinal above $\nu$ for some cutpoint
$\nu>\delta_{i}$ that is not a limit of Woodin cardinals. For any
$\kappa_{i}$-good ordinal $\delta^{*}$ in $\mathcal{M}|\kappa_{i}$ we define
$\mathcal{N}_{\delta^{*}}$ as the result of a hod pair construction777Similar
as in [10, Definition 2.6] but we are not working in a weak background triple.
above $\delta^{*}$ in $\mathcal{M}|\kappa_{i}$ relative to tail strategies of
$\Sigma_{k}$ for $k<i$ until it reaches $i+1$ Woodin cardinals. Note that by
Lemmas 5.8 and 5.13 we have or will have that a tail of $\Sigma_{k-1}$
restricted to $\mathcal{M}$ is in $\mathcal{M}$ for all $k\leq i$.

To keep the notation simple, we illustrate the definition of
$\mathcal{N}_{\delta^{*}}$ at $\kappa_{1}$ and $\kappa_{2}$. The definition
for $\kappa_{i}$, $i>2$ is then a straightforward generalization. In what
follows, whenever we say that we “add a strategy” we mean that the strategy
gets added to the model in a fine structural sense as in [10]. We decided to
not go into the fine structural details here in order to be able to focus on
the new ideas in this construction. We refer the interested reader to [10,
Chapter 1].

So let $\delta^{*}$ be $\kappa_{1}$-good. Then $\mathcal{N}_{\delta^{*}}$ is
given by the following construction: Perform a fully backgrounded construction
as in [6] (without the smallness assumption) using extenders with critical
point above $\delta^{*}$ inside $\mathcal{M}|\kappa_{1}$ until it reaches a
model $\mathcal{R}_{0}$ satisfying

$L[\mathcal{R}_{0}]\vDash\text{``$\mathcal{R}_{0}\cap\operatorname{Ord}$ is
Woodin''.}$

We will argue in Lemma 5.10 that $\mathcal{R}_{0}$ is a
$\Sigma_{\mathcal{M}_{\infty}^{0}}$-iterate of $\mathcal{M}_{\infty}^{0}$
(that itself is a $\Sigma_{0}$-iterate of $\mathcal{P}_{0}$). In particular,
$(\mathcal{R}_{0},\Sigma_{\mathcal{R}_{0}})$ is a hod pair. Therefore, it
makes sense to define $\mathcal{R}_{1}$ as the result of adding
$\Sigma_{\mathcal{R}_{0}}$ to $\mathcal{R}_{0}$. That means, we construct
further relative to $\Sigma_{\mathcal{R}_{0}}$ and obtain $\mathcal{R}_{1}$ as
a $\Sigma_{\mathcal{R}_{0}}$-mouse over $\mathcal{R}_{0}$ such that

$L[\mathcal{R}_{1}]\vDash\text{``$\mathcal{R}_{1}\cap\operatorname{Ord}$ is
Woodin''.}$

We let

$\mathcal{N}_{\delta^{*}}=Lp_{\omega}^{\Sigma_{\mathcal{R}_{0}}}(\mathcal{R}_{1}).$

######  Lemma 5.10.

Let $\delta^{*}$ be a $\kappa_{1}$-good ordinal in $\mathcal{M}$ and let
$\mathcal{R}$ be the result of a fully backgrounded $J^{\vec{E}}$-construction
above $\delta^{*}$ inside $\mathcal{M}|\kappa_{1}$ up to the level where it
reaches a Woodin cardinal. Then $\mathcal{R}$ is a
$\Sigma_{\mathcal{M}_{\infty}^{0}}$-iterate of $\mathcal{M}_{\infty}^{0}$ and
$\Sigma_{\mathcal{R}}\upharpoonright\mathcal{M}\in\mathcal{M}$.

###### Proof.

Working in $\mathcal{M}$, compare $\mathcal{M}_{\infty}^{0}$ (via
$\Sigma_{\mathcal{M}_{\infty}^{0}}$) against the construction of
$\mathcal{R}$. Recall that $\mathcal{R}$ is constructed in
$\mathcal{M}|\kappa_{1}$ and
$\Sigma_{\mathcal{M}_{\infty}^{0}}\upharpoonright\mathcal{M}\in\mathcal{M}$ by
Lemma 5.8. By stationarity and universality of fully backgrounded
constructions (cf. [10, Lemmas 2.11 and 2.13]), $\mathcal{M}_{\infty}^{0}$
iterates to a level of the construction of $\mathcal{R}$. As we stop the
construction of $\mathcal{R}$ when it reaches a Woodin cardinal,
$\mathcal{M}_{\infty}^{0}$ in fact iterates to $\mathcal{R}$, as desired. This
immediately gives
$\Sigma_{\mathcal{R}}\upharpoonright\mathcal{M}\in\mathcal{M}$. ∎

Now let $\delta^{*}$ be $\kappa_{2}$-good. Then we obtain
$\mathcal{N}_{\delta^{*}}$ as follows: Start by performing a fully
backgrounded construction above $\delta^{*}$ inside $\mathcal{M}|\kappa_{2}$
until it reaches a model $\mathcal{R}_{0}$ satisfying

$L[\mathcal{R}_{0}]\vDash\text{``$\mathcal{R}_{0}\cap\operatorname{Ord}$ is
Woodin''.}$

As in Lemma 5.10 we get that $\mathcal{R}_{0}$ is a
$\Sigma_{\mathcal{M}_{\infty}^{0}}$-iterate of $\mathcal{M}_{\infty}^{0}$ and
$\Sigma_{\mathcal{R}_{0}}\upharpoonright\mathcal{M}\in\mathcal{M}$. Therefore,
we can let $\mathcal{R}_{1}$ be the result of adding
$\Sigma_{\mathcal{R}_{0}}$ to $\mathcal{R}_{0}$. That means, we let
$\mathcal{R}_{1}$ be the result of a fully backgrounded construction relative
to $\Sigma_{\mathcal{R}_{0}}$ on top of $\mathcal{R}_{0}$ in
$\mathcal{M}|\kappa_{2}$ until

$L[\mathcal{R}_{1}]\vDash\text{``$\mathcal{R}_{1}\cap\operatorname{Ord}$ is
Woodin''.}$

That means, both $\mathcal{R}_{0}\cap\operatorname{Ord}$ and
$\mathcal{R}_{1}\cap\operatorname{Ord}$ are Woodin cardinals in
$L[\mathcal{R}_{1}]$. Note that all extenders in $\mathcal{M}$ that serve as
backgrounds for extenders in $\mathcal{R}_{0}$ or $\mathcal{R}_{1}$ are
themselves backgrounded by extenders in $\mathcal{W}$. This comes from the
fact that the fully backgrounded constructions giving rise to
$\mathcal{R}_{0}$ and $\mathcal{R}_{1}$ are performed in
$\mathcal{M}|\kappa_{2}$ and only use extenders above $\delta^{*}>\delta_{2}$.
We will show in Lemma 5.11 that $\mathcal{R}_{1}$ is a
$\Sigma_{\mathcal{M}_{\infty}^{1}}$-iterate of $\mathcal{M}_{\infty}^{1}$ and
$\Sigma_{\mathcal{R}_{1}}\upharpoonright\mathcal{M}\in\mathcal{M}$. Moreover,
$\Sigma_{\mathcal{R}_{1}}\upharpoonright\mathcal{M}$ agrees with the strategy
$\mathcal{R}_{1}$ inherits from $\mathcal{M}$ (and ultimately from
$\mathcal{W}$) as the result of a fully backgrounded construction. So we can
let $\mathcal{R}_{2}$ be the result of adding $\Sigma_{\mathcal{R}_{1}}$ to
$\mathcal{R}_{1}$. That means, we again construct further relative to
$\Sigma_{\mathcal{R}_{1}}$ and obtain $\mathcal{R}_{2}$ as a
$\Sigma_{\mathcal{R}_{1}}$-premouse above $\mathcal{R}_{1}$ such that

$L[\mathcal{R}_{2}]\vDash\text{``$\mathcal{R}_{2}\cap\operatorname{Ord}$ is
Woodin''.}$

Finally, we let

$\mathcal{N}_{\delta^{*}}=Lp_{\omega}^{\Sigma_{\mathcal{R}_{1}}}(\mathcal{R}_{2}).$

The next lemma provides a version of Lemma 5.10 for $\kappa_{i}$-good ordinals
for $i>1$.

######  Lemma 5.11.

Let $i>0$, let $\delta^{*}$ be a $\kappa_{i}$-good ordinal in $\mathcal{M}$
and let $\mathcal{R}_{i-1}$ be the result of a hod pair construction above
$\delta^{*}$ as described above up to the level where it reaches $i$ Woodin
cardinals. Then $\mathcal{R}_{i-1}$ is a
$\Sigma_{\mathcal{M}_{\infty}^{i-1}}$-iterate of $\mathcal{M}_{\infty}^{i-1}$
and $\Sigma_{\mathcal{R}_{i-1}}\upharpoonright\mathcal{M}\in\mathcal{M}$.

###### Proof.

Suppose inductively that $\mathcal{M}_{\infty}^{i-1}\in\mathcal{M}|\kappa_{i}$
and using Lemmas 5.8 and 5.13
$\Sigma_{\mathcal{M}_{\infty}^{i-1}}\upharpoonright\mathcal{M}\in\mathcal{M}$.
Working in $\mathcal{M}$, compare $\mathcal{M}_{\infty}^{i-1}$ (via
$\Sigma_{\mathcal{M}_{\infty}^{i-1}}$) against the construction of
$\mathcal{R}_{i-1}$ (as the result of a fully backgrounded construction in
$\mathcal{M}$) using Steel’s comparison method for iterating into a
backgrounded strategy in [26]. $\mathcal{R}_{i-1}$ is the result of a fully
backgrounded construction using extenders with critical point above
$\delta^{*}>\delta_{i}>\mathcal{M}_{\infty}^{i-1}\cap\operatorname{Ord}$, so
it does not move in the comparison and $\mathcal{M}_{\infty}^{i-1}$ iterates
to an initial segment of $\mathcal{R}_{i-1}$. Let $\mathcal{Q}$ be the
corresponding iterate of $\mathcal{M}_{\infty}^{i-1}$. Then in fact
$\mathcal{Q}=\mathcal{R}_{i-1}$. As a result of the comparison, the strategies
of $\mathcal{R}_{i-1}$ (inherited by the backgrounded construction) and the
iterate $\mathcal{Q}$ of $\mathcal{M}_{\infty}^{i-1}$ agree. Therefore, not
only $\Sigma_{\mathcal{M}_{\infty}^{i-1}}\upharpoonright\mathcal{M}$ and its
tail strategy $\Sigma_{\mathcal{Q}}\upharpoonright\mathcal{M}$ are in
$\mathcal{M}$ but also $\Sigma_{\mathcal{R}_{i-1}}\upharpoonright\mathcal{M}$.
∎

As in Lemma 5.5, each $\mathcal{N}_{\delta^{*}}$ for a $\kappa_{i}$-good
ordinal $\delta^{*}$ is $\kappa_{i}$-suitable and $\vec{B}$-quasi-iterable for
an appropriate $\vec{B}$. Let $(\mathcal{M}_{\infty}^{i})^{*}$ be the quasi-
limit of the $\mathcal{N}_{\delta^{*}}$’s inside $\mathcal{M}$ for all
$\kappa_{i}$-good ordinals $\delta^{*}$ in $\mathcal{M}|\kappa_{i}$. The
following lemma can be shown analogous to Lemma 5.7.

######  Lemma 5.12.

$\mathcal{M}_{\infty}^{i}=(\mathcal{M}_{\infty}^{i})^{*}$.

As in the previous section for $i=0$, the proof of the successor case in the
proof of [10, Theorem 6.5] (cf. [11, Lemma 1.14]) shows that
$\Sigma_{\mathcal{M}_{\infty}^{i}}$ is in $\mathcal{M}$. We briefly sketch the
modifications of this argument that yield
$\Sigma_{\mathcal{M}_{\infty}^{i}}\upharpoonright\mathcal{M}\in\mathcal{M}$
also in the case $i>0$.

######  Lemma 5.13.

$\Sigma_{\mathcal{M}_{\infty}^{i}}\upharpoonright\mathcal{M}\in\mathcal{M}.$

###### Proof.

As we are no longer working in the derived model of the result of a fully
backgrounded construction to obtain $\mathcal{M}_{\infty}^{i}$ but in the
derived model of the result of an $\mathcal{M}$-construction, we briefly
sketch how to modify the proof of the successor case in the proof of [10,
Theorem 6.5] to obtain Lemma 5.13. We assume that the reader is familiar with
the proof of [10, Theorem 6.5]. Sargsyan’s argument uses universality in the
sense of [10, Lemma 2.13] or [22, Lemma 11.1] to for example show that
$\mathcal{Q}$-structures for iteration trees according to the iteration
strategy in question can be obtained via fully backgrounded constructions
relative to segments of that strategy. This sort of universality goes back to
Mitchell and Schindler’s proof of universality for $K^{c}$ in [5]. To apply
Sargsyan’s argument in our setting we for example need the following claim.

######  Claim.

Let $\mathcal{T}$ be an iteration tree on $\mathcal{M}_{\infty}^{i}$ in
$\mathcal{M}$ of limit length and let
$b=\Sigma_{\mathcal{M}_{\infty}^{i}}(\mathcal{T})$. Suppose $\mathcal{T}$ is
not entirely based on $(\mathcal{M}_{\infty}^{i})^{-}$ and
$\mathcal{Q}(b,\mathcal{T})$ exists. Let $\mathcal{S}$ be the node in
$\mathcal{T}$ such that the fragment of $\mathcal{T}$ from
$\mathcal{M}_{\infty}^{i}$-to-$\mathcal{S}$ is based on
$(\mathcal{M}_{\infty}^{i})^{-}$ and the rest of $\mathcal{T}$ is above
$\mathcal{S}^{-}$. Let $\mathcal{N}$ be the result of a fully backgrounded
construction relative to
$(\Sigma_{\mathcal{M}_{\infty}^{i}})_{\mathcal{S}^{-}}$ in
$\mathcal{Q}_{i}=\mathcal{M}|\delta_{i+1}$. Then

  1. (1)

the comparison of $\mathcal{Q}(b,\mathcal{T})$ and $\mathcal{N}$ in
$\mathcal{Q}_{i}[G]$ is successful, and

  2. (2)

$\mathcal{N}$ is universal when compared with $\mathcal{Q}(b,\mathcal{T})$,
i.e., it wins the comparison.

###### Proof.

Note that (2) follows from (1) by universality of fully backgrounded
constructions as in [10, Lemma 2.13] and [22, Lemma 11.1]. In particular, it
suffices for universality of $\mathcal{N}$ against
$\mathcal{Q}(b,\mathcal{T})$ that the comparison of
$\mathcal{Q}(b,\mathcal{T})$ and $\mathcal{N}$ is successful in
$\mathcal{Q}_{i}[G]$. So we are left with showing (1).

Suppose toward a contradiction that the comparison is not successful and let
$\mathcal{U}_{0}$ and $\mathcal{U}_{1}$ be iteration trees on
$\mathcal{Q}(b,\mathcal{T})$ and $\mathcal{N}$ witnessing this. Let
$G^{*}\supseteq G$ be generic over $\mathcal{W}$. Take a countable
substructure of a sufficiently large initial segment of $\mathcal{W}[G^{*}]$
reflecting this situation and write
$\bar{\mathcal{M}}[\bar{G}],\bar{\mathcal{Q}}(b,\mathcal{T}),\bar{\mathcal{N}},\bar{\mathcal{U}}_{0}$,
and $\bar{\mathcal{U}}_{1}$ for the images of
$\mathcal{M}[G],\mathcal{Q}(b,\mathcal{T}),\mathcal{N},\mathcal{U}_{0}$, and
$\mathcal{U}_{1}$ under the collapse. Let
$\bar{\mathcal{U}}_{1}^{\bar{\mathcal{M}}}$ be the iteration tree on
$\bar{\mathcal{M}}$ obtained by lifting $\bar{\mathcal{U}}_{1}$ onto the
background model. By the iterability proof for $\mathcal{M}$-constructions,
cf. Lemma 6.3, the models in $\bar{\mathcal{U}}_{1}$ can be realized into
$\mathcal{M}[G]$ and we can pick branches via $\Sigma$ by lifting (parts of
the tree) onto $\mathcal{W}[G^{*}]$. As $\Sigma$ can be extended to an
iteration strategy $\Sigma^{G^{*}}$ for trees in $\mathcal{W}[G^{*}]$, this
describes an iteration strategy for $\bar{\mathcal{N}}$. The
$\bar{\mathcal{N}}$-side of the comparison provides $\mathcal{Q}$-structures
for the $\bar{\mathcal{Q}}(b,\mathcal{T})$-side, cf., for example the proof of
[10, Lemma 5.5]. Therefore, the comparison of
$\bar{\mathcal{Q}}(b,\mathcal{T})$ and $\bar{\mathcal{N}}$ is successful. ∎

The rest of the argument in the proof of the successor case in the proof of
[10, Theorem 6.5] can be adapted analogously. We leave the details to the
reader. ∎

###  5.3. Certified extenders

The following definition is crucial for the rest of this paper as it describes
how we want to identify extenders that can serve as witnesses for the
strongness of $\kappa_{i}$ for $i<\omega$.

######  Definition 5.14.

Let $F$ be an extender with critical point $\kappa_{i}$ for some $i<\omega$.
Then we say $F$ is _certified_ iff

$\pi_{F}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{F}(\mathcal{M}_{\infty}^{i,-})},$

for $\Sigma^{i,\infty}=(\Sigma_{i})_{\mathcal{M}_{\infty}^{i,-}}$ the
canonical tail strategy of $\Sigma_{i}$. Here $\mathcal{M}_{\infty}^{i,-}$
denotes $\mathcal{M}_{\infty}^{i}|\delta_{\infty}^{i}$, where
$\delta_{\infty}^{i}$ denotes the largest Woodin cardinal of the
$\kappa_{i}$-suitable premouse $\mathcal{M}_{\infty}^{i}$.

######  Remark.

An extender $F$ on $\mathcal{M}$ with critical point $\kappa_{i}$ is uniquely
determined by $\pi_{F}\upharpoonright\mathcal{M}_{\infty}^{i,-}$ as
$\mathcal{M}_{\infty}^{i,-}\cap\operatorname{Ord}=(\kappa_{i}^{+})^{\mathcal{M}}$
by the following lemma.

######  Lemma 5.15.

Let $i\geq 0$. Then

$\mathcal{M}_{\infty}^{i,-}\cap\operatorname{Ord}=(\kappa_{i}^{+})^{\mathcal{M}}.$

###### Proof.

The lemma follows by the same argument as [13, Lemma 2.7(b)] (cf., also [27,
Lemma 3.38(2)]), using the fact that $\mathcal{M}_{\infty}^{i}$ is a direct
limit in the derived model $M$ of $\mathcal{M}[G]$ for
$G\subseteq\operatorname{Col}(\omega,{<}\kappa_{i})$ and that $\theta_{0}$ of
this derived model is $\omega_{2}^{\mathcal{M}[G]}$, i.e.,
$\theta_{0}^{M}=(\kappa_{i}^{+})^{\mathcal{M}}$. ∎

###  5.4. Preservation of certification under hulls

We first argue that taking hulls preserves the certification of extenders in
case the critical points of the extenders do not get collapsed.

######  Lemma 5.16.

Let $i<\omega$ and let $\bar{\mathcal{W}}$ be the transitive collapse of a
hull of $\mathcal{W}|\Omega$ for some sufficiently large $\Omega$ such that
$\mathcal{W}|\kappa_{i}\subseteq\bar{\mathcal{W}}$. Let

$\pi\colon\bar{\mathcal{W}}\rightarrow\mathcal{W}$

be the uncollapse embedding, $\bar{\mathcal{M}}$ the collapse of
$\mathcal{M}|\Omega$, and $(\bar{\kappa}_{j},\bar{\delta}_{j}\mid j<\omega)$
the preimages of $(\kappa_{j},\delta_{j}\mid j<\omega)$ (so in particular,
$\bar{\kappa}_{j}=\kappa_{j}$ and $\bar{\delta}_{j}=\delta_{j}$ for $j\leq
i$). Then every extender in $\bar{\mathcal{M}}$ with critical point
$\kappa_{i}$ is certified in $\bar{\mathcal{M}}$.

######  Remark.

Note that Lemma 5.16 is not trivial even though we do not collapse the direct
limit $\mathcal{M}^{i}_{\infty}$. The extender we want to certify gets
collapsed as its index gets collapsed, so the certification in the collapse
$\bar{\mathcal{W}}$ uses a different tree on $\mathcal{M}^{i}_{\infty}$ than
the certification in $\mathcal{W}$ does.

###### Proof of Lemma 5.16.

Let $E$ be an $\bar{\mathcal{M}}$-extender with critical point $\kappa_{i}$
and let $F=\pi(E)$ be its preimage under the collapse. We aim to show that

$\pi_{E}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{E}(\mathcal{M}_{\infty}^{i,-})},$

for $\Sigma^{i,\infty}=(\Sigma_{i})_{\mathcal{M}_{\infty}^{i,-}}$. As $F$ is
certified in $\mathcal{M}$, we can find an iteration tree $\mathcal{U}$ on
$\mathcal{M}_{\infty}^{i,-}$ of limit length and a cofinal well-founded branch
$b$ through $\mathcal{U}$ such that
$\pi_{F}(\mathcal{M}_{\infty}^{i,-})=\mathcal{M}_{b}^{\mathcal{U}}$ and

$\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{F}(\mathcal{M}_{\infty}^{i,-})}=\pi_{b}^{\mathcal{U}}.$

In particular, $\mathcal{U}^{\frown}b$ is by $\Sigma^{i,\infty}$. Let
$\bar{\mathcal{U}}$ and $\bar{b}$ be the images of $\mathcal{U}$ and $b$ under
the collapse, i.e., $\pi(\bar{\mathcal{U}},\bar{b})=(\mathcal{U},b)$. Then by
hull condensation (cf. Definition 3.1), $\bar{\mathcal{U}}^{\frown}\bar{b}$ is
by $\Sigma^{i,\infty}$ as well.

By elementarity of $\pi$, we have
$\pi_{E}(\mathcal{M}_{\infty}^{i,-})=\mathcal{M}_{\bar{b}}^{\bar{\mathcal{U}}}$
as $\mathcal{M}_{\infty}^{i,-}$ is fixed by the collapse. So

$\pi_{\bar{b}}^{\bar{\mathcal{U}}}\colon\mathcal{M}_{\infty}^{i,-}\rightarrow\pi_{E}(\mathcal{M}_{\infty}^{i,-}).$

As $\bar{\mathcal{U}}^{\frown}\bar{b}$ is by $\Sigma^{i,\infty}$ this implies

$\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{E}(\mathcal{M}_{\infty}^{i,-})}=\pi_{\bar{b}}^{\bar{\mathcal{U}}}.$

Note that the fact that $F$ is certified allows us to define
$b=\Sigma^{i,\infty}(\mathcal{U})$ via $F$ as follows: $b$ is the unique
branch such that

$\pi_{F}\,"\mathcal{M}_{\infty}^{i,-}=\operatorname{rng}(\pi_{b}^{\mathcal{U}}).$

By elementarity of $\pi$, this implies

$\pi_{E}\,"\mathcal{M}_{\infty}^{i,-}=\operatorname{rng}(\pi_{\bar{b}}^{\bar{\mathcal{U}}}).$

The following claim finishes the proof of Lemma 5.16.

######  Claim.

$\pi_{E}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi_{\bar{b}}^{\bar{\mathcal{U}}}.$

###### Proof.

Suppose toward a contradiction that
$\pi_{E}\upharpoonright\mathcal{M}_{\infty}^{i,-}\neq\pi_{\bar{b}}^{\bar{\mathcal{U}}}$
and let $a$ be minimal (in the canonical well-order of
$\mathcal{M}_{\infty}^{i,-}$) with
$\pi_{E}(a)\neq\pi_{\bar{b}}^{\bar{\mathcal{U}}}(a)$. Say
$\pi_{E}(a)>\pi_{\bar{b}}^{\bar{\mathcal{U}}}(a)$. Then
$\pi_{\bar{b}}^{\bar{\mathcal{U}}}(a)\not\in\operatorname{rng}(\pi_{E}\upharpoonright\mathcal{M}_{\infty}^{i,-})=\pi_{E}\,"\mathcal{M}_{\infty}^{i,-}$
as there is no element of $\mathcal{M}_{\infty}^{i,-}$ that $\pi_{E}$ can map
to $\pi_{\bar{b}}^{\bar{\mathcal{U}}}(a)$ while respecting the well-order of
$\mathcal{M}_{\infty}^{i,-}$. This contradicts
$\pi_{E}\,"\mathcal{M}_{\infty}^{i,-}=\operatorname{rng}(\pi_{\bar{b}}^{\bar{\mathcal{U}}})$.
The argument in the case $\pi_{E}(a)<\pi_{\bar{b}}^{\bar{\mathcal{U}}}(a)$ is
similar. ∎

∎

Now we extend Lemma 5.16 to deal with hulls that collapse the critical point
of the extender in question.

######  Lemma 5.17.

Let $i<\omega$ and let $\bar{\mathcal{W}}$ be the transitive collapse of a
hull of $\mathcal{W}|\Omega$ for some sufficiently large $\Omega$ such that
$\mathcal{W}|\delta_{i}\subseteq\bar{\mathcal{W}}$ and
$|\bar{\mathcal{W}}|<\kappa_{i}$. Let

$\pi\colon\bar{\mathcal{W}}\rightarrow\mathcal{W}$

be the uncollapse embedding, $\bar{\mathcal{M}}$ the collapse of
$\mathcal{M}|\Omega$, and $(\bar{\kappa}_{j},\bar{\delta}_{j}\mid j<\omega)$
the preimages of $(\kappa_{j},\delta_{j}\mid j<\omega)$ (so in particular,
$\bar{\kappa}_{j}=\kappa_{j}$ for $j<i$ and $\bar{\delta}_{j}=\delta_{j}$ for
$j\leq i$). Then every extender in $\bar{\mathcal{M}}$ with critical point
$\kappa_{i}$ is certified in $\bar{\mathcal{M}}$.

###### Proof.

This lemma can be proved similar as Lemma 5.16, so we only describe the
changes that need to be made. It no longer holds that the direct limit
$\mathcal{M}_{\infty}^{i,-}$ does not get collapsed, so let
$\bar{\mathcal{M}}_{\infty}^{i,-}$ be the preimage of
$\mathcal{M}_{\infty}^{i,-}$ under $\pi$. Choose
$E,F,\bar{\mathcal{U}},\mathcal{U},\bar{b},b$ as in the proof of Lemma 5.16.
As $\bar{\mathcal{U}}$ and $\mathcal{U}$ are iteration trees on different
models, i.e., $\bar{\mathcal{M}}_{\infty}^{i,-}$ and
$\mathcal{M}_{\infty}^{i,-}$, we cannot simply use hull condensation to argue
that $\bar{\mathcal{U}}^{\frown}\bar{b}$ is by $\Sigma^{i,\infty}$ as well.
Instead, we consider trees $\bar{\mathcal{T}}$ and $\mathcal{T}$ on
$\mathcal{P}_{i}^{-}$ with last models $\bar{\mathcal{M}}_{\infty}^{i,-}$ and
$\mathcal{M}_{\infty}^{i,-}$ respectively. Now apply hull condensation to the
stacks $\bar{\mathcal{T}}^{\frown}\bar{\mathcal{U}}$ and
$\mathcal{T}^{\frown}\mathcal{U}$ to obtain that
$\bar{\mathcal{T}}^{\frown}\bar{\mathcal{U}}$ is by $\Sigma_{i}$. This allows
us to continue as in the proof of Lemma 5.16 and obtain

$\pi_{E}\upharpoonright\bar{\mathcal{M}}_{\infty}^{i,-}=\pi_{\bar{b}}^{\bar{\mathcal{U}}}=\pi^{\bar{\Sigma}^{i,\infty}}_{\bar{\mathcal{M}}_{\infty}^{i,-},\pi_{E}(\bar{\mathcal{M}}_{\infty}^{i,-})},$

as desired, where
$\bar{\Sigma}^{i,\infty}=(\Sigma_{i})_{\bar{\mathcal{M}}_{\infty}^{i,-}}$. ∎

###  5.5. All extenders on the $\mathcal{M}$-sequence are certified

Similarly as in [11, Lemma 1.23] we have that fully backgrounded extenders
with critical point $\kappa_{i}$ for some $i\geq 0$ are certified.

######  Lemma 5.18.

Let $i\geq 0$ and let $E$ be an extender on $\mathcal{M}$ with critical point
$\kappa_{i}$ that is added by the fully backgrounded part of the construction
of $\mathcal{M}$. Then $E$ is certified.

###### Proof.

Let $F$ be an extender with critical point $\kappa_{i}$ that gets added by the
fully backgrounded part of the construction of $\mathcal{M}$. Say
$\mathcal{N}_{\xi+1}=(J_{\alpha}^{\vec{E}},\in,\vec{E},F)$ and let $F^{*}$ be
the background extender of $F$. Let $\mathcal{U}$ be an iteration tree on
$\mathcal{M}_{\infty}^{i,-}$ of limit length such that if we let
$b=\Sigma^{i,\infty}(\mathcal{U})$,

$\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{F}(\mathcal{M}_{\infty}^{i,-})}=\pi_{b}^{\mathcal{U}}.$

Moreover, let $\mathcal{U}^{*}$ be an iteration tree on
$\mathcal{M}_{\infty}^{i,-}$ of limit length such that if we let
$b^{*}=\Sigma^{i,\infty}(\mathcal{U}^{*})$,

$\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{F^{*}}(\mathcal{M}_{\infty}^{i,-})}=\pi_{b^{*}}^{\mathcal{U}^{*}}.$

Let

$k\colon\operatorname{Ult}(\mathcal{N}_{\xi+1},F)\rightarrow\operatorname{Ult}(\mathcal{W},F^{*})$

be the canonical realization map. Then
$k\upharpoonright\pi_{F}(\mathcal{M}_{\infty}^{i,-})\colon\pi_{F}(\mathcal{M}_{\infty}^{i,-})\rightarrow\pi_{F^{*}}(\mathcal{M}_{\infty}^{i,-})$
makes the diagram in Figure 2 commute.

$\pi_{F^{*}}$$\pi_{F^{*}}(\mathcal{M}_{\infty}^{i,-})$$\mathcal{M}_{\infty}^{i,-}$$\pi_{F}$$\pi_{F}(\mathcal{M}_{\infty}^{i,-})$$k$
Figure 2. Realization into the ultrapower by the background extender.

As $F^{*}$ is an extender on $\mathcal{W}$ and $(\Sigma_{i})^{g}$ is strongly
guided by a generic enumeration of $\vec{\tau}\subseteq\mathcal{P}_{i}$ for
all $g\subseteq\operatorname{Col}(\omega,\mathcal{P}_{i})$,

$\pi_{F^{*}}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi_{b^{*}}^{\mathcal{U}^{*}}.$

We now define a cofinal branch $c$ through $\mathcal{U}$ such that
$k(\mathcal{M}_{c}^{\mathcal{U}})=\mathcal{M}_{b^{*}}^{\mathcal{U}^{*}}$ and
$\pi_{F}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi_{c}^{\mathcal{U}}$.

Let

$X=\\{\alpha<\operatorname{lh}(\mathcal{U})\mid\exists\beta\,\alpha=k(\pi_{F}(\beta))\text{
and }b_{\alpha}^{*}\in\pi_{F^{*}}(\mathcal{N}_{\xi+1})\\}.$

For $\alpha\in X$ let $c_{\alpha}\in\operatorname{Ult}(\mathcal{N}_{\xi+1},F)$
be such that

$k(c_{\alpha})=b_{\alpha}^{*}.$

Let $c=\bigcup_{\alpha\in X}c_{\alpha}$. Then $c$ is as desired and the
following claim finishes the proof of Lemma 5.18.

######  Claim.

$b=c$.

###### Proof.

Both $\mathcal{U}^{\frown}c$ and $\mathcal{U}^{*}{}^{\frown}b^{*}$ are
iteration trees on $\mathcal{M}_{\infty}^{i,-}$ and $\mathcal{U}^{\frown}c$ is
a hull of $\mathcal{U}^{*}{}^{\frown}b^{*}$. As
$\mathcal{U}^{*}{}^{\frown}b^{*}$ is according to $\Sigma^{i,\infty}$, by hull
condensation $\mathcal{U}^{\frown}c$ is according to $\Sigma^{i,\infty}$ as
well and hence

$c=\Sigma^{i,\infty}(\mathcal{U})=b.$

∎

∎

Now we argue that generically countably complete extenders are certified as
well.

######  Lemma 5.19.

Let $i\geq 0$ and let $E$ be an extender on $\mathcal{M}$ with critical point
$\kappa_{i}$ that is added by the generically countably complete part of the
construction of $\mathcal{M}$. Then $E$ is certified.

###### Proof.

Let $E$ be an extender on the sequence of $\mathcal{M}$ with critical point
$\kappa_{i}$ that is added by the generically countably complete part of the
construction of $\mathcal{M}$ and suppose toward a contradiction that $E$ is
not certified. Let $\hat{\mathcal{W}}$ be a countable elementary substructure
of a sufficiently large initial segment $\mathcal{W}|\Omega$ of $\mathcal{W}$.
Let

$\sigma\colon\hat{\mathcal{W}}\rightarrow\mathcal{W}|\Omega$

and suppose that all relevant objects are in the range of $\sigma$. For
notational simplicity write $\mathcal{M}$ for the result of the
$\mathcal{M}$-construction in $\mathcal{W}|\Omega$ and let $\hat{\mathcal{M}}$
be the transitive collapse of $\mathcal{M}$, i.e., the result of the
$\mathcal{M}$-construction in $\hat{\mathcal{W}}$. Say
$\sigma(\hat{\kappa}_{i},\hat{E},\hat{\mathcal{M}}_{\infty}^{i},\hat{\Sigma}^{i,\infty})=(\kappa_{i},E,\mathcal{M}_{\infty}^{i},\Sigma^{i,\infty})$.

$\hat{\mathcal{M}}_{\infty}^{i}$ is countable, suitable at $\hat{\kappa}_{i}$,
and $\vec{A}$-iterable in the derived model $M_{i}$ of
$\mathcal{M}|\delta_{i+1}[G]$ for
$G\subseteq\operatorname{Col}(\omega,{<}\kappa_{i})$ and hence an element of
the direct limit system giving rise to $\mathcal{M}_{\infty}^{i}$. So when
comparing $\hat{\mathcal{M}}_{\infty}^{i}$ via $\hat{\Sigma}^{i,\infty}$ and
$\mathcal{M}_{\infty}^{i}$ via $\Sigma^{i,\infty}$ inside $\mathcal{W}$,
$\hat{\mathcal{M}}_{\infty}^{i}$ iterates to $\mathcal{M}_{\infty}^{i}$.
Moreover,

$\sigma\upharpoonright\hat{\mathcal{M}}_{\infty}^{i,-}=\pi^{\hat{\Sigma}^{i,\infty}}_{\hat{\mathcal{M}}_{\infty}^{i,-},\sigma(\hat{\mathcal{M}}_{\infty}^{i,-})}$

as there are term relations that are strongly guiding
$\hat{\Sigma}^{i,\infty}$. We have that $E=\sigma(\hat{E})$ is generically
countably complete, so by Fact 4.2 there is a realization map

$\tau\colon\operatorname{Ult}(\hat{\mathcal{M}},\hat{E})\rightarrow\mathcal{M}$

such that $\sigma\upharpoonright\hat{\mathcal{M}}=\tau\circ\pi_{\hat{E}}$
where
$\pi_{\hat{E}}\colon\hat{\mathcal{M}}\rightarrow\operatorname{Ult}(\hat{\mathcal{M}},\hat{E})$
denotes the ultrapower embedding. Therefore, $\pi_{\hat{E}}$ respects
$\hat{\Sigma}^{i,\infty}$ and

$\pi_{\hat{E}}\upharpoonright\hat{\mathcal{M}}_{\infty}^{i,-}=\pi^{\hat{\Sigma}^{i,\infty}}_{\hat{\mathcal{M}}_{\infty}^{i,-},\pi_{\hat{E}}(\hat{\mathcal{M}}_{\infty}^{i,-})}.$

So $\hat{E}$ is certified via $\hat{\Sigma}^{i,\infty}$. By elementarity of
$\sigma$, this implies that $E$ is certified via $\Sigma^{i,\infty}$, a
contradiction. ∎

###  5.6. Characterization via operators

Now we argue that we can also characterize extenders using operators in order
to obtain the extenders in a simply definable way (e.g., in
$M_{1}^{\\#,\Sigma_{i}}$ of the current level).

We define universal mice at $\kappa_{i}$ and $\kappa_{i}$-good universal mice
as in [11]. The definitions in [11] generalize to $\kappa_{i}$, $i>0$ by
considering $\mathcal{M}$-constructions instead of fully backgrounded
constructions. Then [11, Lemmas 1.14 and 1.15] generalize and we can show that
good universal mice exist.

As usual, we write

$\mathcal{O}_{\eta}^{\mathcal{Q}}=\bigcup\\{\mathcal{Q}||\beta\mid\rho_{\omega}(\mathcal{Q}||\beta)\leq\eta\text{
and }\eta\text{ is a cutpoint in }\mathcal{Q}\\}$

for premice $\mathcal{Q}$ and ordinals
$\eta<\mathcal{Q}\cap\operatorname{Ord}$.

######  Definition 5.20.

Let $i<\omega$ and let $X\in\mathcal{W}|\delta_{i+1}$ be a premouse such that
$\mathcal{M}|\delta_{i}\unlhd X$. We say
$\mathcal{Q}\subseteq\mathcal{W}|\delta_{i+1}$ is a _universal mouse over $X$
at $\kappa_{i}$_ if $\mathcal{Q}$ is a $(\delta_{i+1},\delta_{i+1})$-iterable
$X$-mouse such that for every $\eta\in(X\cap\operatorname{Ord},\delta_{i+1})$,
$\mathcal{O}_{\eta}^{\mathcal{Q}}\unlhd Lp(\mathcal{Q}|\eta)$ and for a
stationary set of $\xi<\delta_{i+1}$,
$(\xi^{+})^{\mathcal{Q}}=(\xi^{+})^{\mathcal{W}}$.

Before we can define good universal mice, we need some notation. Let
$i<\omega$ and let $X\in\mathcal{W}|\delta_{i+1}$ be a premouse with
$X\unrhd\mathcal{M}|\delta_{i}$. For any $X$-premouse
$\mathcal{Q}\subseteq\mathcal{W}|\delta_{i+1}$, let $\kappa^{\mathcal{Q}}$
denote the least ${<}\delta_{i+1}$-strong cardinal of $\mathcal{Q}$, if it
exists. For any universal mouse $\mathcal{Q}\subseteq\mathcal{W}|\delta_{i+1}$
over $X$ at $\kappa_{i}$, consider the derived model $M_{\mathcal{Q}}$
computed in $\mathcal{Q}[g]$ for a
$\operatorname{Col}(\omega,{<}\kappa^{\mathcal{Q}})$-generic $g$ over
$\mathcal{Q}$. Working in $M_{\mathcal{Q}}$, let

$\mathcal{F}_{\mathcal{Q}}=\\{(\mathcal{N},\vec{A})\mid\mathcal{N}\text{ is
countable, suitable at $\kappa_{i}$, and }\vec{A}\text{-iterable}\\}.$

Write $\mathcal{M}_{\infty}^{\mathcal{Q}}$ for the direct limit of
$\mathcal{F}_{\mathcal{Q}}$ and $\delta_{\infty}^{\mathcal{Q}}$ for the
largest Woodin cardinal of $\mathcal{M}_{\infty}^{\mathcal{Q}}$. Note that
$\mathcal{M}_{\infty}^{\mathcal{Q}}\in\mathcal{Q}$ and let

$\mathcal{R}^{\mathcal{Q}}=\mathcal{M}_{\infty}^{\mathcal{Q}}|((\delta_{\infty}^{\mathcal{Q}})^{+})^{\mathcal{M}_{\infty}^{\mathcal{Q}}}.$

######  Definition 5.21.

Let $i<\omega$ and let $X\in\mathcal{W}|\delta_{i+1}$ be a premouse such that
$\mathcal{M}|\delta_{i}\unlhd X$. We say a universal mouse $\mathcal{Q}$ over
$X$ at $\kappa_{i}$ is a _$\kappa_{i}$ -good universal mouse over $X$_ iff it
has a $(\delta_{i+1},\delta_{i+1})$-iteration strategy $\Phi$ such that
whenever $i\colon\mathcal{Q}\rightarrow\mathcal{N}$ is an embedding obtained
from an iteration according to $\Phi$, $\mathcal{N}$ is a universal mouse,
$\mathcal{R}^{\mathcal{N}}$ is a $\Sigma_{i}$-iterate of $\mathcal{P}_{i}$,
and whenever $g$ is generic over $\mathcal{W}$ for a partial order in
$\mathcal{N}$,

$(\Sigma^{g}_{i})_{\mathcal{R}^{\mathcal{N}}}\upharpoonright\mathcal{N}[g]\in
L[\mathcal{N}[g]].$

The existence of good universal mice follows as in [11, Lemmas 1.14 and 1.15],
cf. also the proof of Lemma 5.13.

######  Lemma 5.22.

Let $i<\omega$ and let $X\unrhd\mathcal{M}|\delta_{i}$ be a premouse with
$X\in\mathcal{W}|\delta_{i+1}$. Fix some ordinal $\eta$ with
$X\cap\operatorname{Ord}<\eta<\delta_{i+1}$ and let
$\mathcal{Q}=\mathcal{Q}^{X,\eta}$ be the result of an
$\mathcal{M}$-construction (or a fully backgrounded construction) above $X$
inside $\mathcal{W}|\delta_{i+1}$ that only uses extenders with critical point
above $\eta$. Let $\Phi$ be the iteration strategy for $\mathcal{Q}$ induced
by the background strategy and let $\mathcal{N}$ be an iterate of
$\mathcal{Q}$ (above $X$) according to $\Phi$ such that the iteration
embedding $i\colon\mathcal{Q}\rightarrow\mathcal{N}$ exists. Then
$\mathcal{N}$ is a $\kappa_{i}$-good universal mouse over $X$.

Now we can define operators $S_{i}$ for each $i>0$ analogous to Sargsyan’s
$S$-operator in [11] that goes back to Steel [22] and show that the operators
are independent of the choice of the good universal mice. This yields another
way of characterizing extenders and we will argue that it gives rise to the
same extenders as our certification defined in Section 5.3.

Let $i<\omega$ and let $\mathcal{Q}_{i}$ be a $\kappa_{i}$-good universal
mouse. Write
$\pi^{\mathcal{Q}_{i}}\colon\mathcal{P}_{i}\rightarrow\mathcal{R}^{\mathcal{Q}}$
for the iteration embedding according to $\Sigma_{i}$. Let
$T_{i}\subseteq\mathcal{P}_{i}$ be the collection of all terms that
$\Sigma_{i}$ guides correctly. Write
$\pi^{\mathcal{Q}_{i}}_{\tau}=\pi^{\mathcal{Q}_{i}}\upharpoonright
H_{\tau}^{\mathcal{P}_{i}}$ for $\tau\in T_{i}$ and

$\mathcal{Q}_{i}^{+}=(\mathcal{Q}_{i}|((\kappa^{\mathcal{Q}_{i}})^{+})^{\mathcal{Q}_{i}},\in,\\{\pi_{\tau}^{\mathcal{Q}_{i}}\\}_{\tau\in
T_{i}}).$

For any $X\in\mathcal{W}|\delta_{i+1}$ and some $\kappa_{i}$-good universal
mouse $\mathcal{Q}_{i}$ over $X$ let

$S_{i}^{\mathcal{Q}_{i}}(X)=Hull_{1}^{\mathcal{Q}_{i}^{+}}(X).$

As in [11, Lemma 1.18] the operator $S_{i}^{\mathcal{Q}_{i}}$ does not depend
on the choice of $\mathcal{Q}_{i}$, so we can simplify the notation and write
$S_{i}(X)$ for $S_{i}^{\mathcal{Q}_{i}}(X)$ for some $\kappa_{i}$-good
universal mouse $\mathcal{Q}_{i}$ over $X$.

######  Definition 5.23.

Let $i<\omega$. We say an $\mathcal{M}$-extender $E$ with critical point
$\kappa_{i}$ is _characterized via $S_{i}$_ iff

$Hull_{1}^{S_{i}(\mathcal{M}|\xi)}(\mathcal{M}|\kappa_{i})=S_{i}(\mathcal{M}|\kappa_{i})$

where $\xi\geq\kappa_{i}^{+}$ is the least $i$-weak cutpoint of
$\operatorname{Ult}(\mathcal{M},E)$ above $\kappa_{i}$, and if $\alpha$ is the
successor of $\xi$ in $\operatorname{Ult}(\mathcal{M},E)$, then $E$ is the
$(\kappa_{i},\alpha)$-extender derived from the uncollapse map

$k\colon S_{i}(\mathcal{M}|\kappa_{i})\rightarrow S_{i}(\mathcal{M}|\xi).$

Write $\xi(E)$ for the ordinal $\xi$ defined above.

######  Lemma 5.24.

Let $i<\omega$. An $\mathcal{M}$-extender $E$ with critical point $\kappa_{i}$
is certified iff it is characterized via the operator $S_{i}$.

###### Proof.

The forward direction, i.e., that any certified extender with critical point
$\kappa_{i}$ is characterized via the operator $S_{i}$, follows from the fact
that if an extender $E$ is according to
$(\Sigma_{i})_{\mathcal{M}_{\infty}^{i,-}}$ it clearly moves term relations
correctly.

For the other direction, suppose $E$ is the extender derived from $k\colon
S_{i}(\mathcal{M}|\kappa_{i})\rightarrow S_{i}(\mathcal{M}|\xi(E))$. Let
$\mathcal{Q}$ be a $\kappa_{i}$-good universal mouse over
$\mathcal{M}|\kappa_{i}$ and let $\mathcal{Q}^{*}$ be a $\kappa_{i}$-good
universal mouse over $\mathcal{M}|\xi(E)$. Consider the direct limits
$\mathcal{R}^{\mathcal{Q}}$ and $\mathcal{R}^{\mathcal{Q}^{*}}$ of suitable
premice in the corresponding derived models. Both, $\mathcal{R}^{\mathcal{Q}}$
and $\mathcal{R}^{\mathcal{Q}^{*}}$, are $\Sigma_{i}$-iterates of
$\mathcal{P}_{i}$. Moreover, $\mathcal{R}^{\mathcal{Q}}$ is a point in the
direct limit system giving rise to $\mathcal{R}^{\mathcal{Q}^{*}}$. In fact,
$\mathcal{R}^{\mathcal{Q}^{*}}$ is a
$(\Sigma_{i})_{\mathcal{R}^{\mathcal{Q}}}$-iterate of
$\mathcal{R}^{\mathcal{Q}}$. Note that $\mathcal{M}|\delta_{i+1}$ is a
$\kappa_{i}$-good universal mouse over $\mathcal{M}|\kappa_{i}$ with
$\mathcal{R}^{\mathcal{M}|\delta_{i+1}}=\mathcal{M}_{\infty}^{i}$. So the fact
that the iteration from $\mathcal{R}^{\mathcal{Q}}$ to
$\mathcal{R}^{\mathcal{Q}^{*}}$ is according to
$(\Sigma_{i})_{\mathcal{R}^{\mathcal{Q}}}$ suffices to argue that
$\pi_{E}\upharpoonright\mathcal{M}_{\infty}^{i,-}=\pi^{\Sigma^{i,\infty}}_{\mathcal{M}_{\infty}^{i,-},\pi_{E}(\mathcal{M}_{\infty}^{i,-})}$.
∎

###  5.7. Properties of the operators

The reason we use the characterization of extenders via operators is that this
allows us to prove that the extenders that can be characterized this way are
easily definable. This is made precise in the following lemma (cf. [11, Lemma
1.22]). The proof is analogous to the proof of [11, Lemma 1.22], so we leave
it to the reader.

######  Lemma 5.25.

Let $\xi$ be a cardinal in $\mathcal{M}$ with
$(\kappa_{i}^{+})^{\mathcal{M}}<\xi<\delta_{i+1}$ for some $i<\omega$. Then

$S_{i}(\mathcal{M}|\xi)\in M_{1}^{\\#,\Sigma_{i}}(\mathcal{M}|\xi).$

##  6\. Proving that it works

###  6.1. Iterability

We now prove that hulls of $\mathcal{M}$ are iterable. The argument will
combine the standard lifting to the background model for fully backgrounded
constructions from [6] with the almost linearity of the iteration tree when
considering extenders with critical point (the image of) some $\kappa_{i}$,
$i<\omega$, and index above (the image of) $\delta_{i+1}$ (cf. [20]).

Our concept of an almost linear iteration tree is a generalization of the one
in [20] as we distinguish two different types of extenders – with and without
a background – and allow arbitrary iteration trees for backgrounded extenders.
We make this more precise now and use the notation from [25] for iteration
trees $\mathcal{T}$ and the associated tree order $T$.

######  Definition 6.1.

Let $\mathcal{N}$ be the result of an $\mathcal{M}$-construction in some
translatable structure or an iterate of the result of such a construction.
Write $(\kappa_{i}^{\mathcal{N}},\delta_{i}^{\mathcal{N}}\mid i<\omega)$ for
the sequence of the specific cardinals $\kappa_{i}$ and $\delta_{i}$,
$i<\omega$, chosen in the definition of the $\mathcal{M}$-construction.

  1. (1)

An extender $E$ on the $\mathcal{N}$-sequence is called _strongness witness_
or _non-backgrounded_ iff the critical point of $E$ is
$\kappa_{i}^{\mathcal{N}}$ for some $i<\omega$ and the index of $E$ is above
$\delta_{i+1}^{\mathcal{N}}$. Otherwise we say $E$ is _backgrounded_.

  2. (2)

An iteration tree $\mathcal{T}$ on $\mathcal{N}$ is called _almost linear on
strongness witnesses_ iff

    1. (a)

for all $\alpha+1<\operatorname{lh}(\mathcal{T})$ such that
$E_{\alpha}^{\mathcal{T}}$ is a strongness witness we have
$\operatorname{pred}_{T}^{*}(\alpha+1)\in[0,\alpha]_{T}$, where
$\operatorname{pred}_{T}^{*}(\alpha+1)$ denotes the unique ordinal $\beta+1$
such that $\mathcal{M}_{\beta+1}^{\mathcal{T}}$ is obtained as an ultrapower
by a strongness witness $E_{\beta}^{\mathcal{T}}$ and
$i_{\beta+1,\operatorname{pred}_{T}(\alpha+1)}\colon\mathcal{M}_{\beta}^{*,\mathcal{T}}\rightarrow\mathcal{M}_{\operatorname{pred}_{T}(\alpha+1)}^{\mathcal{T}}$
is an iteration via backgrounded extenders, and

    2. (b)

any $\alpha<\operatorname{lh}(\mathcal{T})$ has only finitely many
$T$-successors $\beta+1$ such that $E_{\beta}^{\mathcal{T}}$ is a strongness
witness.

The following lemma uses similar ideas as the proof of [20, Lemma 2.4].

######  Lemma 6.2.

Let $\mathcal{N}$ be the result of an $\mathcal{M}$-construction in some
translatable structure. Then any normal iteration tree $\mathcal{T}$ on
$\mathcal{N}$ is almost linear.

###### Proof.

Analogous to the proof of [20, Lemma 2.4], we start with the following claim.

######  Claim.

Let $\mathcal{U}$ be a normal iteration tree on $\mathcal{N}$ of successor
length $\lambda+1$ and let $i+1\in(0,\lambda]_{U}$ be such that
$E=E_{i}^{\mathcal{U}}$ is a strongness witness with critical point
$\kappa_{E}$ and length $\lambda_{E}$. Then there is no strongness witness $F$
on the sequence of $\mathcal{M}_{\lambda}^{\mathcal{U}}$ with critical point
$\kappa_{F}$ and length $\lambda_{F}$ such that

$\kappa_{E}\leq\kappa_{F}<\lambda_{E}\leq\lambda_{F}.$

###### Proof.

Suppose the claim does not hold and let $F$ be a strongness witness on the
sequence of $\mathcal{M}_{\lambda}^{\mathcal{U}}$ with
$\kappa_{E}\leq\kappa_{F}<\lambda_{E}\leq\lambda_{F}$. Then it is not hard to
see that
$\rho_{\omega}(J_{\lambda_{F}}^{\mathcal{M}_{\lambda}^{\mathcal{U}}})\geq\lambda_{E}$,
cf. the subclaim in the proof of [20, Lemma 2.4, Claim 1]. This implies that
$F|\xi\in J_{\lambda_{F}}^{\mathcal{M}_{\lambda}^{\mathcal{U}}}$ for all
$\kappa_{F}<\xi<\lambda_{E}$. In particular, $\kappa_{F}$ is
$\lambda_{E}$-strong in $\mathcal{M}_{\lambda}^{\mathcal{U}}$. Therefore,
$\kappa_{F}$ is $\lambda_{E}$-strong in $\mathcal{M}_{i+1}^{\mathcal{U}}$.
This yields a contradiction as $\mathcal{M}_{i+1}^{\mathcal{U}}$ is an
ultrapower by $E$, so there is no extender with critical point between
$\kappa_{E}$ and $\lambda_{E}$ that overlaps $\lambda_{E}$ in
$\mathcal{M}_{i+1}^{\mathcal{U}}$. ∎

The claim implies that for every normal iteration tree $\mathcal{U}$ on
$\mathcal{N}$ of length $\lambda+2$ and every $i<j+1<\lambda+1$ with
$j+1\in(i,\lambda]_{U}$ and
$i=U-\operatorname{pred}(j+1)=U-\operatorname{pred}(\lambda+1)$ such that
$E_{j}^{\mathcal{U}}$ and $E_{\lambda}^{\mathcal{U}}$ are both strongness
witnesses

$\operatorname{crit}(E_{j}^{\mathcal{U}})>\operatorname{crit}(E_{\lambda}^{\mathcal{U}}).$

Using this observation, a short argument as in the proof of [20, Lemma 2.4]
yields that any normal iteration tree $\mathcal{T}$ on $\mathcal{N}$ is almost
linear. ∎

######  Lemma 6.3.

Let $i<\omega$ and let $\bar{\mathcal{W}}$ be the transitive collapse of a
hull of $\mathcal{W}|\Omega$ for some sufficiently large $\Omega$ such that
$\mathcal{W}|\kappa_{i}\subseteq\bar{\mathcal{W}}$. Let

$\pi\colon\bar{\mathcal{W}}\rightarrow\mathcal{W}$

be the uncollapse embedding, $\bar{\mathcal{M}}$ the collapse of
$\mathcal{M}|\Omega$, and $(\bar{\kappa}_{j},\bar{\delta}_{j}\mid j<\omega)$
the preimages of $(\kappa_{j},\delta_{j}\mid j<\omega)$ (so in particular,
$\bar{\kappa}_{j}=\kappa_{j}$ and $\bar{\delta}_{j}=\delta_{j}$ for $j\leq
i$). Then $\bar{\mathcal{M}}$ is $(\delta_{i+1},\delta_{i+1})$-iterable with
respect to extenders with critical point above $\kappa_{i}$.

###### Proof.

Let $i<\omega$, $\bar{\mathcal{W}}$ and $\bar{\mathcal{M}}$ be as in the
statement of the lemma. For notational simplicity we write $\mathcal{M}$ and
$\mathcal{W}$ instead of $\mathcal{M}|\Omega$ and $\mathcal{W}|\Omega$ as it
will be clear from the context if we refer to the proper class model or the
restriction to $\Omega$. We recursively describe an iteration strategy
$\Lambda$ for $\bar{\mathcal{M}}$ acting on iteration trees of length
${<}\delta_{i+1}$ which use extenders with critical point above $\kappa_{i}$.
The recursive construction will depend on what type of extenders get used –
backgrounded extenders or non-backgrounded strongness witnesses. For
notational simplicity, we describe the iteration strategy $\Lambda$ only for
normal trees and leave it to the reader to generalize the construction to
stacks of normal trees. Simultaneously to the construction of $\Lambda$, we
define a _realization of $\mathcal{T}$ into a backgrounded iteration of
$\mathcal{M}$_. That means, for every normal iteration tree $\mathcal{T}$ on
$\bar{\mathcal{M}}$ we recursively define a stack of iteration trees
$\vec{\mathcal{U}}=(\mathcal{U}_{\alpha}\mid\alpha<\gamma)$ on $\mathcal{M}$
and an embedding

$k\colon\operatorname{lh}(\mathcal{T})\rightarrow\gamma\times\bigcup_{\alpha<\gamma}\operatorname{lh}(\mathcal{U}_{\alpha})$

such that $\mathcal{U}$ only uses fully backgrounded extenders in
$\mathcal{M}$ and for every ordinal $\xi<\operatorname{lh}(\mathcal{T})$ if
$k(\xi)=(\alpha,\eta)$ there is an elementary embedding

$\iota_{\xi}\colon\mathcal{M}_{\xi}^{\mathcal{T}}\rightarrow\mathcal{M}_{\eta}^{\mathcal{U}_{\alpha}}.$

Here,
$\pi\upharpoonright\bar{\mathcal{M}}=\iota_{0}\colon\bar{\mathcal{M}}\rightarrow\mathcal{M}$
and the other embeddings $\iota_{\xi}$ for $\xi>0$ will be obtained as copy or
realization maps. Note that for example in case $\mathcal{T}$ only uses non-
backgrounded extenders, $\gamma=1$, $\operatorname{lh}(\mathcal{U}_{0})=1$ and
all $\iota_{\xi}$ for $0<\xi<\operatorname{lh}(\mathcal{T})$ are realization
maps of $\mathcal{M}_{\xi}^{\mathcal{T}}$ into
$\mathcal{M}=\mathcal{M}_{0}^{\mathcal{U}_{0}}$.

Now we turn to the recursive definition of $\Lambda$ together with
realizations into a backgrounded iteration of $\mathcal{M}$. Let $\mathcal{T}$
be a normal iteration tree of limit length $\lambda<\delta_{i+1}$ on
$\bar{\mathcal{M}}$ according to $\Lambda$ and recall that $\mathcal{T}$ is
almost linear by Lemma 6.2. We distinguish the following cases.

######  Case 1.

$\mathcal{T}$ only uses fully backgrounded extenders.

The cofinal well-founded branches through $\mathcal{T}\upharpoonright\eta$ for
limit ordinals $\eta\leq\lambda$ that get picked by $\Lambda$ are given by the
usual strategy of lifting the iteration tree $\mathcal{T}$ to a tree on the
background model $\bar{\mathcal{W}}$ via the backgrounds of the extenders used
in $\mathcal{T}$ and choosing the cofinal branches according to the pullback
strategy for $\bar{\mathcal{W}}$ of the iteration strategy of $\mathcal{W}$.
This yields canonical cofinal well-founded branches through
$\mathcal{T}\upharpoonright\eta$ for all limit ordinals $\eta\leq\lambda$. Let
$\iota_{\xi}$ for $\xi<\operatorname{lh}(\mathcal{T})$ be the copy maps
according to $\pi\upharpoonright\bar{\mathcal{M}}$ and let
$\mathcal{U}_{0}=\pi^{\mathcal{T}}$ be the corresponding copy tree.

######  Case 2.

$\mathcal{T}$ uses a non-backgrounded extender $E$ in addition to fully
backgrounded extenders.

The iteration tree $\mathcal{T}$ naturally splits into an iteration tree
$\mathcal{T}_{1}$ before applying $E$ and an iteration tree $\mathcal{T}_{2}$
after applying $E$ as backgrounded extenders after $E$ cannot make us go back
to a model before the application of $E$, i.e., a model in $\mathcal{T}_{1}$.
Let $\bar{\mathcal{M}}^{1}=\mathcal{M}_{\eta}^{\mathcal{T}_{1}}$ denote the
model in $\mathcal{T}_{1}$ to which $E$ gets applied and write
$\mathcal{T}_{1}\upharpoonright\bar{\mathcal{M}}^{1}$ for the tree
$\mathcal{T}_{1}\upharpoonright(\eta+1)$ with last model
$\bar{\mathcal{M}}^{1}$. Moreover, let $\iota_{0}\mathcal{T}_{1}$ be the copy
tree and $\iota_{\eta}$ the copy map resulting from copying $\mathcal{T}_{1}$
up to $\bar{\mathcal{M}}^{1}$ onto $\mathcal{M}$, cf. Figure 4 below.

######  Claim 1.

There is a realization map
$\tau\colon\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)\rightarrow\mathcal{M}^{1}$.

###### Proof.

Let $\iota_{\xi}\colon\bar{\mathcal{N}}\rightarrow\mathcal{N}$ be the copy map
resulting of copying $\mathcal{T}_{1}$ onto $\mathcal{M}$ via $\iota_{0}$. Let
$\mathcal{W}^{1}$ and $\mathcal{W}_{\mathcal{N}}$ be the results of lifting
the iteration from $\mathcal{M}$ to $\mathcal{M}^{1}$ as well as the iteration
from $\mathcal{M}^{1}$ to $\mathcal{N}$ onto an iteration of the background
model $\mathcal{W}$.

$\bar{\mathcal{M}}^{1}$$\iota_{\eta}$$\mathcal{M}^{1}$$\bar{\mathcal{N}}\ni
E$$\bar{\mathcal{W}}^{1}$$\bar{\mathcal{W}}_{\mathcal{N}}$$\mathcal{N}$$\mathcal{W}^{1}$$\mathcal{W}_{\mathcal{N}}$$\;\;\pi_{E}$$\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)$$\;\tau$$\iota_{\xi}$
Figure 3. The realization map $\tau$.

As $E$ is non-backgrounded with critical point above $\kappa_{i}$,
$\iota_{\xi}(E)$ is generically countably complete in
$\mathcal{W}_{\mathcal{N}}$ and has critical point at least $\kappa_{i+1}$.
Recall that $|\bar{\mathcal{N}}|<\delta_{i+1}<\kappa_{i+1}$ as
$\lambda=\operatorname{lh}(\mathcal{T})<\delta_{i+1}$. Therefore,
$\iota_{\xi}(E)$ is countably complete in $\mathcal{W}_{\mathcal{N}}[G]$,
where $G$ is $\operatorname{Col}(\omega,\delta_{i+1})$-generic over
$\mathcal{W}_{\mathcal{N}}$ and the length $\lambda_{E}$ of $E$ is countable
in $\mathcal{W}_{\mathcal{N}}[G]$. So if we write
$\kappa=\operatorname{crit}(\iota_{\xi}(E))$ we can fix a function

$\rho\colon\iota_{\xi}\,"[\lambda_{E}]^{{<}\omega}\rightarrow\kappa$

such that for each $a\in[\lambda_{E}]^{{<}\omega}$ and
$A\subseteq[\operatorname{crit}(E)]^{|a|}$,

$A\in
E_{a}\Leftrightarrow\iota_{\xi}(A)\in\iota_{\xi}(E)_{\iota_{\xi}(a)}\Leftrightarrow\rho\,"\iota_{\xi}(a)\in\iota_{\xi}(A).$

Here, note that
$\iota_{\xi}(A)\in\iota_{\xi}(E)_{\iota_{\xi}(a)}\Rightarrow\rho\,"\iota_{\xi}(a)\in\iota_{\xi}(A)$
holds by definition of countable completeness, cf. Definition 4.1, and
$\iota_{\xi}(A)\in\iota_{\xi}(E)_{\iota_{\xi}(a)}\Leftarrow\rho\,"\iota_{\xi}(a)\in\iota_{\xi}(A)$
follows easily as
$\iota_{\xi}(A)\notin\iota_{\xi}(E)_{\iota_{\xi}(a)}\Leftrightarrow[\kappa]^{|a|}\setminus\iota_{\xi}(A)\in\iota_{\xi}(E)_{\iota_{\xi}(a)}\Rightarrow\rho\,"\iota_{\xi}(a)\in[\kappa]^{|a|}\setminus\iota_{\xi}(A)\Leftrightarrow\rho\,"\iota_{\xi}(a)\notin\iota_{\xi}(A).$

Let
$\tau\colon\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)\rightarrow\mathcal{M}^{1}$
be given by

$[a,f]\mapsto\iota_{\eta}(f)(\rho\,"\iota_{\xi}(a)),$

for $a\in[\lambda_{E}]^{{<}\omega}$ and $f\in\bar{\mathcal{M}}^{1}$, cf.
Figure 3. Note that $\tau$ is well-defined, i.e., does not depend on the
choice of $a$ and $f$ representing $[a,f]$. As $\bar{\mathcal{M}}^{1}$ and
$\bar{\mathcal{N}}$ agree on their subsets of $\operatorname{crit}(E)$ as well
as $\mathcal{M}^{1}$ and $\mathcal{N}$ agree on their subsets of
$\kappa=\operatorname{crit}(\iota_{\xi}(E))$, we have
$\iota_{\xi}(A)=\iota_{\eta}(A)$ for all
$A\in\mathcal{P}(\operatorname{crit}(E))\cap\bar{\mathcal{M}}^{1}$.

We now argue that $\tau$ is elementary. Let
$[a,f]\in\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)$ and let $\varphi$ be a
formula. Write

$A=\\{u\in\bar{\mathcal{M}}^{1}\mid\bar{\mathcal{M}}^{1}\vDash\varphi(f(u))\\}.$

Then

$\displaystyle\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)\vDash\varphi([a,f])$
$\displaystyle\text{ iff }A\in E_{a}$ $\displaystyle\text{ iff
}\iota_{\xi}(A)\in\iota_{\xi}(E)_{\iota_{\xi}(a)}$ $\displaystyle\text{ iff
}\rho\,"\iota_{\xi}(a)\in\iota_{\xi}(A)$ $\displaystyle\text{ iff
}\rho\,"\iota_{\xi}(a)\in\iota_{\eta}(A)$ $\displaystyle\text{ iff
}\rho\,"\iota_{\xi}(a)\in\\{u\in\mathcal{M}^{1}\mid\mathcal{M}^{1}\vDash\varphi(\iota_{\eta}(f)(u))\\}$
$\displaystyle\text{ iff
}\mathcal{M}^{1}\vDash\varphi(\iota_{\eta}(f)(\rho\,"\iota_{\xi}(a)))$
$\displaystyle\text{ iff }\mathcal{M}^{1}\vDash\varphi(\tau([a,f])).$

∎

Let $\iota_{\xi+1}$ be the realization map $\tau$ in Claim 1. Then we consider
the copy tree $\iota_{\xi+1}\mathcal{T}_{2}$ and obtain a realization of
$\mathcal{T}$ into a backgrounded iteration of $\mathcal{M}$ as in Figure 4.
Here $\bar{\mathcal{N}}$ is the last model of $\mathcal{T}_{1}$ and
$\mathcal{N}$ is the last model of the copy tree $\iota_{0}\mathcal{T}_{1}$.
Moreover, we write $\iota_{0}\mathcal{T}_{1}\upharpoonright\mathcal{M}^{1}$
for the copy tree of $\mathcal{T}_{1}\upharpoonright\bar{\mathcal{M}}^{1}$
with last model $\mathcal{M}^{1}$.

$\mathcal{M}$$\iota_{0}\mathcal{T}_{1}$$\mathcal{M}^{1}$$\bar{\mathcal{M}}$$\mathcal{T}_{1}$$\bar{\mathcal{M}}^{1}$$\iota_{0}$$\iota_{\eta}$$\;\;\;\;\iota_{\xi+1}\mathcal{T}_{2}$$\mathcal{T}_{1}\;$$\bar{\mathcal{N}}\ni
E$$\iota_{0}\mathcal{T}_{1}\;\;\;\;\;\;$$\mathcal{N}$$\;\;\pi_{E}$$\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)$$\;\iota_{\xi+1}$$\mathcal{T}_{2}$
Figure 4. Realization into a backgrounded iteration of $\mathcal{M}$ when
applying a non-backgrounded extender $E$.

Finally, we define $\Lambda(\mathcal{T})$ to be the canonical branch through
$\mathcal{T}$ (or more precisely through $\mathcal{T}_{2}$) obtained from the
iteration strategy for the background model $\mathcal{W}$ by lifting the stack
of the two trees
$(\iota_{0}\mathcal{T}_{1}\upharpoonright\mathcal{M}^{1},\iota_{\xi+1}\mathcal{T}_{2})$
to a stack of two trees on $\mathcal{W}$.

######  Case 3.

$\mathcal{T}$ uses more than one extender that is not fully backgrounded but
these overlapping extenders are not used cofinally often in the tree.

This case is a straightforward generalization of the previous case. Suppose
for example that $\mathcal{T}$ uses two non-backgrounded extenders $E$ and
$F$. If the second non-backgrounded extender $F$ gets a applied to an earlier
model in the tree, before the first non-backgrounded extender $E$ got used, we
realize the ultrapower into the corresponding iterate of $\mathcal{M}$ and
continue there. This case is illustrated in Figure 5, where we for simplicity
assume that $F$ gets applied to $\bar{\mathcal{M}}$. We leave the technical
details to the reader.

$\mathcal{M}$$\iota_{0}\mathcal{T}_{1}$$\mathcal{M}^{1}$$\bar{\mathcal{M}}$$\mathcal{T}_{1}$$\bar{\mathcal{M}}^{1}$$\iota_{0}$$\;\;\;\;\iota_{\xi+1}\mathcal{T}_{2}$$\mathcal{M}^{2}$$\mathcal{T}_{1}\;$$\bar{\mathcal{N}}\ni
E=E_{\xi}^{\mathcal{T}}$$\iota_{0}\mathcal{T}_{1}\;\;\;\;\;\;$$\mathcal{N}$$\;\;\pi_{E}$$\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)$$\;\iota_{\xi+1}$$\mathcal{T}_{2}$$\bar{\mathcal{M}}^{2}\ni
F$$\;\iota_{\zeta}$$\;\;\pi_{F}$$\operatorname{Ult}(\bar{\mathcal{M}},F)$$\;\iota_{\zeta+1}$$\mathcal{T}_{3}$$\iota_{\zeta+1}\mathcal{T}_{3}\;\;\;\;\;\;\;\;$
Figure 5. Realization into a backgrounded iteration of $\mathcal{M}$ when
applying two non-backgrounded extenders $E$ and $F$ with
$\operatorname{crit}(E)>\operatorname{crit}(F)$.

The cofinal well-founded branch $\Lambda(\mathcal{T})$ through $\mathcal{T}$
(or more precisely through $\mathcal{T}_{3}$) is as in the previous case
obtained by lifting the iteration tree $\iota_{\zeta+1}\mathcal{T}_{3}$ of
$\mathcal{M}$ to the background $\mathcal{W}$.

If we do not go back to an earlier model in the tree, e.g. if the critical
point of the second non-backgrounded extender $F$ is larger than the critical
point of the first non-backgrounded extender $E$ that got used, we continue
similarly as in Case 2 and copy to a stack of length three on $\mathcal{M}$,
cf. Figure 6. We again leave the technical details to the reader.

$\mathcal{M}$$\iota_{0}\mathcal{T}_{1}$$\mathcal{M}^{1}$$\bar{\mathcal{M}}$$\mathcal{T}_{1}$$\bar{\mathcal{M}}^{1}$$\iota_{0}$$\;\;\;\;\iota_{\xi+1}\mathcal{T}_{2}$$\mathcal{M}^{2}$$\mathcal{T}_{1}\;$$\bar{\mathcal{N}}\ni
E=E_{\xi}^{\mathcal{T}}$$\iota_{0}\mathcal{T}_{1}\;\;\;\;\;\;$$\mathcal{N}$$\;\;\pi_{E}$$\operatorname{Ult}(\bar{\mathcal{M}}^{1},E)$$\;\iota_{\xi+1}$$\mathcal{T}_{2}$$\bar{\mathcal{M}}^{2}$$\mathcal{T}_{2}\;\;\;\;$$\bar{\mathcal{N}}^{*}\ni
F=E_{\zeta}^{\mathcal{T}}$$\iota_{\xi+1}\mathcal{T}_{2}\;\;\;\;\;\;\;\;$$\mathcal{N}$$\;\;\;\;\pi_{F}$$\operatorname{Ult}(\bar{\mathcal{M}}^{2},F)$$\;\iota_{\zeta+1}$$\mathcal{T}_{3}$$\;\;\;\;\iota_{\zeta+1}\mathcal{T}_{3}$
Figure 6. Realization into a backgrounded iteration of $\mathcal{M}$ when
applying two non-backgrounded extenders $E$ and $F$ with
$\operatorname{crit}(E)<\operatorname{crit}(F)$.

The cofinal well-founded branch $\Lambda(\mathcal{T})$ through $\mathcal{T}$
(or more precisely through $\mathcal{T}_{3}$) is as in the previous case
obtained by lifting the stack of iteration trees
$(\iota_{0}\mathcal{T}_{1}\upharpoonright\mathcal{M}^{1},\iota_{\xi+1}\mathcal{T}_{2}\upharpoonright\mathcal{M}^{2},\iota_{\zeta+1}\mathcal{T}_{3})$
on $\mathcal{M}$ to the background $\mathcal{W}$. Here
$\iota_{0}\mathcal{T}_{1}\upharpoonright\mathcal{M}^{1}$ and
$\iota_{\xi+1}\mathcal{T}_{2}\upharpoonright\mathcal{M}^{2}$ denote the copy
trees of $\mathcal{T}_{1}\upharpoonright\bar{\mathcal{M}}^{1}$ and
$\mathcal{T}_{2}\upharpoonright\bar{\mathcal{M}}^{2}$ with last models
$\mathcal{M}^{1}$ and $\mathcal{M}^{2}$ respectively.

######  Case 4.

$\mathcal{T}$ uses cofinally many non-backgrounded extenders.

By Lemma 6.2, $\mathcal{T}$ is an almost linear iteration tree. Therefore,
there is a canonical cofinal branch $b$ through $\mathcal{T}$ in this case and
we can let $\Lambda(\mathcal{T})=b$. To continue, we can consider the tree
$\mathcal{T}^{\frown}b$ of length $\lambda+1$ and let $\iota_{\lambda}$ be the
limit of $\iota_{\xi}$ for $\xi\in b$.

This finishes the definition of $\Lambda$. It is now straightforward to check
that $\Lambda$ is an iteration strategy for $\bar{\mathcal{M}}$ above
$\kappa_{i}$ for trees of length ${<}\delta_{i+1}$. ∎

###  6.2. Comparison against a hull

The following lemma will be crucial in the proof that the construction of
$\mathcal{M}$ converges and all $\kappa_{i}$ for $i<\omega$ are fully strong
in $\mathcal{M}$. We inductively assume that all $\kappa_{j}$ for $j<i$ are
strong in $\mathcal{M}$ and use the lemma to prove that $\kappa_{i}$ is strong
in $\mathcal{M}$ in Lemma 6.6.

######  Lemma 6.4.

Let $i<\omega$ and suppose that all cardinals $\kappa_{j}$ for $j<i$ are fully
strong in $\mathcal{M}$. Let $\bar{\mathcal{W}}$ be the transitive collapse of
a hull of $\mathcal{W}|\Omega$ for some sufficiently large $\Omega$ such that
$\mathcal{W}|\kappa_{i}\subseteq\bar{\mathcal{W}}$ and
$|\bar{\mathcal{W}}|<\delta_{i+1}$. Let

$\pi\colon\bar{\mathcal{W}}\rightarrow\mathcal{W}$

be the uncollapse embedding, $\bar{\mathcal{M}}$ the collapse of
$\mathcal{M}|\Omega$, and $(\bar{\kappa}_{j},\bar{\delta}_{j}\mid j<\omega)$
the preimages of $(\kappa_{j},\delta_{j}\mid j<\omega)$ (so in particular,
$\bar{\kappa}_{j}=\kappa_{j}$ and $\bar{\delta}_{j}=\delta_{j}$ for $j\leq
i$). Then there is an iterate $\mathcal{M}^{*}$ of $\bar{\mathcal{M}}$ such
that $\bar{\mathcal{M}}$-to-$\mathcal{M}^{*}$ is non-dropping,
$\mathcal{M}^{*}\unlhd\mathcal{M}$, and if

$i\colon\bar{\mathcal{M}}\rightarrow\mathcal{M}^{*}$

denotes the iteration embedding, the critical point of $i$ is above
$\kappa_{i}$.

###### Proof.

By Lemma 6.3 we have that $\bar{\mathcal{M}}$ is
$(\delta_{i+1},\delta_{i+1})$-iterable with respect to extenders with critical
point above $\kappa_{i}$ and we can compare it against the construction of
$\mathcal{M}$ (cf., e.g. [16] for such a comparison argument for $K^{c}$
constructions). We have $|\bar{\mathcal{M}}|<\delta_{i+1}$ and we compare
$\bar{\mathcal{M}}$ against models $\mathcal{M}_{\xi}$ of the construction of
$\mathcal{M}$ with $|\mathcal{M}_{\xi}|<\delta_{i+1}$. We will see below that
the comparison against the construction of $\mathcal{M}$ will finish
successfully before reaching models $\mathcal{M}_{\xi}$ of size larger than
$\delta_{i+1}$. So the iteration trees we consider on $\bar{\mathcal{M}}$ will
have length at most
$\max(|\bar{\mathcal{M}}|,|\mathcal{M}_{\xi}|)^{+}<\delta_{i+1}$ for ordinals
$\xi$ with $|\mathcal{M}_{\xi}|<\delta_{i+1}$. In addition, we will argue in
Claim 2 below that all extenders used in this comparison on the
$\bar{\mathcal{M}}$-side have critical point above $\kappa_{i}$, so the
iterability given by Lemma 6.3 suffices for $\bar{\mathcal{M}}$. Moreover, we
start with arguing that no disagreement is caused by the $\mathcal{M}$-side,
so the construction of $\mathcal{M}$ does not move in the comparison process.
This also shows that we do not need iterability for $\mathcal{M}$ itself for
the comparison argument.

######  Claim 1.

The construction of $\mathcal{M}$ does not move in the comparison.

###### Proof.

Let $(\mathcal{M}_{\xi},\mathcal{N}_{\xi}\mid\xi\leq\Omega)$ be the models in
the construction of $\mathcal{M}$ and suppose toward a contradiction that the
construction of $\mathcal{M}$ does move in the comparison. Let $E$ be the
first extender on some $\mathcal{N}_{\xi}$ in the construction of
$\mathcal{M}$ that is used. Write $\mathcal{M}^{*}$ for the iterate of
$\bar{\mathcal{M}}$ we have obtained up to this stage in the comparison and
$\mathcal{T}$ for the corresponding iteration tree of length $\lambda+1$ for
some ordinal $\lambda$, i.e.,
$\mathcal{M}_{\lambda}^{\mathcal{T}}=\mathcal{M}^{*}$. As the construction of
$\mathcal{M}$ wins the comparison, there is no drop on the main branch of the
$\bar{\mathcal{M}}$-side and we have an elementary embedding
$i=i_{0\lambda}^{\mathcal{T}}\colon\bar{\mathcal{M}}\rightarrow\mathcal{M}^{*}$.

It is a standard argument that fully backgrounded extenders do not move in
such a comparison (see for example [10, Lemma 2.11] or [16]), so we can
suppose that $\operatorname{crit}(E)\in\\{\kappa_{j}\mid j<\omega\\}$. As
$|\bar{\mathcal{M}}|<\delta_{i+1}$ and $\mathcal{N}_{0}<\delta_{i+1}$ their
comparison lasts at most up to
$\max(|\bar{\mathcal{M}}|,|\mathcal{N}_{0}|)^{+}<\delta_{i+1}$. Repeating this
implies $\operatorname{crit}(E)=\kappa_{j}$ for some $j\leq i$ as
$\bar{\mathcal{M}}$ cannot iterate up to $\delta_{i+1}<\kappa_{i+1}$. In fact,
$\operatorname{crit}(E)=\kappa_{j}$ for $j<i$ and
$\delta_{j+1}<index(E)<\delta_{i+1}$ as by the definition of
$\mathcal{M}$-constructions, cf. Definition 4.3, extenders with critical point
$\kappa_{j}$ that are indexed below $\delta_{j+1}$ are fully backgrounded.

We start by arguing that if there is an extender $F$ in $\mathcal{M}^{*}$ with
the same index as $E$ then $F$ is certified via the same tail strategy of
$\Sigma_{j}$ as $F$ and hence $E=F$, contradicting the assumption that $E$ is
used because of a disagreement. To see that $F$ is certified by the same
strategy certifying $E$, we consider different cases depending on how the
iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$ looks like. Let
$\bar{\Sigma}$ denote the iteration strategy of $\bar{\mathcal{W}}$ obtained
as the collapse of $\Sigma$.

######  Case 1.

On the main branch in the iteration from $\bar{\mathcal{M}}$ to
$\mathcal{M}^{*}$ all extenders used are fully backgrounded, i.e., for every
$E_{\alpha}^{\mathcal{T}}$, $\alpha\in[0,\lambda]_{T}$, and every $l<\omega$,

$\operatorname{crit}(E_{\alpha}^{\mathcal{T}})\neq
i_{0\alpha}^{\mathcal{T}}(\kappa_{l}).$

As the construction of $\mathcal{M}$ did not move up to this point of the
comparison, this means that disagreements were only caused by fully
backgrounded extenders. By Steel’s comparison theorem [26, Theorem 4.10],
coiterating against the construction by least disagreement of backgrounded
extenders not only iterates away extender disagreements between
$\bar{\mathcal{M}}$ and $\mathcal{M}$ but also strategy disagreements between
the models. All extenders used in the iteration from $\bar{\mathcal{M}}$ to
$\mathcal{M}^{*}$ have critical point above $\kappa_{i}$ as
$\bar{\mathcal{M}}|\kappa_{i}=\mathcal{M}|\kappa_{i}$ and none of the
backgrounded extenders overlaps $\kappa_{i}$. Write $\kappa_{k}^{*}$ for the
image of $\bar{\kappa}_{k}$ in $\mathcal{M}^{*}$ for all $k<\omega$ and note
that $\kappa_{k}^{*}=\kappa_{k}$ for all $k\leq i$. Recall that
$\operatorname{crit}(E)=\kappa_{j}<\kappa_{i}$, so the extender $F$ on
$\mathcal{M}^{*}$ has critical point $\kappa_{j}^{*}=\kappa_{j}$.

We can at the same time consider the iterate $\mathcal{W}^{*}$ of
$\bar{\mathcal{W}}$ via the background extenders and maintain the property
that $\mathcal{M}^{*}$ is the result of an $\mathcal{M}$-construction inside
$\mathcal{W}^{*}$. In particular, $F$ is generically countably complete and
hence by Lemma 5.19 certified in $\mathcal{W}^{*}$ as it has critical point
$\kappa_{j}=\kappa_{j}^{*}<\kappa_{i}$ and index above $\kappa_{i}$.

As a result of the comparison, not only $\mathcal{M}^{*}$ and $\mathcal{M}$
agree below the index of $F$ but also the strategies of their backgrounds
$\mathcal{W}^{*}$ and $\mathcal{W}$. So in particular they have the same tail
strategy $(\Sigma_{j})_{\mathcal{M}_{\infty}^{j,-}}$ of $\Sigma_{j}$. As $E$
and $F$ are both certified via this strategy, this implies $E=F$.

######  Case 2.

The iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$ uses in addition
to fully backgrounded extenders a single extender $G$ that is not fully
backgrounded.

We aim to show that $F$ is certified via the same tail strategy of
$\Sigma_{j}$ that certifies $E$. For notational simplicity we suppose that the
iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$ only uses $G$, i.e.,
$\mathcal{M}^{*}=\operatorname{Ult}(\bar{\mathcal{M}},G)$. Taking additional
iterations via fully backgrounded extenders into account is a straightforward
generalization of the argument below so we leave that to the reader. Write
$(\kappa_{i}^{*},\delta_{i}^{*}\mid i<\omega)$ for the image of
$(\kappa_{i},\delta_{i}\mid i<\omega)$ under $\pi_{G}$.

Suppose toward a contradiction that $F$ is not certified via a tail strategy
of $\Sigma_{j}$. Let $\overline{\overline{\mathcal{W}}}$ be a countable hull
of $\bar{\mathcal{W}}$ and hence in particular a countable hull of
$\mathcal{W}|\Omega$, let

$\bar{\pi}\colon\overline{\overline{\mathcal{W}}}\rightarrow\bar{\mathcal{W}}$

and

$\sigma\colon\overline{\overline{\mathcal{W}}}\rightarrow\mathcal{W}$

be the hull embeddings and $k$ an embedding such that
$\sigma=k\circ\bar{\pi}$. Suppose all relevant objects in $\bar{\mathcal{W}}$
and $\mathcal{W}$ are in the range of $\bar{\pi}$ and $\sigma$ respectively.
Let $\overline{\overline{\mathcal{M}}}$ be the $\bar{\pi}$-collapse of
$\bar{\mathcal{M}}$, $\bar{\mathcal{M}}^{*}$ the $\bar{\pi}$-collapse of
$\mathcal{M}^{*}$ and let $\bar{\pi}(\bar{G},\bar{F})=(G,F)$. Write
$\overline{\overline{\Sigma}}$ for the strategy of
$\overline{\overline{\mathcal{W}}}$, i.e.,
$\sigma(\overline{\overline{\Sigma}})=\Sigma$.

$\mathcal{M}$$\bar{\mathcal{M}}$$\operatorname{Ult}(\bar{\mathcal{M}},G)=\mathcal{M}^{*}$$\overline{\overline{\mathcal{M}}}$$\operatorname{Ult}(\overline{\overline{\mathcal{M}}},\bar{G})=\bar{\mathcal{M}}^{*}$$\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})$$k$$\bar{\pi}$$\sigma$$\tau_{\bar{G}}^{\bar{\mathcal{M}}}\;\;\;$$\;\;\;\tau_{\bar{G}}^{\mathcal{M}}$$\;\;\;\tau$
Figure 7. Realizing $\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})$.

######  Subclaim.

There is a realization map
$\tau\colon\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})\rightarrow\mathcal{M}$.

###### Proof.

We have that $G$ is generically countably complete in $\bar{\mathcal{W}}$. In
particular, $\operatorname{Ult}(\overline{\overline{\mathcal{M}}},\bar{G})$
can be realized into $\bar{\mathcal{M}}$. Let
$\tau_{\bar{G}}^{\bar{\mathcal{M}}}\colon\operatorname{Ult}(\overline{\overline{\mathcal{M}}},\bar{G})\rightarrow\bar{\mathcal{M}}$
be the realization map and let

$\tau_{\bar{G}}^{\mathcal{M}}=k\circ\tau_{\bar{G}}^{\bar{\mathcal{M}}}$

be a realization of
$\operatorname{Ult}(\overline{\overline{\mathcal{M}}},\bar{G})$ into
$\mathcal{M}$, cf. Figure 7. Write $H=\tau_{\bar{G}}^{\mathcal{M}}(\bar{F})$.
Recall that $\operatorname{crit}(E)=\kappa_{j}$ and $index(E)>\delta_{j+1}$ in
$\mathcal{M}$. As $\mathcal{M}^{*}$ results from a comparison against the
construction of $\mathcal{M}$, $\mathcal{M}^{*}$ and $\mathcal{M}$ agree below
the index of $E$. In particular, $\kappa_{j}^{*}=\kappa_{j}$ and
$\delta_{j+1}^{*}=\delta_{j+1}$. Therefore, in
$\mathcal{M}^{*}=\operatorname{Ult}(\bar{\mathcal{M}},G)$,
$\operatorname{crit}(F)=\kappa_{j}^{*}$ and $F$ is indexed above
$\delta_{j+1}^{*}$. By elementarity, the analogous statements hold for
$\bar{F}$ in $\operatorname{Ult}(\overline{\overline{\mathcal{M}}},\bar{G})$.
Therefore, the critical point of $H$ is some $\kappa_{i}$, $i<\omega$, and $H$
is indexed in $\mathcal{M}$ above $\delta_{i+1}$. This yields that $H$ is
generically countably complete in $\mathcal{M}$ so there is a realization map
$\tau\colon\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})\rightarrow\mathcal{M}$,
as desired. ∎

Using the previous subclaim we can argue as in the proof of Lemma 5.19 to
obtain that $\bar{F}$ is certified via a tail of
$\overline{\overline{\Sigma}}_{j}$. By elementarity of $\sigma$, this
contradicts our assumption that $F$ is not certified via a tail of
$\Sigma_{j}$.

######  Case 3.

The iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$ uses in addition
to fully backgrounded extenders finitely many extenders that are not fully
backgrounded.

This is a straightforward generalization of the previous case using the fact
that we can still find realization maps for iterates of countable
substructures as in the iterability proof, cf. Lemma 6.3.

######  Case 4.

We are at a limit stage of the iteration and cofinally many times below this
stage we applied an extender with critical point the image of some
$\bar{\kappa}_{i}$, $i<\omega$.

Let $(E_{i}\mid i<\gamma)$ be the sequence of extenders we applied that have
critical point the image of some $\bar{\kappa}_{i}$ for $i<\omega$. Recall
that the extenders need to be of increasing length and note that if there is
some $i<j<\gamma$ such that
$\operatorname{crit}(E_{i})>\operatorname{crit}(E_{j})$ and $E_{j}$ is the
first extender where this happens after stage $i$, then $E_{j}$ gets applied
to $\bar{\mathcal{M}}$. As we are only interested in the extenders on the main
branch leading from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$, we might as well
assume that the sequence $(\operatorname{crit}(E_{i})\mid i<\gamma)$ is
nondecreasing.

######  Subcase 4.1.

The critical points of all $E_{i}$ are images of the same $\bar{\kappa}_{k}$
for some fixed $k<\omega$.

The interesting case here is $\operatorname{crit}(F)=\kappa_{k}^{*}$ as in
this case we need a small argument to define the direct limit system that
gives rise to the direct limit $(\mathcal{M}_{\infty}^{k})^{*}$ to
characterize $F$ in $\mathcal{M}^{*}$. As we have seen in Section 5, in
$\bar{\mathcal{M}}$,

$\displaystyle\text{for all }\bar{\kappa}_{k}\text{-good ordinals
}\delta^{*}<\delta^{**},$ $\displaystyle\mathcal{N}_{\delta^{**}}\text{ is}$
$\displaystyle\text{ an iterate of }\mathcal{N}_{\delta^{*}}\text{ via a tail
strategy of }\bar{\Sigma}_{k}.$

By elementarity, the same statement holds in $\mathcal{M}^{*}$. This allows us
to define $(\mathcal{M}_{\infty}^{k})^{*}$, the direct limit system at
$\kappa_{k}^{*}$ in $\mathcal{M}^{*}$, as the quasi-limit of
$\mathcal{N}_{\delta^{*}}$ for all $\kappa_{k}^{*}$-good ordinals $\delta^{*}$
in $\mathcal{M}^{*}|\kappa_{k}^{*}$. By agreement in the comparison process,
$\mathcal{M}^{*}|\kappa_{k}^{*}=\mathcal{M}|\kappa_{k}$ and
$(\mathcal{M}_{\infty}^{k})^{*}=\mathcal{M}_{\infty}^{k}$.

Now we can finish the argument in this subcase by arguing as in Case 2:
Consider a countable hull $\overline{\overline{\mathcal{W}}}$ of
$\bar{\mathcal{W}}$ and let $\bar{\mathcal{M}}^{*}$, $\bar{F}$ be the
collapses of $\mathcal{M}^{*}$, $F$. By the iterability proof, cf. the proof
of Lemma 6.3, there is an elementary embedding
$\iota\colon\bar{\mathcal{M}}^{*}\rightarrow\mathcal{M}^{*}$. Hence, we can
use the same argument as in Case 2 to argue that
$\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})$ can be realized and
therefore $\bar{F}$ is certified.

######  Subcase 4.2.

The critical points of the $E_{i}$ are increasing, i.e., not all $E_{i}$ have
the same critical point, but the critical points are not cofinal in the images
of the $\bar{\kappa}_{i}$.

Then the sequence $(E_{i}\mid i<\gamma)$ will on a tail arise from extenders
with critical points that are images of the same $\kappa_{k}$ for some
$k<\omega$. In fact, we can split the sequence $(E_{i}\mid i<\gamma)$ into
finitely many blocks, each block consisting of extenders with critical points
that are images of the same $\bar{\kappa}_{k}$ for some $k<\omega$. Now the
argument is a straightforward generalization of Subcase 4.1 and Case 3.

######  Subcase 4.3.

The critical points of the $E_{i}$ are cofinal in the images of the
$\bar{\kappa}_{i}$, i.e., in $\mathcal{M}^{*}$ the supremum of the critical
points of the $E_{i}$ is equal to the supremum of the images of
$\bar{\kappa}_{i}$.

In this case the index of $F$ is above the supremum of $\kappa_{k}^{*}$ for
$k<\omega$ and the critical point of $F$ is $\kappa_{j}^{*}=\kappa_{j}$ by
agreement between $\mathcal{M}^{*}$ and $\mathcal{M}$. As in Case 2 we can
consider a countable hull $\overline{\overline{\mathcal{W}}}$ of
$\bar{\mathcal{W}}$ and let $\bar{\mathcal{M}}^{*}$, $\bar{F}$ be the
collapses of $\mathcal{M}^{*}$, $F$. By the iterability proof, cf. the proof
of Lemma 6.3, there is an elementary embedding
$\iota\colon\bar{\mathcal{M}}^{*}\rightarrow\mathcal{M}^{*}$. Hence, we can
use the same argument as in Case 2 to argue that
$\operatorname{Ult}(\bar{\mathcal{M}}^{*},\bar{F})$ can be realized and
therefore $\bar{F}$ is certified.

We finally argue that the reason $E$ is used on the $\mathcal{M}$-side of the
comparison cannot be that there is no extender $F$ with the same index as $E$
on $\mathcal{M}^{*}$. So say $E=E_{\alpha}^{\mathcal{M}}$ and suppose toward a
contradiction that $E_{\alpha}^{\mathcal{M}^{*}}=\emptyset$. Recall that
$\operatorname{crit}(E_{\alpha}^{\mathcal{M}})=\kappa_{j}<\kappa_{i}$.

######  Subclaim.

The iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$ does not use any
extenders with critical point ${\leq}\kappa_{j}$ on the main branch.

###### Proof.

Suppose toward a contradiction that there is some extender $F$ with critical
point $\kappa_{l}\leq\kappa_{j}$ that gets used on the main branch in the
iteration from $\bar{\mathcal{M}}$ to $\mathcal{M}^{*}$.

######  Case 1.

$F$ is used as there is no extender with the same index on the corresponding
model $\mathcal{N}_{\zeta}$ in the construction of $\mathcal{M}$.

Recall that we suppose that $\kappa_{l}$ is fully strong in $\mathcal{M}$. So
there is some extender $E_{\beta}^{\mathcal{M}}$ on the sequence of
$\mathcal{M}$ with critical point $\kappa_{l}$ and index $\beta$ above the
index of $F$ (in the $\mathcal{M}^{*}$-sequence). Let $\beta$ be minimal with
this property and write $\alpha$ for the $\mathcal{M}^{*}$-index of $F$. Then
$\alpha$ is a cutpoint in
$\operatorname{Ult}(\mathcal{M},E_{\beta}^{\mathcal{M}})$ and the
$(\kappa_{l},\alpha)$-extender $E$ derived from the ultrapower embedding
$\pi^{\mathcal{M}}_{E_{\beta}^{\mathcal{M}}}$ is on the $\mathcal{M}$-sequence
by the initial segment condition. So $E_{\alpha}^{\mathcal{M}}=E$ and $E$ is
on the sequence of $\mathcal{N}_{\nu}$ for some $\nu\geq\zeta$. Therefore,
even if $F$ is used in the comparison against $\mathcal{N}_{\zeta}$ it will
not be used on the main branch in the comparison of $\bar{\mathcal{M}}$
against the construction of $\mathcal{M}$ as this part of the coiteration gets
collapsed when moving on to the comparison with $\mathcal{N}_{\nu}$. So we
might and will assume that the iteration from $\bar{\mathcal{M}}$ to
$\mathcal{M}^{*}$ involves comparisons against $\mathcal{N}_{\nu}$ or larger
models from the construction of $\mathcal{M}$.

######  Case 2.

$F$ is used as there is an extender $G$ with the same index on the
corresponding model $\mathcal{N}_{\zeta}$ in the construction of $\mathcal{M}$
and $F\neq G$.

In this case we can argue that $F$ and $G$ are both certified with the same
critical point $\kappa_{l}$. Therefore, $F=G$ and hence this case cannot
occur. ∎

By our hypothesis, $\kappa_{j}$ is strong in $\bar{\mathcal{M}}$. Hence by the
subclaim $\kappa_{j}$ is strong in $\mathcal{M}^{*}$, so there is some
extender $E_{\beta}^{\mathcal{M}^{*}}$ with critical point $\kappa_{j}$ and
$\beta>\alpha$. Let $\beta>\alpha$ be minimal such that
$E_{\beta}^{\mathcal{M}^{*}}$ is an extender with critical point $\kappa_{j}$.
Then $\alpha$ is a cutpoint in
$\operatorname{Ult}(\mathcal{M}^{*},E_{\beta}^{\mathcal{M}^{*}})$ and the
$(\kappa_{j},\alpha)$-extender $F$ derived from the ultrapower embedding
$\pi_{E_{\beta}^{\mathcal{M}^{*}}}^{\mathcal{M}^{*}}$ is on the sequence of
$\mathcal{M}^{*}$ by the initial segment condition. I.e.,