###### Proof of .

###### On Indestructible Strongly Guessing Models

###### Abstract.

In [14] we defined and proved the consistency of the principle ${\rm
GM}^{+}(\omega_{3},\omega_{1})$ which implies that many consequences of strong
forcing axioms hold simultaneously at $\omega_{2}$ and $\omega_{3}$. In this
paper we formulate a strengthening of ${\rm GM}^{+}(\omega_{3},\omega_{1})$
that we call ${\rm SGM}^{+}(\omega_{3},\omega_{1})$. We also prove, modulo the
consistency of two supercompact cardinals, that ${\rm
SGM}^{+}(\omega_{3},\omega_{1})$ is consistent with ZFC. In addition to all
the consequences of ${\rm GM}^{+}(\omega_{3},\omega_{1})$, the principle ${\rm
SGM}^{+}(\omega_{3},\omega_{1})$, together with some mild cardinal arithmetic
assumptions that hold in our model, implies that any forcing that adds a new
subset of $\omega_{2}$ either adds a real or collapses some cardinal. This
gives a partial answer to a question of Abraham [1] and extends a previous
result of Todorčević [15] in this direction.

###### Key words and phrases:

Guessing model, Indestructible guessing model, Magidor model, Virtual model,
Side conditions.

###### 2020 Mathematics Subject Classification:

03E35, 03E55, 03E57, 03E65.

##  1\. Introduction

One of the driving themes of research in set theory in recent years has been
the search for higher forcing axioms. Since most applications of strong
forcing axioms such as the Proper Forcing Axiom (PFA) and Martin’s Maximum
(MM) can be factored through some simple, but powerful combinatorial
principles, it is natural to look for higher cardinal versions of such
principles. One such principle is ${\rm ISP}(\omega_{2})$ which was formulated
by Viale and Weiß [18]. They derived it from PFA and showed that it has a
number of structural consequences on the set-theoretic universe. For instance,
it implies the tree property at $\omega_{2}$, and the failure of
$\square(\lambda)$, for regular $\lambda\geq\omega_{2}$. Moreover, Kruger [10]
showed that it implies the Singular Cardinal Hypothesis (SCH). A strengthening
of ${\rm ISP}(\omega_{2})$ was studied by Cox and Krueger [6]. They showed
that this strengthening, which we refer to as ${\rm
SGM}(\omega_{2},\omega_{1})$, is consistent with $2^{\aleph_{0}}>\aleph_{2}$.
It is therefore a natural candidate for a generalization to higher cardinals.
In [14] we introduced one such generalization, ${\rm
GM}^{+}(\omega_{3},\omega_{1})$, and showed that if there are two supercompact
cardinals then there is a generic extension $V[G]$ of the universe in which
${\rm GM}^{+}(\omega_{3},\omega_{1})$ holds. In addition to ${\rm
ISP}(\omega_{2})$ the principle ${\rm GM}^{+}(\omega_{3},\omega_{1})$ implies
that the tree property also holds at $\omega_{3}$, and that the
approachability ideal $I[\omega_{2}]$ restricted to ordinals of cofinality
$\omega_{1}$ is the non stationary ideal restricted to this set. The
consistency of the latter was previously shown by Mitchell [13] starting from
a model with a greatly Mahlo cardinal. In the current paper we further
strengthen the principle ${\rm GM}^{+}(\omega_{3},\omega_{1})$ in the spirit
of Cox and Krueger [6]. We call this new principle ${\rm
SGM}^{+}(\omega_{3},\omega_{1})$. We show its consistency by a modification of
the forcing notion from [14], again starting from two supercompact cardinals.
As a new application we show that ${\rm SGM}^{+}(\omega_{3},\omega_{1})$,
together with some mild cardinal arithmetic assumptions which hold in our
model, implies that any forcing that adds a new subset of $\omega_{2}$ either
adds a real or collapses some cardinal. This gives a partial answer to a
question of Abraham [1] and extends a previous result of Todorčević [15] in
this direction.

The paper is organized as follows. In §2 we recall some preliminaries and
background facts. In §3 we state the main theorem and derive the application
of ${\rm SGM}^{+}(\omega_{3},\omega_{1})$ to the problem of Abraham mentioned
above. In §4 we review the definitions on virtual models from [14]. in §5 we
describe the main forcing and prove the relevant facts about it. The pieces
are then put together in §6 to derive the main theorem of this paper. We warn
the reader that the current paper relies heavily on the concepts and results
from [14], hence having access to this paper is necessary for the full
understanding of our results.

##  2\. Preliminaries

### Special trees

Let $T=(T,\leq_{T})$ be a tree of height $\omega_{1}$. Recall that $T$ is
called _special_ if there is a function $f:T\to\omega$, such that for every
$s<_{T}t$ in $T$, we have $f(s)\neq f(t)$. If $T$ is a tree of height
$\omega_{1}$, the standard forcing $\mathbb{S}(T)$ to generically specialize
$T$ consists of finite partial specializing functions, ordering by reverse
inclusion. If $T$ has no cofinal branches then $\mathbb{S}(T)$ has the
countable chain condition, see [2]. We say that $T$ is _weakly special_ if
there is a function $f:T\to\omega$ such that whenever $s,t,u\in T$ and
$f(s)=f(t)=f(u)$, if $s<_{T}t$ and $s<_{T}u$ then $t$ and $u$ are comparable.
Baumgartner [3, Theorem 7.3] proved that for a tree $T$ of height $\omega_{1}$
which has no branches of length $\omega_{1}$, $T$ is special iff $T$ is weakly
special. He also showed [3, Theorem 7.5] that if every tree of height and size
$\omega_{1}$ which has no branches of length $\omega_{1}$ is special then
every tree of height and size $\omega_{1}$ which has at most $\omega_{1}$
branches is weakly special. We recall the following well known fact, see [6,
Proposition 4.3] for a proof.

######  Proposition 2.1.

If $V\subseteq W$ are models of set theory with
$\omega_{1}^{V}=\omega_{1}^{W}$, $T\in V$ is a tree of height $\omega_{1}$
which is weakly special in $V$, then every branch of $T$ in $W$ of length
$\omega_{1}$ is in $V$.

###### .

∎

### Guessing models

Throughout this paper by a model $M$, we mean a set or a class such that
$(M,\in)$ satisfies a sufficient fragment of ${\rm ZFC}$. For a set or class
$M$, we say that a set $x\subseteq M$ is bounded in $M$ if there is $y\in M$
such that $x\subseteq y$. Let us call a transitive model $R$ powerful if it
includes every set bounded in $R$. We say that $x$ is _guessed_ in $M$ if
there is $x^{*}\in M$ with $x^{*}\cap M=x\cap M$. Suppose that $\gamma$ is an
$M$-regular cardinal. We say $x$ is _$\gamma$ -approximated_ in $M$ if for
every $a\in M$ with $|a|^{M}<\gamma$, we have $a\cap x\in M$. We say $M$ is
_$\gamma$ -guessing in a model_ $N$, if for any $x\in N$ bounded in $M$, if
$x$ is $\gamma$-approximated in $M$, then $x$ is guessed in $M$.

######  Definition 2.2 ([17]).

Let $\gamma$ be a regular cardinal. A set $M$ is said to be $\gamma$-guessing
if $M$ is $\gamma$-guessing in $V$.

Recall form [9] that a pair $(M,N)$ of transitive models with $\gamma\in
M\subseteq N$ has the $\gamma$-approximation property, if $M$ is
$\gamma$-guessing in $N$.

######  Theorem 2.3 (Krueger [10]).

Let $M_{0}\subseteq M_{1}\subseteq M_{2}$ be transitive models. If
$(M_{0},M_{1})$ and $(M_{1},M_{2})$ have the $\omega_{1}$-approximation
property, then so does $(M_{0},M_{2})$.

###### .

∎

######  Theorem 2.4 (Cox–Krueger [5]).

Suppose $R$ is a powerful model, and $M\prec R$. Let also $\gamma$ be a
regular cardinal in $M$ with $\gamma\subseteq M$. Then the following are
equivalent.

  1. (1)

$M$ is a $\gamma$-guessing model.

  2. (2)

The pair $(\overline{M},V)$ has the $\gamma$-approximation property.

###### .

∎

### Indestructible and special guessing models

The notion of an indestructible guessing model was introduced by Cox and
Krueger [6].

######  Definition 2.5.

An $\omega_{1}$-guessing model $M$ of size $\omega_{1}$ is _indestructible_ if
$M$ remains $\omega_{1}$-guessing in any outer universe with the same
$\omega_{1}$.

Let us first recall that a set $X$ of size $\omega_{1}$ is _internally
unbounded_ if $X\cap\mathcal{P}_{\omega_{1}}(X)$ is cofinal in
$\mathcal{P}_{\omega_{1}}(X)$, and is _internally club_ if
$X\cap\mathcal{P}_{\omega_{1}}(X)$ contains a closed unbounded subset of
$\mathcal{P}_{\omega_{1}}(X)$. It is easily seen that if $M$ is an
$\omega_{1}$-sized elementary submodel of a powerful model $R$, then $M$ is
internally unbounded (I.U.) if and only if there is an I.U. sequence for $M$,
i.e., a $\subseteq$-increasing and $\in$-sequence $\langle
M_{\alpha}:\alpha<\omega_{1}\rangle$ of countable sets with union $M$, and
that $M$ is internally club (I.C.) if and only if there is an I.C. sequence
for $M$, i.e. $\subseteq$-continuous I.U. sequence for $M$.

It was shown by Krueger [10] that if $R$ is a powerful model and $M\prec R$
with $\omega_{1}\subseteq M$ is an $\omega_{1}$-guessing model of size
$\omega_{1}$, then $M$ is internally unbounded. Let us fix an I.U. sequence
$\langle M_{\alpha}:\alpha<\omega_{1}\rangle$ for an I.U.
$\omega_{1}$-guessing model $M$ of size $\omega_{1}$. For every
$\alpha<\omega_{1}$, let

$T_{\alpha}(M)\coloneqq\\{(z,f)\in M:f:z\cap M_{\alpha}\rightarrow 2\\}.$

Let also

$T(M)=\bigcup_{\alpha<\omega_{1}}T_{\alpha}(M).$

We order $T(M)$ by $(z,f)\leq(z^{\prime},f^{\prime})$ if and only if
$z=z^{\prime}$ and $f\subseteq f^{\prime}$. Notice that $T(M)\subseteq M$, and
hence $|T(M)|\leq\omega_{1}$. Clearly $T(M)$ is a tree of height $\omega_{1}$.
Assume that $b$ is an uncountable branch through $T(M)$. Then there is $z\in
M$ such that, for all $(z^{\prime},f^{\prime})\in b$, we have $z^{\prime}=z$.
Let

$F=\bigcup\\{f:(z,f)\in b\\}.$

Then $F:z\cap M\mapsto 2$ is a function which is $\omega_{1}$-approximated in
$M$. Since $M$ is an $\omega_{1}$-guessing model, there is $F^{*}\in M$ with
$F\\!\upharpoonright\\!(z\cap M)=F^{*}\\!\upharpoonright\\!(z\cap M)$. Given
$F^{*}$ we can recover the branch $b$, therefore $T(M)$ has at most
$\omega_{1}$ branches. Now, by Proposition 2.1, we have the following [6,
Proposition 4.4].

######  Proposition 2.6 (Cox-Krueger [6]).

Suppose $M$ is an I.U. $\omega_{1}$-guessing model of cardinality
$\omega_{1}$. Then there is a tree $T$ of size and height $\omega_{1}$ with
$\omega_{1}$ cofinal branches such that $T$ being weakly special implies that
$M$ is indestructibly $\omega_{1}$-guessing.

###### .

∎

### Strong properness

######  Definition 2.7 (Mitchell [12]).

Let $\mathbb{P}$ be a forcing notion and $A$ a set. We say that
$p\in\mathbb{P}$ is $(A,\mathbb{P})$-strongly generic if for every $q\leq p$
there is a condition $q\\!\upharpoonright\\!A\in A$ such that any $r\in A$
with $r\leq q\\!\upharpoonright\\!A$ is compatible with $q$.

######  Definition 2.8.

Let $\mathbb{P}$ be a forcing notion, and $\mathcal{S}$ be a collection of
sets. We say that $\mathbb{P}$ is strongly proper for $\mathcal{S}$ if for
every $A\in\mathcal{S}$ and $p\in A\cap\mathbb{P}$, there an
$(A,\mathbb{P})$-strongly generic condition $q\leq p$.

A forcing notion $\mathbb{P}$ is called strongly proper, if for every
sufficiently large regular cardinal $\theta$, there is a club $\mathcal{C}$ in
$\mathcal{P}_{\omega_{1}}(H_{\theta})$ such that $\mathbb{P}$ is strongly
proper for $\mathcal{C}$.

######  Definition 2.9.

A forcing notion $\mathbb{P}$ has the $\gamma$-approximation property if for
every $V$-generic filter $G\subseteq\mathbb{P}$, the pair $(V,V[G])$ has the
$\gamma$-approximation property.

The following are standard.

######  Lemma 2.10.

Every strongly proper forcing has the $\omega_{1}$-approximation property.

###### .

∎

######  Lemma 2.11.

Suppose $R$ is a powerful model, and $M\prec R$ is an $\omega_{1}$-guessing
model. Assume $\mathbb{P}$ is a forcing with the $\omega_{1}$-approximation
property. Then, $M$ remains $\omega_{1}$-guessing in $V^{\mathbb{P}}$.

###### Proof.

It follows easily from Theorem 2.3 and Theorem 2.4. ∎

### Magidor models

The following definition is motivated by Magidor’s reformulation of
supercompactness, see [11].

######  Definition 2.12 (Magidor model, [14]).

We say that a model $M$ is a $\kappa$-Magidor model if, letting $\overline{M}$
be the transitive collapse of $M$ and $\pi$ the collapsing map,
$\overline{M}=V_{\bar{\gamma}}$, for some $\bar{\gamma}<\kappa$ with ${\rm
cof}(\bar{\gamma})\geq\pi(\kappa)$, and $V_{\pi(\kappa)}\subseteq M$. If
$\kappa$ is clear from the context, we omit it.

######  Proposition 2.13.

Suppose $\kappa$ is supercompact and $\mu>\kappa$ with ${\rm
cof}(\mu)\geq\kappa$. Then the set of $\kappa$-Magidor models is stationary in
$\mathcal{P}_{\kappa}(V_{\mu})$.

###### Proof.

See [14, Proposition 2.21]. ∎

### Some lemmata

######  Lemma 2.14 (Neeman [7]).

Suppose that $M$ is a $0$-guessing model which is sufficiently elementary in
some transitive model $A$. Suppose $\mathbb{P}\in M$ is a forcing, and that
$p\in\mathbb{P}$. Then, if $p$ is $(M,\mathbb{P})$-generic it is
$(M,\mathbb{P})$-strongly generic.

###### .

∎

######  Lemma 2.15.

Let $\mathbb{P}\in H_{\theta}$ be a forcing with the
$\omega_{1}$-approximation property. Suppose that $M\prec H_{\theta}$ is
countable and $\mathbb{P}\in M$. Let $Z\in M$ and let $f:M\cap Z\rightarrow 2$
be a function which is not guessed in $M$. If $p\in\mathbb{P}$ is
$(M,\mathbb{P})$-generic, then

$p\Vdash``\check{f}\text{ is not guessed in }M[\dot{G}]."$

###### Proof.

Pick a $V$-generic filter $G\subseteq\mathbb{P}$ containing $p$. Assume
towards a contradiction that for some function $g\in M[G]$,
$g\\!\upharpoonright\\!{M\cap Z}=f$. Suppose $x\in[Z]^{\omega}\cap M$. Then
$x\subseteq M$ and hence $g\restriction{x}=f\restriction{x}\in V$. Since
$M[G]\cap V=M$ we have that $g\restriction{x}\in M$. Now by elementarity of
$M[G]$ in $H_{\theta}[G]$ we have that $g$ is is countably approximated in
$V$. Since $\mathbb{P}$ has the $\omega_{1}$-approximation property it follows
that $g\in V$ and hence $g\in M$. Therefore, $f$ is guessed in $M$, a
contradiction. ∎

######  Lemma 2.16.

Suppose $\mathbb{P}$ is a forcing notion. Assume that $\dot{f}:Z\rightarrow 2$
is a $\mathbb{P}$-name for a function. Suppose that $M$ is a model with
$\dot{f},Z\in M$. Let $p\in\mathbb{P}$ be an $(M,\mathbb{P})$-generic
condition that forces $\dot{f}\restriction{M\cap Z}=\check{g}$, for some
function $g$. If $g$ is guessed in $M$, then $p$ decides $\dot{f}$.

###### Proof.

Suppose that $h\in M$ is a function such that $h\\!\upharpoonright\\!M\cap
Z=g$, we shall show that $p\Vdash``\dot{f}=\check{h}"$. Fix $q\leq p$. We
claim that there is a condition below $q$ forcing $\dot{f}=\check{h}$. If not,
the set

$D=\\{r\in\mathbb{P}:\exists\zeta\in Z\text{ such that
}r\Vdash\check{h}(\zeta)\neq\dot{f}(\zeta)\\}$

is pre-dense below $q$ and belongs to $M$. Since $q$ is
$(M,\mathbb{P})$-generic, we can find some $r\in D\cap M$ compatible with $q$.
Thus there exists $\zeta\in M\cap Z$ such that
$r\Vdash``\dot{f}(\zeta)\neq\check{h}(\zeta)=\check{g}(\zeta)"$. This is a
contradictions, as $r$ is also compatible with $q$, and $q$ forces
$\dot{f}(\zeta)=\check{g}(\zeta)$. ∎

######  Lemma 2.17.

Assume that $M\prec H_{\theta}$ is countable, and that $Z\in M$. Suppose that
$z\mapsto f_{z}$ is a function on $[Z]^{\omega}$ in $M$ so that $f_{z}$ is a
function with $z\subseteq{\rm dom}(f_{z})$. Let $f:M\cap Z\rightarrow 2$ be a
function that is not guessed in $M$. Let $B\in M$ be a cofinal subset of
$[Z]^{\omega}$. Then there is $B^{*}\in M$ which is a cofinal subset of $B$
such that for every $z\in B^{*}$, $f_{z}\nsubseteq f$.

###### Proof.

For each $\zeta\in Z$, let

$A_{\zeta}^{\epsilon}=\\{z\in B:f_{z}(\zeta)=\epsilon\\},\text{ where
}\epsilon=0,1.$

Notice that the sequence

$\langle A_{\zeta}^{\epsilon}:\zeta\in N\cap Z,\epsilon\in\\{0,1\\}\rangle$

belongs to $M$. We are done if there is some $\zeta\in Z$ such that both
$A_{\zeta}^{0}$ and $A_{\zeta}^{1}$ are cofinal in $B$, as then one can find,
by elementarity, such $\zeta\in M\cap Z$, and pick $A_{\zeta}^{1-f(\zeta)}$.
Therefore, let us assume that for every $\zeta\in Z$, there is a unique
$\epsilon\in\\{0,1\\}$ such that $A_{\zeta}^{\epsilon}$ is cofinal in $B$.
Define now $h$ on $Z$ by letting $h(\zeta)$ be $\epsilon$ if and only if
$A_{\zeta}^{\epsilon}$ is cofinal in $[Z]^{\omega}$. Clearly $h$ belongs to
$M$. But $h\restriction{M\cap Z}\neq f$, since $f$ is not guessed in $M$.
Hence, there must exist $\zeta\in M\cap Z$ such that $h(\zeta)\neq f(\zeta)$,
which in turn implies that $A_{\zeta}^{1-f(\zeta)}$ is a cofinal subset of $B$
and belongs to $M$. Let $B^{*}$ be $A_{\zeta}^{1-f(\zeta)}$. ∎

######  Definition 2.18.

Let $T$ be a tree of height $\omega_{1}$. We say that $p\in\mathbb{S}(T)$ if
$p$ is a partial function, ${\rm dom}(p)$ is a finite subset of $T$, $p:{\rm
dom}(p)\to\omega$, and if $x,y\in{\rm dom}(p)$ are distinct and $p(x)=p(y)$,
then $x$ and $y$ are incomparable in $T$. The order on $\mathbb{S}(T)$ is
reverse inclusion.

It is well known that if $T$ has no uncountable branches then $\mathbb{S}(T)$
has the ccc. We also need the following fact which was shown by Chodounský-
Zapletal [4]. We provide a proof for completeness.

######  Proposition 2.19 (Chodounský-Zapletal [4]).

Suppose $T$ is a tree of height $\omega_{1}$ without uncountable branches.
Then $\mathbb{S}(T)$ has the $\omega_{1}$-approximation property.

###### Proof.

Suppose $\mu$ is a cardinal and $\dot{f}$ is an $\mathbb{S}(T)$-name for a
function from $\mu$ to $2$ which is countably approximated in $V$. It suffices
to show that some condition in $\mathbb{S}(T)$ decides $\dot{f}$. Fix a
sufficiently large regular cardinal $\theta$ and a countable $M\prec
H(\theta)$ containing all the relevant objects. Since $\dot{f}$ is forced to
be countably approximated in $V$ we can pick a condition $p\in\mathbb{S}(T)$
and a function $g:M\cap\mu\to 2$ such that $p$ forces that
$\dot{f}\\!\upharpoonright\\!(M\cap\mu)=\check{g}$. Since $\mathbb{S}(T)$ is
ccc, $p$ is $(M,\mathbb{S}(T))$-generic. Therefore, by Lemma 2.16 we may
assume that $g$ is not guessed in $M$. Let $d={\rm dom}(p)$, let $d_{0}=d\cap
M$, and $p_{0}=p\\!\upharpoonright\\!d_{0}$. Let $n=|d\setminus d_{0}|$. Let
$C$ be the collection of $x\in[T\cup\mu]^{\omega}$ such that $d_{0}\subseteq
x$ and $T\cap x$ is an initial segment of $T$ under $\leq_{T}$. If $x\in C\in
M$ then there is a condition $p_{x}\leq p_{0}$, and a function $g_{x}:\mu\cap
x\to 2$ such that $p_{x}\\!\upharpoonright\\!x=p_{0}$, $|p\setminus p_{0}|=n$,
and $p_{x}$ forces that $\dot{f}\\!\upharpoonright\\!(x\cap\mu)$ equals
$g_{x}$. This is witnessed by the condition $p$. By elementarity of $M$ such a
pair $(p_{x},g_{x})$ exists, for all $x\in C$. Moreover, we can pick the
assignment $x\mapsto(p_{x},g_{x})$ in $M$. Let $d_{x}={\rm dom}(p_{x})$. By
Lemma 2.17 we can find $C^{*}\in M$, a cofinal subset of $C$ under inclusion,
such that $g_{x}\nsubseteq g$, for all $x\in C^{*}$. This implies that $p_{x}$
are $p$ are incompatible, for all $x\in C^{*}\cap M$. Let us fix an
enumeration $d\setminus d_{0}=\\{t(0),\ldots,t(n-1)\\}$, and, for each $x\in
C^{*}$, an enumeration $d_{x}\setminus
d_{0}=\\{t_{x}(0),\ldots,t_{x}(n-1)\\}$.

######  Claim 2.20.

Let $B\in M$ be a cofinal subset of $C^{*}$, and $i,j<n$. Then there is
$B^{*}\in M$, a cofinal subset of $B$, such that, for every $x\in B^{*}\cap
M$, $t(i)$ and $t_{x}(j)$ are incomparable.

###### Proof.

For $t\in T$, let ${\rm pred}_{T}(t)$ be the set of predecessors of $t$ in
$T$, and let $h_{t}$ be the characteristic function of ${\rm pred}_{T}(t)$ as
a subset of $T$. Let $h=h_{t(i)}\\!\upharpoonright\\!(T\cap M)$. For $x\in B$
let $h_{x}=h_{t_{x}(j)}\\!\upharpoonright\\!(T\cap x)$. Suppose first that $h$
is not guessed in $M$. Then by Lemma 2.17 we can find $B^{*}\in M$, a cofinal
subset of $B$, such that $h_{x}\nsubseteq h$, for all $x\in B^{*}$. This
implies that if $x\in B^{*}\cap M$ then $t(i)$ and $t_{x}(j)$ do not have the
same predecessors in $T\cap x$ and hence are incomparable, as desired. Suppose
now that $h$ is guessed in $M$ and let $k\in M$ be a function such that
$k\\!\upharpoonright\\!(T\cap M)=h$. Since $h^{-1}(1)$ is a chain and is
closed downwards in $T$, the same is true for $k^{-1}(1)$. $T$ has no
uncountable chains, so $k^{-1}(1)$ is countable and hence we must have
$k^{-1}(1)\subseteq M$. Moreover $k^{-1}(1)={\rm pred}_{T}(t(i))\cap M$. By
elementarity of $M$ there is $s\in M$ such that $k^{-1}(1)={\rm pred}_{T}(s)$.
Now, let $B^{*}$ be the set of $x\in B$ such that $s\in x$. Then $B^{*}\in M$
and is a cofinal subset of $B$. If $x\in B^{*}\cap M$, then since
$t_{x}(j)\notin x$, it follows that $t_{x}(j)\notin{\rm pred}_{T}(s)$. It
follows that $t_{x}(j)$ is also below $t(i)$. Also, $t_{x}(j)$ cannot be above
$t(i)$ in $T$, since then $t_{x}(j)$ and all its predecessors in $T$ are in
$M$ and $t(i)$ is not in $M$. It follows that $B^{*}$ is as required. ∎

Now, applying the above claim repeatedly for all pairs $i,j<n$ we can find a
cofinal subset $B$ of $C^{*}$ such that, for all $x\in B\cap M$, every element
of $d_{x}\setminus d_{0}$ is incomparable with every element of $d\setminus
d_{0}$. It follows that if $x\in B\cap M$ then $p_{x}\cup p\in\mathbb{S}(T)$,
which is a contradiction.

∎

##  3\. The main theorems

We recall the guessing model principle introduced by Viale and Weiß [18]. We
let ${\rm GM}(\kappa,\gamma)$ be the statement that ${\rm
GM}(\kappa,\gamma,H_{\theta})$ holds, for all sufficiently large regular
$\theta$, where ${\rm GM}(\kappa,\gamma,H_{\theta})$ states that

$\mathfrak{G}_{\kappa,\gamma}(H_{\theta})=\\{M\in{\mathcal{P}}_{\kappa}(H_{\theta}):M\prec
H_{\theta}\mbox{ and }M\mbox{ is $\gamma$-guessing}\\}.$

is stationary in ${\mathcal{P}}_{\kappa}(H_{\theta})$

######  Definition 3.1 (Cox–Krueger [6]).

$\rm SGM(\omega_{2},\omega_{1})$ state that for every sufficiently large
regular cardinal $\theta$ the following set is stationary in
$\mathcal{P}_{\omega_{2}}(H_{\theta})$

$\mathfrak{G}_{\omega_{2},\omega_{1}}(H_{\theta})=\\{M\in{\mathcal{P}}_{\omega_{2}}(H_{\theta}):M\prec
H_{\theta}\mbox{ and }M\mbox{ is indestructibly $\omega_{1}$-guessing}\\}.$

Notice that it was proved in [6] that Suslin’s Hypothesis follows from $\rm
SGM(\omega_{2},\omega_{1})$, but not form $\rm GM(\omega_{2},\omega_{1})$.

Let us now recall the notion of a strongly $\gamma$-guessing model introduced
and studied by the authors in [14].

######  Definition 3.2.

Let $\gamma\leq\kappa$ be regular uncountable cardinals. A model $M$ of
cardinality $\kappa^{+}$ is called strongly $\gamma$-guessing if it is the
union of an $\in$-increasing chain $\langle M_{\xi}:\xi<\kappa^{+}\rangle$ of
$\gamma$-guessing models of cardinality $\kappa$ such that
$M_{\xi}=\bigcup\\{M_{\eta}:\eta<\xi\\}$, for every $\xi$ of cofinality
$\kappa$.

It was proved in Remark 1.14 of [14] that every strongly $\gamma$-guessing
model is $\gamma$-guessing.

######  Definition 3.3.

${\rm GM}^{+}(\kappa^{++},\gamma)$ is the statement that ${\rm
GM}^{+}(\kappa^{++},\gamma,H_{\theta})$ holds, for all sufficiently large
regular $\theta$, where ${\rm GM}^{+}(\kappa^{++},\gamma,H_{\theta})$ states
that

$\mathfrak{G}^{+}_{\kappa^{++},\gamma}(H_{\theta})=\\{M\in{\mathcal{P}}_{\kappa^{++}}(H_{\theta}):M\prec
H_{\theta}\mbox{ and }M\mbox{ is strongly $\gamma$-guessing}\\}.$

is stationary in ${\mathcal{P}}_{\kappa^{++}}(H_{\theta})$.

######  Definition 3.4.

A model $M$ of cardinality $\omega_{2}$ is indestructibly strongly
$\omega_{1}$-guessing if it is the union of an increasing chain $\langle
M_{\xi}:\xi<\omega_{2}\rangle$ of indestructibly $\omega_{1}$-guessing models
of cardinality $\omega_{1}$ such that
$M_{\xi}=\bigcup\\{M_{\eta}:\eta<\xi\\}$, for every $\xi$ of cofinality
$\omega_{1}$.

######  Definition 3.5.

The principle ${\rm SGM}^{+}(\omega_{3},\omega_{1})$ states that
$\mathfrak{S}^{+}_{\omega_{3},\omega_{1}}(H_{\theta})$ is stationary, for all
large enough $\theta$, where

$\mathfrak{S}^{+}_{\omega_{3},\omega_{1}}(H_{\theta})\coloneqq\\{M\in\mathcal{P}_{\omega_{3}}(H_{\theta}):M\prec
H_{\theta}\text{ is indestructibly and strongly
}\omega_{1}\text{-guessing}\\}.$

######  Definition 3.6.

For a regular cardinal $\kappa$, we let $\rm{MP(\kappa^{+})}$ denote the
statement that every forcing which adds a new subset of $\kappa^{+}$ whose
initial segments are in the ground model collapses some cardinal $\leq
2^{\kappa}$.

Todorčević [15] showed that if every tree of size and height $\omega_{1}$ with
at most $\omega_{1}$ cofinal branches is weakly special and
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ then $\rm MP(\omega_{1})$. The principle
was studied also by Golshani and Shelah in [8], where they showed that ${\rm
MP}(\kappa^{+})$ is consistent, for every regular cardinal $\kappa$.

######  Proposition 3.7 (Cox–Krueger [6]).

If $\rm SGM(\omega_{2},\omega_{1})$ and $2^{\aleph_{0}}<\aleph_{\omega_{1}}$
then ${\rm MP}(\omega_{1})$ holds.

###### .

∎

We shall prove that $\rm SGM^{+}(\omega_{3},\omega_{1})$ implies ${\rm
MP}(\omega_{2})$, and since ${\rm SGM}(\omega_{2},\omega_{1})$ follows from
${\rm SGM}^{+}(\omega_{3},\omega_{1})$, we obtain the simultaneous consistency
of ${\rm MP}(\omega_{1})$ and ${\rm MP}(\omega_{2})$.

######  Theorem 3.8.

Suppose that $V\subseteq W$ are transitive models of $\rm ZFC$. Assume that
$\rm SGM^{+}(\omega_{3},\omega_{1})$ and $2^{\omega_{1}}<\aleph_{\omega_{2}}$
hold in $V$. Suppose that $W$ has a subset of $\omega^{V}_{2}$ which does not
belong to $V$. Then either
$\mathcal{P}^{V}(\omega_{1})\neq\mathcal{P}^{W}(\omega_{1})$ or some
$V$-cardinal $\leq 2^{\omega_{1}}$ is no longer a cardinal in $W$.

###### Proof.

Let $x\in W\setminus V$ be a subset of $\omega^{V}_{2}$. Assume that
$\mathcal{P}^{V}(\omega_{1})=\mathcal{P}^{W}(\omega_{1})$. We shall show that
some cardinal $\leq 2^{\omega_{1}}$ is no longer a cardinal in $W$. Since
$\mathcal{P}^{V}(\omega_{1})=\mathcal{P}^{W}(\omega_{1})$, every initial
segment of $x$ belongs to $V$. Letting now
$\mathfrak{X}=\\{x\cap\gamma:\gamma<\omega_{2}\\}$, we have that
$\mathfrak{X}$ is bounded in $V$. Assume towards a contradiction that every
$V$-cardinal $\leq 2^{\omega_{1}}$ remains cardinal in $W$. Work in $W$, and
let $\mu\geq\omega_{2}$ be the least cardinal such that there is a set $M$ in
$V$ of cardinality $\mu$ such that $M\cap\mathfrak{X}$ is of size
$\omega_{2}$. Thus $\mu\leq 2^{\omega_{1}}$. We claim that $\mu=\omega_{2}$.
Suppose that $\mu>\omega_{2}$ and $M$ is a witness for that, then one can work
in $V$ and write $M$ as the union of an increasing sequence $\langle
M_{\xi}:\xi<{\rm cof^{V}}(\mu)\rangle$ of subsets of $M$ in $V$ whose size are
less than $\mu$. Since $\mu\leq 2^{\omega_{1}}<\aleph_{\omega_{2}}$ and every
cardinal $\leq 2^{\omega_{1}}$ is a cardinal in $W$, ${\rm cof}^{W}(\mu)={\rm
cof}^{V}(\mu)\neq\omega_{2}$. Thus either $\mu$ is of cofinality at most
$\omega_{1}$, and then by the pigeonhole principle, there is $\xi<{\rm
cof}(\mu)$ such that $|M_{\xi}\cap\mathfrak{X}|=\omega_{2}$, or $\mu$ is
regular, and then there is some $\xi<{\rm cof}(\mu)$ such that
$M\cap\mathfrak{X}\subseteq M_{\xi}$. In either case we obtain a contradiction
since $|M_{\xi}|<\mu$. Therefore, $\mu=\omega_{2}$. Let $M$ be a witness for
$\mu=\omega_{2}$, and let $\mathfrak{X}^{\prime}=M\cap\mathfrak{X}$. Notice
that $V\models|M|=\omega_{2}$. Since $M$ is in $V$ and $V$ satisfies $\rm
SGM^{+}(\omega_{3},\omega_{1})$, one can cover $M$ by an indestructibly
strongly $\omega_{1}$-guessing model $N$ of size $\omega_{2}$. Working in $W$,
$x$ is countably approximated in $N$, since if $\gamma\in N\cap\omega_{2}$,
then there is $\gamma^{\prime}>\gamma$ in $N$ such that
$x\cap\gamma^{\prime}\in\mathfrak{X}^{\prime}\subseteq N$, and hence
$x\cap\gamma\in N$. On the other hand $N$ is a guessing model in $W$ by $\rm
SGM^{+}(\omega_{3},\omega_{1})$ in $V$ and both $\omega_{1}$ and $\omega_{2}$
are cardinals in $W$. Thus $x$ is guessed in $N$, but then $x$ must be in $N$
as $\omega_{2}\subseteq N$. Therefore, $x$ is in $V$, which is a
contradiction! ∎

The following corollaries are immediate.

######  Corollary 3.9.

Suppose that $V\subseteq W$ are transitive models of $\rm ZFC$. Assume in $V$,
$\rm SGM^{+}(\omega_{3},\omega_{1})$, $2^{\omega_{0}}<\aleph_{\omega_{1}}$ and
$2^{\omega_{1}}<\aleph_{\omega_{2}}$ hold. Suppose that $W$ has a new subset
of $\omega^{V}_{2}$. Then either $W$ contains a real which is not in $V$ or
some cardinal $\leq 2^{\omega_{1}}$ in $V$ is collapsed in $W$.

###### .

∎

######  Corollary 3.10.

Assume $\rm SGM^{+}(\omega_{3},\omega_{1})$ as well as
$2^{\aleph_{0}}<\aleph_{\omega_{1}}$ and $2^{\aleph_{1}}<\aleph_{\omega_{2}}$.
Then the principles ${\rm MP}(\omega_{1})$ and ${\rm MP}(\omega_{2})$ hold.

###### .

∎

Our main theorem reads as follows.

######  Theorem 3.11.

Assume there are two supercompact cardinals. Then there is a generic extension
$V[G]$ of $V$ in which ${\rm SGM}^{+}(\omega_{3},\omega_{1})$ holds as well as
$2^{\aleph_{0}}=2^{\aleph_{1}}=2^{\aleph_{2}}=\aleph_{3}$.

###### .

∎

##  4\. An overview of virtual models

We review the theory of virtual models from the authors’ paper [14].

### The general setting

We consider the language $\mathcal{L}$ obtained by adding a single constant
symbol $c$ and a predicate $U$ to the standard language of set theory. Let us
say that an $\mathcal{L}$-structure $\mathcal{A}=(A,\in,\kappa,U)$ is
_suitable_ if $A$ is a transitive set and $\mathcal{A}\models``\mathrm{ZFC}"$
in the expanded language, where $\kappa=c^{\mathcal{A}}$ is an uncountable
regular in $\mathcal{A}$. For a suitable structure $\mathcal{A}$ and an
ordinal $\alpha\in A\setminus\kappa$, we let $\mathcal{A}_{\alpha}=(A\cap
V_{\alpha},\in,\kappa,U\cap V_{\alpha})$. Finally, we let

$E_{\mathcal{A}}=\\{\alpha\in{\rm
ORD}^{A}:\mathcal{A}_{\alpha}\prec\mathcal{A}\\}.$

Note that $E_{A}$ is a closed subset of ${\rm ORD}^{\mathcal{A}}$. It is not
definable in $\mathcal{A}$, but $E_{\mathcal{A}}\cap\alpha$ is uniformly
definable in $\mathcal{A}$ with parameter $\alpha$, for each $\alpha\in
E_{\mathcal{A}}$. For $\alpha\in E_{\mathcal{A}}$, we let ${\rm
next}_{\mathcal{A}}(\alpha)$ be the least ordinal in $E_{\mathcal{A}}$ above
$\alpha$, if such an ordinal exists. Otherwise, we leave ${\rm
next}_{\mathcal{A}}(\alpha)$ undefined. We shall often abuse notation and
refer to the structure $\mathcal{A}$ by $A$.

######  Lemma 4.1.

Suppose $M$ is an elementary submodel of a suitable structure $A$, and that
$\alpha\in E_{A}$. If $(M\cap{\rm ORD}^{A})\setminus\alpha\neq\varnothing$,
then ${\rm min}(M\cap{\rm ORD}^{A}\setminus\alpha)\in E_{A}$.

###### Proof.

See [16, Lemma 1.1] or [14, Lemma 3.1]. ∎

If $M$ is a submodel of a suitable structure $A$ and $X$ is a subset of $A$,
let the ${\rm Hull}$ operation be defined by

${\rm Hull}(M,X)=\\{f(\bar{x}):f\in M,\bar{x}\in X^{<\omega},f\mbox{ is a
function, and }\bar{x}\in{\rm dom}(f)\\}.$

For $\alpha\in E_{A}$, we let ${\mathscr{A}}_{\alpha}$ denote the set of all
transitive $\mathcal{L}$-structures $\mathcal{H}\in A$ that are elementary
extensions of $\mathcal{A}_{\alpha}$ and have the same cardinality as
$A_{\alpha}$. Note that if $\mathcal{H}\in\mathscr{A}_{\alpha}$ and
$\alpha\in\mathcal{H}$, then $E_{\mathcal{H}}\cap\alpha=E_{A}\cap\alpha$. For
$\mathcal{H}\in\mathscr{A}_{\alpha}$, we will refer to $A_{\alpha}$ as the
standard part of $\mathcal{H}$. Note that if $\mathcal{H}$ has nonstandard
elements then $\alpha\in E_{\mathcal{H}}$.

######  Lemma 4.2.

Suppose $M$ is an elementary submodel of a suitable structure $A$, and that
$X\subseteq A$. Let $\delta=\sup(M\cap{\rm ORD}^{A})$, and suppose that $X\cap
A_{\delta}$ is nonempty. Then ${\rm Hull}(M,X)$ is the least elementary
submodel of $A$ containing $M$ and $X\cap A_{\delta}$ as subsets.

###### Proof.

See [16, Lemma 1.2] or [14, Lemma 3.3]. ∎

### Virtual models in $V_{\lambda}$

Assume that $\kappa<\lambda$ are supercompact and inaccessible cardinals,
respectively. Consider the $\mathcal{L}$-structure

$V_{\lambda}\coloneqq(V_{\lambda},\kappa,\in,U).$

Let $E=E_{V_{\lambda}}$ and ${\rm next}(\alpha)={\rm
next}_{V_{\lambda}}(\alpha)$.

######  Definition 4.3.

Suppose $\alpha\in E$. We let $\mathscr{V}_{\alpha}$ denote the collection of
all substructures $M$ of $V_{\lambda}$ of size less than $\kappa$ such that,
if we let $A={\rm Hull}(M,V_{\alpha})$, then $A\in\mathscr{A}_{\alpha}$ and
$M$ is an elementary submodel of $A$. The members of $\mathscr{V}_{\alpha}$
are called $\alpha$-models.

We write $\mathscr{V}_{<\alpha}$ for $\bigcup\\{\mathscr{V}_{\gamma}:\gamma\in
E\cap\alpha\\}$ and $\mathscr{V}$ for $\mathscr{V}_{<\lambda}$. The
collections $\mathscr{V}_{\leq\alpha}$ and $\mathscr{V}_{\geq\alpha}$ are
defined in the obvious way. For $M\in\mathscr{V}$, $\eta(M)$ denotes the
unique ordinal $\alpha$ such that $M\in\mathscr{V}_{\alpha}$. Note that if
$M\in\mathscr{V}_{\alpha}$, then $\sup(M\cap{\rm ORD})\geq\alpha$. In general,
$M$ is not elementary in $V_{\lambda}$, in fact, this only happens if
$M\subseteq V_{\alpha}$. In this case we will say that $M$ is a standard
$\alpha$-model. We refer to members of $\mathscr{V}$ as virtual models. We
also refer to members of $\mathscr{V}^{A}$, for some suitable structure $A$
with $A\subseteq V_{\lambda}$, as general virtual models.

Suppose $M,N\in\mathscr{V}$ and $\alpha\in E$. We say that an isomorphism
$\sigma:M\rightarrow N$ is an $\alpha$-isomorphism if it has an extension to
an isomorphism $\bar{\sigma}:{\rm Hull}(M,V_{\alpha})\rightarrow{\rm
Hull}(N,V_{\alpha})$. We say that $M$ and $N$ are $\alpha$-isomorphic and
write $M\cong_{\alpha}N$ if there is an $\alpha$-isomorphism between them.
Note that if $\sigma$ and $\bar{\sigma}$ exist, they are unique. It is clear
that $\cong_{\alpha}$ is an equivalence relation, for every $\alpha\in E$. If
$\alpha<\beta$ are in $E$, then for every $\beta$-model $M$ there is a
canonical representative of the $\cong_{\alpha}$-equivalence class of $M$
which is an $\alpha$-model.

######  Definition 4.4.

Suppose $\alpha$ and $\beta$ are members of $E$ and $M$ is a $\beta$-model.
Let $\overline{{\rm Hull}(M,V_{\alpha})}$ be the transitive collapse of ${\rm
Hull}(M,V_{\alpha})$, and let $\pi$ be the collapsing map. We define
$M\\!\restriction\\!\alpha$ to be $\pi[M]$, i.e. the image of $M$ under the
collapsing map of ${\rm Hull}(M,V_{\alpha})$.

Note that if $A\in\mathscr{A}_{\alpha}$ then
$\mathscr{V}_{\alpha}^{A}\subseteq\mathscr{V}_{\alpha}$. Therefore, if
$A,B\in\mathscr{A}_{\alpha}$, $M\in\mathscr{V}^{A}$, and
$N\in\mathscr{V}^{B}$, we can still write $M\cong_{\alpha}N$ if
$M\restriction\alpha=N\restriction\alpha$. It is straightforward to check that
if $\alpha\leq\beta$ are in $E$, and $M\in\mathscr{V}$. Then
$(M\\!\restriction\\!\beta)\\!\restriction\\!\alpha=M\\!\restriction\\!\alpha$.
For each $\alpha\in E$, the virtual version of the membership relation is
defined as follows. Suppose $M,N\in\mathscr{V}$ and $\alpha\in E$. We write
$M\in_{\alpha}N$ if there is $M^{\prime}\in N$ with
$M^{\prime}\in\mathscr{V}^{N}$ such that $M^{\prime}\cong_{\alpha}M$. If this
happens, we say that $M$ is $\alpha$-in $N$.

######  Lemma 4.5.

Let $\alpha\leq\beta$ be in $E$. Suppose $M,N\in\mathscr{V}_{\geq\beta}$ and
$M\in_{\beta}N$. Then
$M\\!\restriction\\!\alpha\in_{\alpha}N\\!\restriction\\!\alpha$.

###### Proof.

See [14, Proposition 3.15]. ∎

######  Definition 4.6.

For $\alpha\in E$, we let $\mathscr{C}_{\alpha}$ denote the collection of
countable models in $\mathscr{V}_{\alpha}$, and we let $\mathscr{U}_{\alpha}$
be the collection of all $M\in\mathscr{V}_{\alpha}$ that are $\kappa$-Magidor
models.

The collections $\mathscr{C}_{<\alpha}$, $\mathscr{C}_{\leq\alpha}$,
$\mathscr{C}_{\geq\alpha}$, $\mathscr{U}_{<\alpha}$,
$\mathscr{U}_{\leq\alpha}$, and $\mathscr{U}_{\geq\alpha}$ are defined
similarly. We write $\mathscr{C}$ for $\mathscr{C}_{<\lambda}$, and
$\mathscr{U}$ for $\mathscr{U}_{<\lambda}$, $\mathscr{C}_{\rm st}$ for the
collection of standard models in $\mathscr{C}$, and $\mathscr{U}_{\rm st}$ for
the standard models in $\mathscr{U}$. Note that both classes $\mathscr{C}$ and
$\mathscr{U}$ are closed under projections.

######  Lemma 4.7.

$\mathscr{C}_{\rm st}$ contains a club in
${\mathcal{P}}_{\omega_{1}}(V_{\lambda})$, and $\mathscr{U}_{\rm st}$ is
stationary in $\mathcal{P}_{\kappa}(V_{\lambda})$.

###### Proof.

See [14, Propositions 3.19 and 3.24]. ∎

######  Definition 4.8.

Let $M\in\mathscr{V}$. We say that $M$ is active at $\alpha\in E$ if
$\eta(M)\geq\alpha$ and ${\rm Hull}(M,V_{\kappa_{M}})\cap E\cap\alpha$ is
unbounded in $E\cap\alpha$, where $\kappa_{M}=\sup(M\cap\kappa)$. We say that
$M$ is strongly active at $\alpha$ if $\eta(M)\geq\alpha$ and $M\cap
E\cap\alpha$ is unbounded in $E\cap\alpha$.

One can easily see that a Magidor model is active at $\alpha$ if and only if
it is strongly active at $\alpha$.

######  Notation 4.9.

Let $M\in\mathscr{V}$. Let $a(M)=\\{\alpha\in E:M\mbox{ is active at
}\alpha\\}$ and $\alpha(M)=\max(a(M))$.

Note that $a(M)$ is a closed subset of $E$ of size at most $|{\rm
Hull}(M,V_{\kappa_{M}})|$. We now recall the definition of the meet from [14]
that is a virtual version of intersection only defined for two models of
different types. Suppose $N\in\mathscr{U}$ and $M\in\mathscr{C}$. Let
$\overline{N}$ be the transitive collapse of $N$, and let $\pi$ be the
collapsing map. Note that if $\overline{N}\in M$, then $\overline{N}\cap M$ is
a countable elementary submodel of $\overline{N}$. Then $\overline{N}\cap
M\in\overline{N}$ since $\overline{N}$ is closed under countable sequence.
Note that $\pi^{-1}(\overline{N}\cap M)=\pi^{-1}[\overline{N}\cap M]$, and
this model is elementary in $N$.

######  Definition 4.10.

Suppose $N\in\mathscr{U}$ and $M\in\mathscr{C}$. Let $\alpha=\max(a(N)\cap
a(M))$. We will define $N\land M$ if $N\in_{\alpha}M$. Let $\overline{N}$ be
the transitive collapse of $N$, and let $\pi$ be the collapsing map. Set

$\eta=\sup(\sup(\pi^{-1}[\overline{N}\cap M]\cap{\rm ORD})\cap
E\cap(\alpha+1)).$

We define the meet of $N$ and $M$ to be $N\land
M\coloneqq\pi^{-1}[\overline{N}\cap M]\restriction\eta$.

It was proved in [14] that $N\land M\in\mathscr{C}_{\eta}$ is an $\eta$-model,
and moreover if $N\land M$ is active at $\gamma$, then $(N\land
M)\\!\upharpoonright\\!\gamma=N\\!\upharpoonright\\!\gamma\land
M\\!\upharpoonright\\!\gamma$. More intersection-like properties of $\land$
are found on [14, Pages 12–15].

######  Notation 4.11.

Let $\alpha\in E$ and let $\mathcal{M}$ be a set of virtual models. We let

$\mathcal{M}\restriction\alpha=\\{M\restriction\alpha:M\in\mathcal{M}\\}\mbox{
and }\mathcal{M}^{\alpha}=\\{M\restriction\alpha:\leavevmode\nobreak\
M\in\mathcal{M}\text{ is active at }\alpha\\}.$

######  Definition 4.12.

Let $\alpha\in E$ and let $\mathcal{M}$ be a subset of
$\mathscr{U}\cup\mathscr{C}$. We say $\mathcal{M}$ is an $\alpha$-chain if for
all distinct $M,N\in\mathcal{M}$, either $M\in_{\alpha}N$ or $N\in_{\alpha}M$,
or there is a $P\in\mathcal{M}$ such that either $M\in_{\alpha}P\in_{\alpha}N$
or $N\in_{\alpha}P\in_{\alpha}M$.

######  Lemma 4.13.

Suppose $\alpha\in E$ and $\mathcal{M}$ is a finite subset of
$\mathscr{U}\cup\mathscr{C}$. Then $\mathcal{M}$ is an $\alpha$-chain if and
only if there is an enumeration $\\{M_{i}:i<n\\}$ of $\mathcal{M}$ such that
$M_{0}\in_{\alpha}\cdots\in_{\alpha}M_{n-1}$.

###### Proof.

See [14, Proposition 3.41]. ∎

##  5\. Virtual models as side conditions

We recall the definition of the pure side conditions forcing with decorations
introduced in [14]. We state its basic properties and refer the reader to the
above paper for proofs.

######  Definition 5.1 (pure side conditions).

Suppose $\alpha\in E$. We say that $p=\mathcal{M}_{p}$ belongs to
$\mathbb{M}^{\kappa}_{\alpha}$ if:

  1. (1)

$\mathcal{M}_{p}\subseteq\mathscr{C}_{\leq\alpha}\cup\mathscr{U}_{\leq\alpha}$
is finite closed under meets, and

  2. (2)

$\mathcal{M}_{p}^{\delta}$ is a $\delta$-chain, for all $\delta\in
E\cap(\alpha+1)$.

We let $\mathcal{M}_{q}\leq\mathcal{M}_{p}$ if for all $M\in\mathcal{M}_{p}$
there is $N\in\mathcal{M}_{q}$ such that $N\restriction{\eta(M)}=M$. Finally,
let
$\mathbb{M}^{\kappa}_{\lambda}=\bigcup\\{\mathbb{M}^{\kappa}_{\alpha}:\alpha\in
E\\}$ with the same order.

Suppose $\mathcal{M}_{p}\in\mathbb{M}^{\kappa}_{\lambda}$. Let

$\mathcal{L}(\mathcal{M}_{p})=\\{M\restriction\alpha:M\in\mathcal{M}_{p}\text{
and }\alpha\in a(M)\\}.$

We say that $M\in\mathcal{L}(\mathcal{M}_{p})$ is $\mathcal{M}_{p}$-free if
every $N\in\mathcal{M}_{p}$ with $M\in_{\eta(M)}N$ is strongly active at
$\eta(M)$. Let $\mathcal{F}(\mathcal{M}_{p})$ denote the set of all
$M\in\mathcal{L}(\mathcal{M}_{p})$ that are $\mathcal{M}_{p}$-free. Note that
a model $M\in\mathcal{L}(\mathcal{M}_{p})$ that is not $\mathcal{M}_{p}$-free
is not $\mathcal{M}_{q}$-free, for any $\mathcal{M}_{q}\leq\mathcal{M}_{p}$.

######  Definition 5.2 (side conditions with decorations).

Suppose $\alpha\in E\cup\\{\lambda\\}$. We say that a pair
$p=(\mathcal{M}_{p},d_{p})$ belongs to $\mathbb{P}^{\kappa}_{\alpha}$ if
$\mathcal{M}_{p}\in\mathbb{M}^{\kappa}_{\alpha}$, $d_{p}$ is a finite partial
function from $\mathcal{F}(\mathcal{M}_{p})$ to
${\mathcal{P}}_{\omega}(V_{\kappa})$ such that

  * $(\divideontimes)$

if $M\in{\rm dom}(d_{p})$, $N\in\mathcal{M}_{p}$, and $M\in_{\eta(M)}N$, then
$d_{p}(M)\in N$.

We say that $q\leq p$ if $\mathcal{M}_{q}\leq\mathcal{M}_{p}$ in
$\mathbb{M}^{\kappa}_{\alpha}$, and, for every $M\in{\rm dom}(d_{p})$, there
exists some $\gamma\in E\cap(\eta(M)+1)$ with $M\restriction\gamma\in{\rm
dom}(d_{q})$ such that $d_{p}(M)\subseteq d_{q}(M\restriction\gamma)$.

We refer to $d_{p}$ as the decoration of $p$. The order on
$\mathbb{P}^{\kappa}_{\lambda}$ is transitive. We will say that $q$ is
stronger than $p$ if $q$ forces that $p$ belongs to the generic filter, in
order words, any $r\leq q$ is compatible with $p$. We write $p\sim q$ if each
of $p$ and $q$ is stronger than the other. We identify equivalent conditions,
often without saying it. Our forcing does not have greatest lower bounds, but
if $p$ and $q$ do have a greatest lower bound we will denote it by $p\land q$.
To be precise we should refer to $p\land q$ as the $\sim$-equivalence class of
a greatest lower bound, but we ignore this point since it should not cause any
confusion. Note that if $p\in\mathbb{P}^{\kappa}_{\alpha}$ and
$M\in\mathcal{M}_{p}$ is a $\delta$-model that is not active at $\delta$, we
may replace $M$ by $M\restriction{\alpha(M)}$ and we get an equivalent
condition. Thus, if $\alpha\in E$ and ${\rm cof}(\alpha)\geq\kappa$, then
$\mathbb{P}^{\kappa}_{\alpha}$ is forcing equivalent to
$\bigcup\\{\mathbb{P}^{\kappa}_{\gamma}:\gamma\in E\cap\alpha\\}$. Suppose
$\alpha,\beta\in E$ and $\alpha\leq\beta$. For every
$p\in\mathbb{P}^{\kappa}_{\beta}$, we let
$\mathcal{M}_{p\restriction\alpha}=\mathcal{M}_{p}\\!\upharpoonright\\!\alpha$
and
$d_{p\restriction\alpha}=d_{p}\restriction{\mathcal{F}(\mathcal{M}_{p}\\!\upharpoonright\\!\alpha)}$.
It is easily seen that
$p\\!\upharpoonright\\!\alpha=(\mathcal{M}_{p\restriction\alpha},d_{p\restriction\alpha})\in\mathbb{P}^{\kappa}_{\alpha}$.
The following is straightforward.

######  Lemma 5.3.

Suppose that $\alpha,\beta\in E$ with $\alpha\leq\beta$. Let
$p\in\mathbb{P}^{\kappa}_{\beta}$ and let $q\in\mathbb{P}^{\kappa}_{\alpha}$
be such that $q\leq p\restriction\alpha$. Then there exists
$r\in\mathbb{P}^{\kappa}_{\beta}$ such that $r\leq p,q$.

###### Proof.

We let $\mathcal{M}_{r}=\mathcal{M}_{p}\cup\mathcal{M}_{q}$. Note that
$\mathcal{M}_{r}$ is closed under meets. We define $d_{r}$ by letting
$d_{r}(M)=d_{q}(M)$ if $M\in{\rm dom}(d_{q})$, and $d_{r}(M)=d_{p}(M)$ if
$M\in{\rm dom}(d_{p})$ with $\eta(M)>\alpha$. It is straightforward that $r$
is as required. ∎

######  Remark 5.4.

The condition $r$ from the previous lemma is the greatest lower bound of $p$
and $q$, so we will let $r\coloneqq p\land q$.

######  Definition 5.5.

Assume $p\in\mathbb{P}^{\kappa}_{\lambda}$, and that
$M\in\mathscr{C}\cup\mathscr{U}$ be such that $p\in M$. Let
$\mathcal{M}_{p^{M}}$ be the closure of $\mathcal{M}_{p}\cup\\{M\\}$ under
meets, and let $d_{p^{M}}$ be defined on

${\rm dom}(d_{p^{M}})=\\{N\\!\upharpoonright\\!\delta:N\in{\rm
dom}(d_{p})\text{ and }\delta={\rm sup}(M\cap\eta(N))\\},$

by letting $d_{p^{M}}(N\\!\upharpoonright\\!\delta)=d_{p}(N)$.

######  Lemma 5.6.

$p^{M}$ is the weakest condition extending $p$ with $M\in\mathcal{M}_{p^{M}}$.

###### Proof.

[14, Lemma 4.11]. ∎

######  Notation 5.7.

For virtual models $N,M$, we set $\alpha(N,M)=\max(a(N)\cap a(M))$.

We are now about to give the restriction of a condition to a given model. We
start with Magidor models.

######  Definition 5.8.

Let $p\in\mathbb{P}^{\kappa}_{\lambda}$. Assume that
$M\in\mathcal{L}(\mathcal{M}_{p})$ is a Magidor model. For
$N\in\mathcal{M}_{p}$, we let $N\restriction M=N\restriction{\alpha(N,M)}$ if
$\kappa_{N}<\kappa_{M}$, otherwise $N\restriction M$ is undefined. Let
$p\restriction M$ be defined by

$\mathcal{M}_{p\restriction M}=\\{N\restriction M:N\in\mathcal{M}_{p}\\}\mbox{
and }d_{p\restriction M}=d_{p}\restriction{({\rm dom}(d_{p})\cap M)}.$

######  Proposition 5.9.

Suppose $p\in\mathbb{P}^{\kappa}_{\lambda}$ and
$M\in\mathcal{L}(\mathcal{M}_{p})$ is a Magidor model. Then $p\restriction
M\in\mathbb{P}^{\kappa}_{\lambda}\cap M$ and $p\leq p\restriction M$.
Moreover, if $q\in M\cap\mathbb{P}^{\kappa}_{\lambda}$ extends $p\restriction
M$. Then $q$ is compatible with $p$ and the meet $p\land q$ exists.

###### Proof.

See [14, Lemmata 4.13 and 4.14]. ∎

As a corollary we have that the forcing $\mathbb{P}^{\kappa}_{\lambda}$ is
strongly proper for $\mathscr{U}$. We now define an analogue of
$p\\!\upharpoonright\\!M$ for countable models
$M\in\mathcal{L}(\mathcal{M}_{p})$. Now, suppose
$p\in\mathbb{P}^{\kappa}_{\alpha}$ and $M\in\mathcal{M}_{p}$ is either a
countable standard model or a countable model with $\alpha\in M$. Suppose
$N\in\mathcal{M}_{p}$ and $N\in_{\gamma}M$, where $\gamma=a(M)\cap a(N)$. If
$\gamma\in M$, then $N\\!\upharpoonright\\!\gamma\in M$, but it may be that
$\gamma\notin M$, but then, if we let
$\gamma^{*}=\min(M\cap\lambda\setminus\gamma)$, we have that $\gamma^{*}\in
E\cap M$ by Lemma 4.1. Thus there is a $\gamma^{*}$-model $N^{*}\in M$ which
is $\gamma$-isomorphic to $N\\!\upharpoonright\\!\gamma$. Moreover, such
$N^{*}$ is unique111 See the paragraph above Definition 3.21 of [14].

######  Definition 5.10.

Let $p\in\mathbb{P}^{\kappa}_{\alpha}$. Assume that
$M\in\mathcal{L}(\mathcal{M}_{p})$ is a countable model which is either
standard or contains $\alpha$. Suppose that $N\in\mathcal{M}_{p}$, and let
$\gamma=\alpha(M,N)$. If $N\in_{\gamma}M$, we let
$\gamma^{*}=\min(M\cap\lambda\setminus\gamma)$. We define $N\restriction M$ to
be the unique $\gamma^{*}$-model $N^{*}\in M$ such that
$N^{*}\cong_{\alpha}N$. Otherwise, $N\restriction M$ is undefined. Let

$\mathcal{M}_{p\restriction M}=\\{N\restriction M:N\in\mathcal{M}_{p}\\}\text{
and}$ ${\rm dom}(d_{p\restriction M})=\\{N\restriction M:N\in{\rm
dom}(d_{p})\text{ and }N\in_{\eta(N)}M\\}.$

If $N\in{\rm dom}(d_{p})$ and $N\in_{\eta(N)}M$, we then let $d_{p\restriction
M}(N\restriction M)=d_{p}(N)$. Let also

$p\restriction M\coloneqq(\mathcal{M}_{p\restriction M},d_{p\restriction M}).$

######  Remark 5.11.

Suppose $N\in{\rm dom}(d_{p})$ and let $\eta=\eta(N)$. If $N\in_{\eta}M$ then
$M$ is strongly active at $\eta$ since $N$ is $\mathcal{M}_{p}$-free. If
$\eta\in M$ then we put $N$ in ${\rm dom}(d_{p\restriction M})$ and keep the
same decoration at $N$. If $\eta\notin M$ we lift $N$ to the least level
$\eta^{*}$ of $M$ above $\eta$, we put the resulting model $N^{*}$ in ${\rm
dom}(d_{p\restriction M})$ and copy the decoration of $N$ to $N^{*}$. If
$P\in\mathcal{M}_{p}$ is such that $P\restriction\eta=N$ then $(P\restriction
M)\restriction{\eta^{*}}=N^{*}$. Moreover, from $N^{*}$ we can recover $N$ as
$N^{*}\restriction{\sup(\eta^{*}\cap M)}$. Thus, the function
$d_{p\restriction M}$ is well defined. Note also that $p\restriction M\in M$.

######  Proposition 5.12.

Let $p\in\mathbb{P}^{\kappa}_{\alpha}$ and let
$M\in\mathscr{C}\cup\mathscr{U}$ such that $\alpha\in M$ and
$M\restriction\alpha\in\mathcal{M}_{p}$. Then $p\restriction
M\in\mathbb{P}^{\kappa}_{\alpha}$, and for any
$q\in\mathbb{P}^{\kappa}_{\alpha}\cap M$ with $q\leq p\restriction M$, $p$ and
$q$ are compatible, and the meet $p\land q$ exists.

###### Proof.

See [14, Proposition 4.14 and Lemma 4.26]. ∎

Note that if $p,q\in\mathbb{P}^{\kappa}_{\lambda}$ are such that $q\leq p$ and
$M\in\mathcal{M}_{p}$ then $q\restriction M\leq p\restriction M$. It follows
from Lemmas 5.6 and 5.12 that $\mathbb{P}^{\kappa}_{\lambda}$ is strongly
proper for $\mathscr{C}_{\rm st}$, and that $\mathbb{P}^{\kappa}_{\alpha}$ is
strongly proper for the collection of all models $M\in\mathscr{C}$ with
$\alpha\in M$.

For a filter $F$ in $\mathbb{P}^{\kappa}_{\lambda}$, we set
$\mathcal{M}_{F}\coloneqq\bigcup\\{\mathcal{M}_{p}:\leavevmode\nobreak\ p\in
F\\}$, and for a $V$-generic filter $G$ in $\mathbb{P}^{\kappa}_{\lambda}$, we
let $G_{\alpha}=G\cap\mathbb{P}^{\kappa}_{\alpha}$, for all $\alpha\in E$. The
following is straightforward.

######  Lemma 5.13.

For every $\delta\in E$ with ${\rm cof}(\delta)<\kappa$,
$\mathcal{M}_{G}^{\delta}$ is a $\delta$-chain.

###### Proof.

See [14, Proposition 4.31]. ∎

######  Proposition 5.14.

Let $\delta\in E$ with ${\rm cof}(\delta)<\kappa$. Suppose
$M\in\mathcal{M}_{G}^{\delta}$ is a Magidor model that is not the least model
in $\mathcal{M}_{G}^{\delta}$. Then

$M\cap V_{\delta}=\bigcup\\{Q\cap V_{\delta}:Q\in_{\delta}M\mbox{ and
}Q\in\mathcal{M}_{G}^{\delta}\\}.$

###### Proof.

See [14, Proposition 4.32]. ∎

######  Proposition 5.15.

$\mathbb{P}^{\kappa}_{\lambda}$ collapses all uncountable cardinals below
$\kappa$ to $\omega_{1}$, and those between $\kappa$ and $\lambda$ to
$\kappa$.

###### Proof.

See [14, Theorems 4.33 and 4.34]. ∎

Notice $\mathbb{P}^{\kappa}_{\lambda}$ is $\lambda$-c.c., see Proposition
5.31.

######  Definition 5.16.

Suppose $G\subseteq\mathbb{P}^{\kappa}_{\lambda}$ is $V$-generic, and that
$\alpha\in E$ is of cofinality less than $\kappa$. We let
$C_{\alpha}(G)=\\{\kappa_{M}:M\in\mathcal{M}_{G}^{\alpha}\\}$.

######  Proposition 5.17.

Let $G$ be a $V$-generic filter over $\mathbb{P}^{\kappa}_{\lambda}$. Then
$C_{\alpha}(G)$ is a club in $\kappa$, for all $\alpha\in E$ of cofinality
${<}\kappa$. Moreover, if $\alpha<\beta$ then $C_{\beta}(G)\setminus
C_{\alpha}(G)$ is bounded in $\kappa$.

###### Proof.

See [14, Lemma 4.37]. ∎

### The iteration

For a condition $\mathcal{M}_{p}\in\mathbb{M}^{\kappa}_{\lambda}$ and an
ordinal $\gamma\in E$, we let

$\mathcal{M}_{p}(\gamma)\coloneqq\\{M\in\mathcal{M}^{{\rm
next}(\gamma)}_{p}:\gamma\in M\\}.$

In other words, $\mathcal{M}_{p}(\gamma)$ consists of those models in
$\mathcal{M}_{p}{\restriction{{\rm next}(\gamma)}}$ that are strongly active
at ${\rm next}(\gamma)$.

Let $E^{+}=E\cup\\{\alpha+1:\alpha\in E\\}$. We will define by induction the
poset $\mathbb{Q}^{\kappa}_{\alpha}$, for $\alpha\in E^{+}\cup\\{\lambda\\}$.
Let ${\rm Fn}(\omega_{1},\omega)$ denote the poset of finite partial functions
from $\omega_{1}$ to $\omega$, ordered under reverse inclusion. Conditions in
$\mathbb{Q}^{\kappa}_{\alpha}$ will be tripes of the form
$p=(\mathcal{M}_{p},d_{p},w_{p})$, where
$(\mathcal{M}_{p},d_{p})\in\mathbb{P}^{\kappa}_{\alpha}$, and $w_{p}$ is a
finite function with ${\rm dom}(w_{p})\subseteq E\cap\alpha$, such that
$w_{p}(\gamma)\in{\rm Fn}(\omega_{1},\omega)$, for all $\gamma\in{\rm
dom}(w_{p})$. If $p$ is such a triple and $\gamma<\alpha$ is in $E$ we let
$p\\!\upharpoonright\\!\gamma$ denote the triple
$(\mathcal{M}_{p}\\!\upharpoonright\\!\gamma,d_{p}\\!\upharpoonright\\!\gamma,w_{p}\\!\upharpoonright\\!\gamma)$,
where
$(\mathcal{M}_{p}\\!\upharpoonright\\!\gamma,d_{p}\\!\upharpoonright\\!\gamma)$
is defined as in $\mathbb{P}^{\kappa}_{\alpha}$ and
$w_{p}\\!\upharpoonright\\!\gamma$ is the restriction of $w_{p}$ to ${\rm
dom}(w_{p})\cap\gamma$.

######  Definition 5.18.

For $\alpha\in E$, we let $\mathbb{Q}^{\kappa}_{\alpha}$ consist of triples
$p=(\mathcal{M}_{p},d_{p},w_{p})$, where

  1. (1)

$(\mathcal{M}_{p},d_{p})\in\mathbb{P}^{\kappa}_{\alpha}$.

  2. (2)

$w_{p}$ is a finite function with ${\rm dom}(w_{p})\subseteq E\cap\alpha$ such
that, if $\gamma\in{\rm dom}(w_{p})$, then $U(\gamma)$ is a
$\mathbb{Q}^{\kappa}_{\gamma}$-term for a relation on $\omega_{1}$ such that
$\dot{T}_{\gamma}=(\check{\omega}_{1},U(\gamma))$ is forced to be a tree of
height $\omega_{1}$ without uncountable branches, $w_{p}(\gamma)\in{\rm
Fn}(\omega_{1},\omega)$, and

$p\\!\upharpoonright\\!\gamma\Vdash_{\mathbb{Q}^{\kappa}_{\gamma}}\check{w}_{p}(\gamma)\in\dot{\mathbb{S}}(\dot{T}_{\gamma}).$

For conditions $p,q\in\mathbb{Q}^{\kappa}_{\alpha}$, we say $p$ is stronger
than $q$ and write $p\leq q$, if and only if,

  1. (1)

$(\mathcal{M}_{p},d_{p})\leq(\mathcal{M}_{q},d_{q})$ in
$\mathbb{P}^{\kappa}_{\alpha}$,

  2. (2)

${\rm dom}(w_{p})\supseteq{\rm dom}(w_{q})$, and $w_{q}(\gamma)\subseteq
w_{p}(\gamma)$, for every $\gamma\in{\rm dom}(w_{q})$.

We let $\mathbb{Q}^{\kappa}_{\alpha+1}$ denote the set of triples $p$ as
above, but with with ${\rm dom}(w_{p})\subseteq\alpha+1$. Let
$\mathbb{Q}^{\kappa}_{\lambda}=\bigcup_{\alpha\in
E}\mathbb{Q}^{\kappa}_{\alpha}$ with the same order.

Note that if $\alpha={\rm min}(E)$ then
$\mathbb{Q}^{\kappa}_{\alpha}=\mathbb{P}^{\kappa}_{\alpha}$. Also, for
$\alpha\in E$, $\mathbb{Q}^{\kappa}_{\alpha+1}$ is forcing equivalent to
$\mathbb{Q}^{\kappa}_{\alpha}\ast\dot{\mathbb{S}}(\dot{T}_{\alpha})$, if
$U(\alpha)$ has the right form, otherwise
$\mathbb{Q}^{\kappa}_{\alpha+1}=\mathbb{Q}^{\kappa}_{\alpha}$. For
$p\in\mathbb{Q}^{\kappa}_{\lambda}$ and $\alpha\in E$, we let

$p\\!\upharpoonright\\!(\alpha+1)=(\mathcal{M}_{p}\\!\upharpoonright\\!\alpha,d_{p}\\!\upharpoonright\\!\alpha,w_{p}\\!\upharpoonright\\!(\alpha+1)).$

The order is transitive and whenever $p\in\mathbb{Q}^{\kappa}_{\lambda}$ and
$\alpha\in E^{+}$, then
$p\\!\upharpoonright\\!\alpha\in\mathbb{Q}^{\kappa}_{\alpha}$. Moreover, if
$p\leq q$ then $p\\!\upharpoonright\\!\alpha\leq
q\\!\upharpoonright\\!\alpha$.

######  Proposition 5.19.

Suppose $p\in\mathbb{Q}^{\kappa}_{\beta}$, and that $\alpha\leq\beta$ are in
$E^{+}$. If $q$ is a condition in $\mathbb{Q}^{\kappa}_{\alpha}$ extending
$p\restriction\alpha$, then $p$ is compatible with $q$ in
$\mathbb{Q}^{\kappa}_{\beta}$.

###### Proof.

Let $(\mathcal{M}_{r},d_{r})$ be
$(\mathcal{M}_{p},d_{p})\land(\mathcal{M}_{q},d_{q})$ as defined in Lemma 5.3.
Let also $w_{r}$ be defined on ${\rm dom}(w_{p})\cup{\rm dom}(w_{q})$ by

$w_{r}(\gamma)=\left\\{\begin{array}[]{cl}w_{q}(\gamma)&\textrm{if
}\gamma<\alpha,\\\ w_{p}(\gamma)&\textrm{if
}\gamma\geq\alpha.\end{array}\right.$

It is easily seen that

$\mathcal{M}_{r}(\gamma)=\left\\{\begin{array}[]{cl}\mathcal{M}_{q}(\gamma)&\textrm{if
}\gamma\leq\alpha,\\\ \mathcal{M}_{p}(\gamma)&\textrm{if
}\gamma>\alpha.\end{array}\right.$

It is evident that $r$ is a condition which extends $p$ and $q$. ∎

######  Remark 5.20.

The condition $r$ from the previous lemma is the greatest lower bound of $p$
and $q$, so we will denote it by $r\coloneqq p\land q$.

The following corollary is immediate.

######  Corollary 5.21.

For every $\alpha\leq\beta$ in $E^{+}\cup\\{\lambda\\}$,
$\mathbb{Q}^{\kappa}_{\alpha}$ is a complete suborder of
$\mathbb{Q}^{\kappa}_{\beta}$.

###### .

∎

######  Proposition 5.22.

Let $p\in\mathbb{Q}^{\kappa}_{\lambda}$ and let
$M\in\mathscr{C}\cup\mathscr{U}$ be such that $p\in M$. Then there is a
condition $p^{M}\leq p$ with $M\in\mathcal{M}_{p^{M}}$.

###### Proof.

Let $\mathcal{M}_{p^{M}}$ and $d_{p^{M}}$ be defined as in Definition 5.5. We
let $p^{M}=(\mathcal{M}_{p^{M}},d_{p^{M}},w_{p})$. It is clear that $p^{M}$ is
the required condition. ∎

######  Proposition 5.23.

Let $\alpha\in E$. Suppose $M\in\mathscr{C}\cup\mathscr{U}$ with
$\eta(M)>\alpha$ and $\alpha\in M$. Let $p$ be a condition such that
$M\\!\upharpoonright\\!\alpha\in\mathcal{M}_{p}$. Then

  1. (1)

if $p\in\mathbb{Q}^{\kappa}_{\alpha}$ then $p$ is
$(M,\mathbb{Q}^{\kappa}_{\alpha})$-generic,

  2. (2)

if $p\in\mathbb{Q}^{\kappa}_{\alpha+1}$ then $p$ is
$(M,\mathbb{Q}^{\kappa}_{\alpha+1})$-generic.

###### Proof.

First note that under our assumptions both $\mathbb{Q}^{\kappa}_{\alpha}$ and
$\mathbb{Q}^{\kappa}_{\alpha+1}$ belong to $M$.
$\mathbb{Q}^{\kappa}_{\alpha+1}$ is either equal to
$\mathbb{Q}^{\kappa}_{\alpha}$ or is isomorphic to
$\mathbb{Q}^{\kappa}_{\alpha}\ast\dot{\mathbb{S}}(\dot{T}_{\alpha})$. Since
$\dot{\mathbb{S}}(\dot{T}_{\alpha})$ is forced to be ccc, (2) follows from
(1). If $\alpha={\rm min}(E)$ then $\mathbb{Q}^{\kappa}_{\alpha}$ is
isomorphic to $\mathbb{P}^{\kappa}_{\alpha}$, and by Proposition 5.12, it is
then strongly proper for all models $M\in\mathscr{C}\cup\mathscr{U}$ with
$\eta(M)>\alpha$.

Suppose now that $\alpha$ is not the least element of $E$ and (2) holds for
all $\beta<\alpha$. Let $D\in M$ be a dense subset of
$\mathbb{Q}^{\kappa}_{\alpha}$. We may assume without loss of generality that
$p\in D$. Let $\beta={\rm max}({\rm dom}(w_{p})\cap M)$. Pick a $V$-generic
filter $G_{\beta+1}$ on $\mathbb{Q}^{\kappa}_{\beta+1}$ such that
$p\restriction{\beta+1}\in G_{\beta+1}$. By elementarity of $M[G_{\beta+1}]$
in ${\rm Hull}(M,V_{\eta(M)})[G_{\beta+1}]$, there is a condition $q\in D\cap
M[G_{\beta+1}]$ satisfying the following.

  1. (1)

$(\mathcal{M}_{q},d_{q})\leq(\mathcal{M}_{p},d_{p})\restriction M$,

  2. (2)

$M\cap{\rm dom}(w_{p})\subseteq{\rm dom}(w_{q})$, and

  3. (3)

$q\restriction{\beta+1}\in G_{\beta+1}$.

By the inductive assumption $p\restriction\beta+1$ is
$(M,\mathbb{Q}^{\kappa}_{\beta+1})$-generic, and hence $M[G_{\beta+1}]\cap
V=M$. Therefore, we can find such $q$ in $M$. We show that $p$ and $q$ are
compatible. First, by Proposition 5.12, we have that $(\mathcal{M}_{q},d_{q})$
and $(\mathcal{M}_{p},d_{p})$ are compatible in $\mathbb{P}^{\kappa}_{\alpha}$
and the meet $(\mathcal{M}_{q},d_{q})\wedge(\mathcal{M}_{p},d_{r})$ exists.
Let us denote this meet by $(\mathcal{M},d)$. Let us also fix $r\in
G_{\beta+1}$ extending $p\restriction{\beta+1}$ and $q\restriction{\beta+1}$.
Let $(\mathcal{M}_{s},d_{s})$ be the meet
$(\mathcal{M},d)\wedge(\mathcal{M}_{r},d_{r})$ as defined in Lemma 5.3. We now
define $w_{s}$ on ${\rm dom}(w_{r})\cup{\rm dom}(w_{p})\cup{\rm dom}(w_{q})$
by letting:

$w_{s}(\gamma)=\begin{cases}w_{r}(\gamma)&\mbox{if }\gamma\in{\rm
dom}(w_{r})\\\ w_{q}(\gamma)&\mbox{if }\gamma\in{\rm dom}(w_{q})\setminus{\rm
dom}(w_{r}),\\\ w_{p}(\gamma)&\mbox{if }\gamma\in{\rm
dom}(w_{p})\setminus({\rm dom}(w_{r})\cup{\rm dom}(w_{q})).\end{cases}$

It is easy to see that $s=(\mathcal{M}_{s},d_{s},w_{s})$ is a condition in
$\mathbb{Q}^{\kappa}_{\alpha}$ extending $p$ and $q$. ∎

######  Proposition 5.24.

Suppose that $\lambda^{*}>\lambda$ is a sufficiently large regular cardinal.
Let $p\in\mathbb{Q}^{\kappa}_{\lambda}$. Suppose that $M^{*}\prec
H_{\lambda^{*}}$ with $\kappa,\lambda\in M^{*}$ is such that $M\coloneqq
M^{*}\cap V_{\lambda}\in\mathcal{M}_{p}$. Then $p$ is
$(M^{*},\mathbb{Q}^{\kappa}_{\lambda})$-generic.

###### Proof.

Observe that $\eta={\rm sup}(M^{*}\cap\lambda)$ is in $E$, and that $M$ is an
$\eta$-model. The rest is as in the proof of Proposition 5.23. ∎

######  Corollary 5.25.

For every $\alpha\in E^{+}\cup\\{\lambda\\}$, $\mathbb{Q}^{\kappa}_{\alpha}$
is proper.

###### .

∎

######  Corollary 5.26.

For every $\alpha\in E^{+}\cup\\{\lambda\\}$, $\mathbb{Q}^{\kappa}_{\alpha}$
is strongly proper for $\mathscr{U}$.

###### Proof.

This follows from Propositions 5.23 and 2.14. ∎

######  Definition 5.27.

Let $\alpha\in E$ and let $G_{\alpha}$ be $V$-generic over
$\mathbb{Q}^{\kappa}_{\alpha}$. In the model $V[G_{\alpha}]$ let
$\mathscr{C}_{\rm st}[G_{\alpha}]$ denote the set of all $M\in\mathscr{C}_{\rm
st}$ such that $\eta(M)>\alpha$, $\alpha\in M$ and
$M\\!\upharpoonright\\!\alpha\in\mathcal{M}^{\alpha}_{G_{\alpha}}$.

As in [14, Lemma 5.2] we have the following.

######  Proposition 5.28.

${\mathscr{C}}_{\rm st}[G_{\alpha}]$ is a stationary subset of
${\mathcal{P}}_{\omega_{1}}(V_{\lambda})$ in the model $V[G_{\alpha}]$.

###### .

∎

Let $\alpha\in E^{+}$. Assume that $G_{\alpha}$ is a $V$-generic filter on
$\mathbb{Q}^{\kappa}_{\alpha}$. One can form the quotient forcing
$\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha}$. Suppose that also
$M\in\mathscr{C}_{>\alpha}$,
$M\\!\upharpoonright\\!\alpha\in\mathcal{M}^{\alpha}_{G_{\alpha}}$, and $p\in
M\cap\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha}$. It is then easily seen that
$p^{M}$ belongs to $\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha}$. By an argument
as in Proposition 5.23 we have the following.

######  Proposition 5.29.

Let $\lambda^{*}>\lambda$ be a sufficiently large regular cardinal. Suppose
that $\alpha\in E$, and $G_{\alpha}\subseteq\mathbb{Q}^{\kappa}_{\alpha}$ is a
$V$-generic filter. Let $p\in\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha}$.
Suppose $M^{*}\prec H_{\lambda^{*}}$ contains all the relevant objects, and
$M=M^{*}\cap V_{\lambda}$ belongs to $\mathcal{M}_{p}$. Then $p$ is
$(M^{*}[G_{\alpha}],\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha})$-generic.

###### .

∎

Suppose $\alpha\in E^{+}$ and let $G_{\alpha}$ be $V$-generic over
$\mathbb{Q}^{\kappa}_{\alpha}$. In $V[G_{\alpha}]$ fix a large regular
cardinal $\lambda^{*}$ and let $\mathscr{S}$ be the collection of all
countable models of the form $M[G_{\alpha}]$ where $M\prec
H_{\lambda^{*}}^{V}$, contains the relevant objects, and $M\cap
V_{\lambda}\in\mathscr{C}_{\rm st}[G_{\alpha}]$. Now, by Propositions 5.28 and
5.29 we have the following.

######  Corollary 5.30.

$\mathscr{S}$ is stationary in
${\mathcal{P}}_{\omega_{1}}([H_{\lambda^{*}}[G_{\alpha}])$ and
$\mathbb{Q}^{\kappa}_{\lambda}/G_{\alpha}$ is $\mathscr{S}$-proper in
$V[G_{\lambda}]$.

###### .

∎

######  Proposition 5.31.

$\mathbb{Q}^{\kappa}_{\lambda}$ satisfies the $\lambda$-c.c.

###### Proof.

Assume that $A\subseteq\mathbb{Q}^{\kappa}_{\lambda}$ is of size $\lambda$.
For each $p\in\mathbb{P}^{\kappa}_{\lambda}$, let
$a(p)=\bigcup\\{a(M):M\in\mathcal{M}_{p}\\}$ which is a closed subset of $E$
of size ${<}\kappa$. By a standard $\Delta$-system argument, we can find a
subset $B$ of $A$ of size $\lambda$ so that there are $a$ and $d$ subsets of
$E$ such that $a(p)\cap a(q)=a$ and ${\rm dom}(w_{p})\cap{\rm dom}(w_{q})=d$,
for all distinct $p,q\in B$. Note that $a$ is closed, and if we let
$\gamma=\max(a)$ then $\gamma\in E$. Since $B$ has size $\lambda$, by a simple
counting argument, we may assume there is
$\mathcal{M}\in\mathbb{M}^{\kappa}_{\gamma}$ such that
$\mathcal{M}_{p}\restriction\gamma=\mathcal{M}$ and that $w_{p}\restriction
d=w_{q}\restriction d$, for all $p\in B$. Now, pick distinct $p,q\in B$, and
define $\mathcal{M}_{r}=\mathcal{M}_{p}\cup\mathcal{M}_{q}$, $d_{r}=d_{p}\cup
d_{q}$, and also $w_{r}=w_{p}\cup w_{q}$. Let
$r=(\mathcal{M}_{r},d_{r},w_{r})$. It is straightforward to check that
$r\in\mathbb{Q}^{\kappa}_{\lambda}$ and $r\leq p,q$. ∎

Putting everything together, we have the following.

######  Corollary 5.32.

$\mathbb{Q}^{\kappa}_{\lambda}$ preserves $\omega_{1}$, $\kappa$ and
$\lambda$, and forces that $\kappa=\omega^{V[G_{\lambda}]}_{2}$ and
$\lambda=\omega^{V[G_{\lambda}]}_{3}$.

###### Proof.

The preservation of $\omega_{1}$ and $\kappa$ is guaranteed by Corollary 5.25
and Corollary 5.26, respectively. It is easily seen that
$\mathbb{P}^{\kappa}_{\lambda}$ is a complete suborder of
$\mathbb{Q}^{\kappa}_{\lambda}$, and hence Propositions 5.15 and 5.31 imply
that in generic extensions by $\mathbb{Q}^{\kappa}_{\lambda}$,
$\kappa=\omega_{2}$ and $\lambda=\omega_{3}$. ∎

######  Proposition 5.33.

For every $\alpha\in E^{+}\cup\\{\lambda\\}$, $\mathbb{Q}^{\kappa}_{\alpha}$
has the $\omega_{1}$-approximation property.

###### Proof.

We proceed by induction. Suppose $\alpha\in E$ and we have established that
$\mathbb{Q}^{\kappa}_{\alpha}$ has the $\omega_{1}$-approximation property.
Recall that $\mathbb{Q}^{\kappa}_{\alpha+1}$ is either
$\mathbb{Q}^{\kappa}_{\alpha}$ or is isomorphic to
$\mathbb{Q}^{\kappa}_{\alpha}\ast\dot{\mathbb{S}}(\dot{T}_{\alpha})$, for some
$\dot{T}_{\alpha}$ which is a name for a tree of size and height $\omega_{1}$
without uncountable branches. Then by Proposition 2.19
$\dot{\mathbb{S}}(\dot{T}_{\alpha})$ is forced to have the
$\omega_{1}$-approximation property over $V[G_{\alpha}]$, where $G_{\alpha}$
is a generic over ${\mathbb{Q}^{\kappa}_{\alpha}}$. By Theorem 2.3 we conclude
that $\mathbb{Q}^{\kappa}_{\alpha+1}$ has the $\omega_{1}$-approximation
property. Recall also that $\mathbb{Q}^{\kappa}_{{\rm min}(E)}$ is strongly
proper for $\mathscr{C}$, by Proposition 5.12, and hence has the
$\omega_{1}$-approximation property by Lemma 2.10.

Suppose now that $\alpha$ is not the least element of $E$ and that the
conclusion holds for every ordinal in $E^{+}\cap\alpha$. Let $\dot{f}$ be a
$\mathbb{Q}^{\kappa}_{\alpha}$-name for a function $\mu\to 2$ which is forced
by a condition $p$ to be countably approximated in $V$. We may assume that
$\mu$ is a cardinal in $V$. Suppose $\lambda^{*}>\mu,\lambda$ is a
sufficiently large regular cardinal. Pick a countable $M^{*}\prec
H_{\lambda^{*}}$ containing the relevant objects. Thus $M=M^{*}\cap
V_{\lambda}$ is a standard virtual model. Let $q\leq p^{M}$ be a condition
which decides $\dot{f}\restriction{M^{*}\cap\mu}$, and forces it to be equal
to some function $g:M^{*}\cap\mu\rightarrow 2$ which is in $V$. By Proposition
5.24 $q$ is $(M^{*},\mathbb{Q}^{\kappa}_{\alpha})$-generic. By Lemma 2.16, it
is enough to show that $g$ is guessed in $M^{*}$. Assume towards a
contradiction that $g$ is not guessed in $M^{*}$. Let $\gamma={\rm max}({\rm
dom}(w_{q})\cap M)$, and pick some $V$-generic filter $G_{\gamma+1}$ on
$\mathbb{Q}^{\kappa}_{\gamma+1}$ such that
$(q\\!\upharpoonright\\!\gamma,w_{q}(\gamma))\in G_{\gamma+1}$. Working in
$V[G_{\gamma+1}]$, we show that in $M^{*}[G_{\gamma+1}]$ there is an
assignment $x\mapsto(q_{x},g_{x})$ on $[\mu]^{\omega}\cap V$ with the
following properties.

  1. (1)

$q_{x}\in\mathbb{Q}^{\kappa}_{\alpha}$,

  2. (2)

$(\mathcal{M}_{q_{x}},d_{q_{x}})\leq(\mathcal{M}_{q},d_{q})\restriction M$,

  3. (3)

$q_{x}\restriction{\gamma+1}\in G_{\gamma+1}$,

  4. (4)

$g_{x}:x\to 2$, $g_{x}\in V$,

  5. (5)

$q_{x}\Vdash\dot{f}\restriction x=\check{g}_{x}$.

If $x\in[\mu]^{\omega}\cap M^{*}$ then $x\mapsto(q,g\\!\upharpoonright\\!x)$
satisfies all the above properties. By elementarity of $M^{*}[G_{\gamma+1}]$
in $H_{\lambda^{*}}[G_{\gamma+1}]$, such an assignment exists for all
$x\in[\mu]^{\omega}\in V$. Moreover, by elementarity again there is such an
assignment in $M^{*}[G_{\gamma+1}]$. Now, by the inductive assumption and
Lemma 2.15, $g$ is not guessed in $M^{*}[G_{\gamma+1}]$. Since
$\mathbb{Q}^{\kappa}_{\alpha}$ is proper, $[\mu]^{\omega}\cap V$ is a cofinal
in $[\mu]^{\omega}$. By Lemma 2.17 there is $B\in M^{*}[G_{\gamma+1}]$, a
cofinal subset of $[\mu]^{\omega}\cap V$, such that for every $x\in B\cap
M^{*}[G_{\gamma+1}]$, $g_{x}\nsubseteq g$, and hence $q_{x}$ is incompatible
with $q$. On the other hand, conditions (1)-(3) above enable us to amalgamate
$q_{x}$ and $q$, as in the proof of Proposition 5.23. Thus, for every such
$x\in B\cap M^{*}[G_{\gamma+1}]$, $q_{x}$ and $q$ are in fact compatible, and
hence we get a contradiction. ∎

One can use Proposition 5.29 and the proof of Proposition 5.33 to prove the
following.

######  Proposition 5.34.

Suppose $\alpha,\beta\in E^{+}\cup\\{\lambda\\}$ and $\alpha<\beta$. Suppose
that $G_{\alpha}$ is a $V$-generic filter over $\mathbb{Q}^{\kappa}_{\alpha}$.
Then $\mathbb{Q}^{\kappa}_{\beta}/G_{\alpha}$ has the
$\omega_{1}$-approximation property over $V[G_{\alpha}]$.

###### .

∎

The following corollary is immediate.

######  Corollary 5.35.

Suppose $\alpha,\beta\in E^{+}\cup\\{\lambda\\}$ and $\alpha<\beta$. Let
$G_{\beta}$ be a $V$-generic filter over $\mathbb{Q}^{\kappa}_{\beta}$, and
let $G_{\alpha}=G_{\beta}\cap\mathbb{Q}^{\kappa}_{\alpha}$.Then the pairs
$(V,V[G_{\alpha}])$ and $(V[G_{\alpha}],V[G_{\beta}])$ have the
$\omega_{1}$-approximation property.

###### .

∎

### Quotients by Magidor models

The first lemma of this subsection states that in this case
$\mathbb{Q}^{\kappa}_{\lambda}$ is strongly proper for $\mathscr{U}$ in a
canonical fashion.

######  Definition 5.36.

Let $p\in\mathbb{Q}^{\kappa}_{\alpha}$. Assume
$N\in\mathcal{L}(\mathcal{M}_{p})$ is a Magidor model. We let

$p\restriction N=(\mathcal{M}_{p\restriction N},d_{p\restriction
N},w_{p}\restriction N).$

It is clear that $p\restriction N$ belongs to
$\mathbb{Q}^{\kappa}_{\alpha}\cap N$.

######  Lemma 5.37.

Let $p,q\in\mathbb{Q}^{\kappa}_{\alpha}$. Assume $N\in\mathcal{M}_{p}$ is a
Magidor model. Suppose that $q\leq p\restriction N$ and $q\in N$. Then $q$ is
compatible with $p$.

###### Proof.

We define a common extension of $p,q$, say $r$. Let
$(\mathcal{M}_{r},d_{r})=(\mathcal{M}_{p},d_{p})\land(\mathcal{M}_{q},d_{q})$,
${\rm dom}(w_{r})={\rm dom}(w_{p})\cup{\rm dom}(w_{q})$, and that

$w_{r}(\gamma)=\begin{cases}w_{q}(\gamma)&\mbox{if }\gamma\in N,\\\
w_{p}(\gamma)&\mbox{if }\gamma\notin N.\end{cases}$

Observe that for $\gamma\in{\rm dom}(w_{p})$ with $\gamma\notin N$, we have
$\mathcal{M}_{r}(\gamma)=\mathcal{M}_{p}(\gamma)$. On the other hand, if
$\gamma\in N$, then $\gamma$ belongs to ${\rm dom}(w_{q})$, and that
$q\\!\upharpoonright\\!\gamma$ forces that
$w_{q}(\gamma)\leq_{\dot{\mathbb{S}}_{\gamma}}w_{p}(\gamma)$. It follows from
Proposition 5.9 that $r$ is a common extension of $p$ and $q$. ∎

######  Remark 5.38.

As before, we let the above $r$ be the meet of $p$ and $q$, and we denote it
by $p\land q$.

Suppose $N\in\mathscr{U}$ and $\alpha\in E$. We set

$\mathbb{Q}^{\kappa}_{\alpha}\restriction
N\coloneqq\\{p\in\mathbb{Q}^{\kappa}_{\alpha}:N\\!\upharpoonright\\!\alpha\in\mathcal{M}_{p}\\}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{Q}^{\kappa}_{\alpha,N}\coloneqq\mathbb{Q}^{\kappa}_{\alpha}\cap N.$

By Lemma 5.37,
$\mathbf{1}_{N}\coloneqq(\\{N\\!\upharpoonright\\!\alpha\\},\varnothing,\varnothing)$
is $(N,\mathbb{Q}^{\kappa}_{\alpha})$-strongly generic, and hence the mapping
$p\mapsto p^{N\restriction\alpha}$ completely embeds
$\mathbb{Q}^{\kappa}_{\alpha,N}$ into
$\mathbb{Q}^{\kappa}_{\alpha}\restriction N$. Moreover, if $p$ is a condition
in $\mathbb{Q}^{\kappa}_{\alpha}\restriction N$, then $p\restriction
N\in\mathbb{Q}^{\kappa}_{\alpha,N}$ is so that if
$q\in\mathbb{Q}^{\kappa}_{\alpha,N}$ extends $p\restriction N$, then $p$ and
$q$ are compatible, and in fact the meet $p\land q$ exists.

######  Lemma 5.39.

Assume that $\alpha\leq\eta(N)$ is in $E$. Let
$p\in\mathbb{Q}^{\kappa}_{\lambda}$, and let
$N\in\mathcal{L}(\mathcal{M}_{p})$ be a Magidor model. Then
$p\restriction\alpha\restriction N=(p\restriction
N)\\!\upharpoonright\\!\alpha$.

###### Proof.

It is enough to show that if $M\in\mathcal{M}_{p}$, then $(M\restriction
N)\\!\upharpoonright\\!\alpha=M\\!\upharpoonright\\!\alpha\restriction N$.
This is of course clear from the definition of $M\restriction N$. ∎

For a $V$-generic filter
$G_{\alpha,N}\subseteq\mathbb{Q}^{\kappa}_{\alpha}\cap N$, we can form the
following quotient forcing

$\mathbb{R}^{N}_{\alpha}\coloneqq\mathbb{Q}^{\kappa}_{\alpha}\restriction
N/G_{\alpha,N}.$

######  Definition 5.40.

Let $G_{\alpha,N}$ be a $V$-generic filter over
$\mathbb{Q}^{\kappa}_{\alpha}\cap N$. Let $\mathscr{C}_{\rm st}[G_{\alpha,N}]$
denote the set of all $M\in\mathscr{C}_{\rm st}$ such that $\alpha,N\in M$ and
$(N\land M)\restriction{\alpha}\in\mathcal{M}_{G_{\alpha,N}}$.

Notice that $\mathscr{C}_{\rm st}[G_{\alpha,N}]$ is stationary in
$\mathcal{P}_{\omega_{1}}(V_{\lambda})$ in the model $V[G_{\alpha,N}]$,
however we do not need this and so we avoid supplying a proof.

######  Lemma 5.41 (Factorization Lemma).

Suppose $N\in\mathscr{U}$, and that $\alpha\leq\beta\leq\eta(N)$ are in $E$.
Let $G_{\alpha,N}\subseteq\mathbb{Q}^{\kappa}_{\alpha}\cap N$ be a $V$-generic
filter. Then, in $V[G_{\alpha,N}]$,

$\begin{split}\rho:\mathbb{Q}^{\kappa}_{\beta}\restriction
N/G_{\alpha,N}&\longrightarrow\mathbb{R}^{N}_{\alpha}\times(\mathbb{Q}^{\kappa}_{\beta}\cap
N)/G_{\alpha,N}\\\ p&\longmapsto(p\\!\upharpoonright\\!\alpha,p\restriction
N)\end{split}$

is a projection.

###### Proof.

We first observe that $\rho$ is a well-defined function since $p\leq
p\\!\upharpoonright\\!\alpha,p\restriction N$, preserves the order, and
respects the maximal conditions. Assume that

$p\in\mathbb{Q}^{\kappa}_{\beta}\restriction N/G_{\alpha,N}\mbox{ and
}(r,s)\in\mathbb{Q}^{\kappa}_{\alpha}\restriction{N}/G_{\alpha,N}\times(\mathbb{Q}^{\kappa}_{\beta}\cap
N)/G_{\alpha,N}\mbox{ with }$
$(r,s)\geq(p\\!\upharpoonright\\!\alpha,p\restriction N).$

Observe that $r\restriction N,s\\!\upharpoonright\\!\alpha\in G_{\alpha,N}$,
and fix a common extension $\bar{q}\in G_{\alpha,N}$ of them. We have

$\bar{q}\land r\leq\bar{q}\leq s\restriction\alpha\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{q}\land s\leq\bar{q}\leq
r\restriction N.$

By Propositions 5.19 and 5.37, $(\bar{q}\land r)\land s=(\bar{q}\land s)\land
r$. Letting $q\coloneqq(\bar{q}\land r)\land s$, we have

$q\\!\upharpoonright\\!\alpha=(\bar{q}\land r)\leq r\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ q\restriction N=(\bar{q}\land s)\leq
s.$

It is easily seen that

$q\\!\upharpoonright\\!\alpha\restriction N=(q\restriction
N)\\!\upharpoonright\\!\alpha=\bar{q}\in G_{\alpha,N}.$

Thus

$q\in\mathbb{P}^{\kappa}_{\beta}/G_{\alpha,N}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
(q\\!\upharpoonright\\!\alpha,q\restriction N)\leq(r,s).$

It remains to show that $p$ and $q$ are compatible. We define a common
extension of them, say $t$. Let
$\mathcal{M}_{t}=\mathcal{M}_{p}\cup\mathcal{M}_{q}$. To see that this is a
condition in $\mathbb{M}^{\kappa}_{\alpha}$, pick $\delta\in E$. We then have

$\mathcal{M}_{t}^{\delta}=\begin{cases}\mathcal{M}_{q}^{\delta}&\mbox{if
}\delta\leq\alpha,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ (since
$\mathcal{M}_{p}^{\delta}\subseteq\mathcal{M}_{r}^{\delta}\subseteq\mathcal{M}_{q}^{\delta}$)}\\\
&\\\ \mathcal{M}_{p}^{\delta}&\mbox{if }\delta>\alpha.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ (since
$\mathcal{M}_{q}^{\delta}\subseteq\mathcal{M}_{s}^{\delta}\subseteq\mathcal{M}_{p}^{\delta}$)}\end{cases}$

In either case, $\mathcal{M}_{t}^{\delta}$ is an $\in_{\delta}$-chain. To
define $d_{t}$ on ${\rm dom}(d_{p})\cup{\rm dom}(d_{q})$, we let

$d_{t}(P)=\begin{cases}d_{p}(P)&\mbox{if }\eta(P)>\alpha\text{ and }P\notin
N,\\\ d_{q}(P)&\mbox{otherwise}.\end{cases}$

Every model in ${\rm dom}(d_{t})$ is $\mathcal{M}_{t}$-free. This is very
similar to the proofs of Lemma 3.14 of [14]. We sketch a proof. If
$\eta(P)\leq\alpha$, then $P$ is in $\mathcal{M}_{q\restriction\alpha}$, and
thus if $Q\in\mathcal{M}_{t}$ is such that $P\in_{\eta(P)}Q$, then
$Q\\!\upharpoonright\\!\alpha\in\mathcal{M}_{q\restriction\alpha}$. Now
$Q\\!\upharpoonright\\!\alpha$, and consequently $Q$ is strongly active at
$\eta(P)$.

On the other hand if $\eta(P)>\alpha$ is in $N$, then $P$ is in
$\mathcal{M}_{s}$, and thus if $Q\in\mathcal{M}_{t}$ is such that
$P\in_{\eta(P)}Q$, then if $Q$ is not in $N$, it must be in $\mathcal{M}_{p}$,
but $s\leq p\restriction N$, and hence $Q$ is strongly active at $\eta(P)$.
Similarly, one can prove that $d_{t}$ satisfies $(\divideontimes)$ in
Definition 5.2. We define $w_{t}$ on ${\rm dom}(w_{p})\cup{\rm dom}(w_{q})$ as
follows. For each $\gamma\in{\rm dom}(w_{t})$, we let

$w_{t}(\gamma)=\left\\{\begin{array}[]{cl}w_{p}(\gamma)&\textrm{if
}\gamma>\alpha\text{ and }\gamma\notin N,\\\
w_{q}(\gamma)&\textrm{otherwise}.\end{array}\right.$

We observe that $w_{t}$ is well-defined thanks to the definition of $q$.
Notice that also if $\gamma<\alpha$ is in ${\rm dom}(w_{t})$, then
$\mathcal{M}_{t}(\gamma)=\mathcal{M}_{q}(\gamma)$ and
$w_{t}(\gamma)=w_{q}(\gamma)$. On the other hand if $\gamma\geq\alpha$, then
$\mathcal{M}_{t}(\gamma)=\mathcal{M}_{p}(\gamma)$. Thus $t$ is a condition in
$\mathbb{Q}^{\kappa}_{\beta}\restriction N$ that extends $p$ and $q$. By an
easy calculation we have that $t\\!\upharpoonright\\!\alpha\restriction
N=\bar{q}\in G_{\alpha,N}$. Thus $t$ is a condition in
$\mathbb{Q}^{\kappa}_{\beta}\restriction N/G_{\alpha,N}$. ∎

######  Lemma 5.42.

Suppose $N\in\mathscr{U}$ and that $\alpha\leq\beta\leq\eta(N)$ are in $E$.
Let $G_{\beta,N}$ be a $V$-generic filter over $\mathbb{Q}^{\kappa}_{\beta}$.
Then in $V[G_{\beta,N}]$, $\mathbb{R}^{N}_{\alpha}$ is a complete suborder of
$\mathbb{R}^{N}_{\beta}$.

###### Proof.

Assume $p\in\mathbb{R}^{N}_{\beta}$ and that
$q\in\mathbb{Q}^{\kappa}_{\alpha}\restriction N$ extends
$p\\!\upharpoonright\\!\alpha$. Let $r=p\land q$ in
$\mathbb{Q}^{\kappa}_{\beta}$. Thus
$r\in\mathbb{Q}^{\kappa}_{\beta}\restriction N$, and we have

$r{\restriction N}=p\restriction N\land q\restriction N\in G_{\beta,N}.$

∎

######  Lemma 5.43.

Assume $N$ is a Magidor model, $M\in\mathscr{C}_{\rm st}[G_{\alpha,N}]$, and
that $p\in\mathbb{R}^{N}_{\alpha}\cap M$. Then
$p^{M\\!\upharpoonright\\!\alpha}$ is an
$(M[G_{\alpha,N}],\mathbb{R}^{N}_{\alpha})$-generic condition.

###### Proof.

By the proof of Lemma 4.7 of [14], we have

$(\mathcal{M}_{p},d_{p})^{M}\restriction{N}=\big{(}(\mathcal{M}_{p},d_{p})\restriction{N}\big{)}^{N\land
M}.$

On the other hand, $N\land M$ is in $\mathcal{M}_{G_{\alpha,N}}$, which in
turn implies that

$\big{(}(\mathcal{M}_{p\restriction{N}},d_{p\restriction{N}})^{N\land
M},\varnothing\big{)}\in G_{\alpha,N}.$

Therefore, we must have that

$p^{M}\restriction{N}=(\mathcal{M}^{N\land
M}_{p\restriction{N}},d_{p\restriction{N}}^{N\land M},w_{p}\restriction{N})\in
G_{\alpha,N}.$

Now $p^{M}\in\mathbb{R}^{N}_{\alpha}$, and that $p^{M}$ is
$(M,\mathbb{Q}^{\kappa}_{\alpha}\\!\upharpoonright\\!N)$-generic. Therefore
$p\\!\upharpoonright\\!N$ is $(M,\mathbb{Q}^{\kappa}_{\alpha}\cap N)$-generic,
and forces

$p\restriction{N}\Vdash``\check{p}\mbox{ is
}(M[\dot{G}_{\alpha,N}],\mathbb{R}^{N}_{\alpha})\mbox{-generic}".$

Since $p\\!\upharpoonright\\!N$ is in $G_{\alpha,N}$, we have $p$ is
$(M[G_{\alpha,N}],\mathbb{R}^{N}_{\alpha})$-generic. ∎

We now state our key lemma.

######  Lemma 5.44.

Suppose $N\in\mathscr{U}$ and that $\gamma\in a(N)$. Let $G_{\gamma,N}$ be a
$V$-generic filter over $\mathbb{Q}^{\kappa}_{\gamma}\cap N$. Then in
$V[G_{\gamma,N}]$, $\mathbb{R}^{N}_{\gamma}$ has the
$\omega_{1}$-approximation property.

###### Proof.

We prove by induction that for every $\beta\leq\gamma$,
$\mathbb{R}^{N}_{\beta}$ has the $\omega_{1}$-approximation property over
$V[G_{\gamma,N}]$, where in the definition of $\mathbb{R}^{N}_{\beta}$, we use
$G_{\gamma,N}\cap\mathbb{Q}^{\kappa}_{\beta}$. Thus fix $\beta$, and assume
that the conclusion of the lemma holds for every ordinal in $E\cap\beta$.

Suppose that $\dot{f}$ is an $\mathbb{R}^{N}_{\beta}$-term forced by some
condition $p$ to be a function on some ordinal $\mu$ that is countably
approximated in $V[G_{\gamma,N}]$. One can pick a countable model $M$
elementary in $V_{\delta}$ with $\delta>\lambda$ such that
$E,\beta,\gamma,N,p,\mu$ and $\dot{f}$ belong to $M$ and

$\big{(}N\land(M\cap
V_{\lambda})\big{)}\\!\upharpoonright\\!\gamma\in\mathcal{M}_{G_{\gamma,N}}.$

Let

$M_{\gamma}\coloneqq(M\cap V_{\lambda})\restriction\gamma\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ M_{\beta}\coloneqq
M_{\gamma}\restriction\beta.$

We may then use Lemma 5.43 to extend $p$ to a condition $p^{M_{\beta}}$ in
$\mathbb{R}^{N}_{\beta}$ so that $M_{\beta}\in\mathcal{M}_{p^{M_{\beta}}}$.
Since $p$ forces that $\dot{f}$ is $\omega_{1}$-approximated in
$V[G_{{\gamma},N}]$, there is a condition $q\leq p^{M_{\beta}}$ which decides
the values of $\dot{f}\\!\upharpoonright\\!M$. Therefore,

$\exists\leavevmode\nobreak\ g:M\cap\mu\rightarrow 2\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{ such that
}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
q\Vdash``\check{g}=\dot{f}\restriction M".$

By Lemmas 5.42 and 5.43, $p^{M_{\beta}}$ is
$(M_{\gamma}[G_{\gamma,N}],\mathbb{R}^{N}_{\beta})$-generic. Hence, it is also
$(M[G_{\gamma,N}],\mathbb{R}^{N}_{\beta})$-generic.

Now by Lemma 2.16, we are done if $g$ is guessed in $M[G_{\gamma,N}]$. Thus we
may assume towards a contradiction that $g$ is not guessed in
$M[G_{\gamma,N}]$, and we let

$\alpha\coloneqq{\rm max}({\rm dom}(w_{q})\cap M).$

Assume that $G^{N}_{\alpha}$ is a $V[G_{\gamma,N}]$-generic filter on
$\mathbb{R}^{N}_{\alpha}$ containing $q\\!\upharpoonright\\!\alpha$, where
again we use

$G_{\alpha,N}=G_{\gamma,N}\cap\mathbb{Q}^{\kappa}_{\alpha}$

to form the quotient $\mathbb{R}^{N}_{\alpha}$.

Notice that Lemmas 5.41 and 5.43 imply that $q\\!\upharpoonright\\!\alpha$ is
$(M[G_{\gamma,N}],\mathbb{R}^{N}_{\alpha})$-generic, and on the other hand by
the inductive hypothesis, $\mathbb{R}^{N}_{\alpha}$ has the
$\omega_{1}$-approximation property over $V[G_{\gamma,N}]$. Consequently,
using Lemma 2.17, the function $g$ is not guessed in
$M[G_{\gamma,N}][G^{N}_{\alpha}]$. Since $G^{N}_{\alpha}$ is a
$V[G_{\gamma,N}]$-generic filter on $\mathbb{R}^{N}_{\alpha}$, the
factorization lemma implies that

$V[G_{\gamma,N}][G^{N}_{\alpha}]=V[G^{N}_{\alpha}][G_{\gamma,N}].$

######  Claim 5.45.

The pair $(V[G^{N}_{\alpha}],V[G^{N}_{\alpha}][G_{\gamma,N}])$ has the
$\omega_{1}$-approximation property.

###### Proof.

Let $G^{N}_{\gamma}$ be a $V[G^{N}_{\alpha}][G_{\gamma,N}]$-generic filter on
$\mathbb{R}^{N}_{\gamma}$. Notice that both $G^{N}_{\alpha}$ and
$G^{N}_{\gamma}$ are also $V$-generic filters on
$\mathbb{Q}^{\kappa}_{\gamma}$ and $\mathbb{Q}^{\kappa}_{\alpha}$,
respectively. Now, Corollary 5.35 implies that the pair

$(V[G^{N}_{\alpha}],V[G^{N}_{\gamma}])$

has the $\omega_{1}$-approximation property, and so does the pair

$(V[G^{N}_{\alpha}],V[G^{N}_{\alpha}][G_{\gamma,N}]).$

∎

Notice that if $\alpha\in N$, then $\mathbb{S}_{\alpha}$ is already
interpreted by $G_{\gamma,N}$, and moreover there is a generic filter on ${\rm
val}_{G_{\gamma,N}}(\dot{\mathbb{S}}_{\alpha})$ in $V[G_{\gamma,N}]$. Thus in
this case conditions in $\dot{\mathbb{S}}_{\alpha}$ cannot prevent us from
amalgamating conditions in $\mathbb{R}^{N}_{\beta}$ i.e if $p^{\prime}$ and
$q^{\prime}$ are conditions in $\mathbb{R}^{N}_{\beta}$ such that
$\alpha\in{\rm dom}(w_{p^{\prime}})\cap{\rm dom}(w_{q^{\prime}})$, we do know
that $w^{G_{\gamma,N}}_{p^{\prime}}(\alpha)$ and
$w^{G_{\gamma,N}}_{q^{\prime}}(\alpha)$ are compatible. Let $\mathbb{Q}$ be
the the interpretation of $\dot{\mathbb{S}}_{\alpha}$ under $G^{N}_{\alpha}$
if $\alpha$ is not in $N$, and otherwise let it be the trivial forcing, and
let also, for every $p^{\prime}\in\mathbb{R}^{N}_{\beta}$,
$z_{p^{\prime}}=w^{G^{N}_{\alpha}}_{p^{\prime}}(\alpha)$ if $\alpha\notin N$,
and otherwise let it be an $\mathbb{R}^{N}_{\alpha}$-name in $N$ for the
maximal condition of the trivial forcing. These conventions are necessary to
avoid giving two almost identical arguments.

######  Claim 5.46.

The poset $\mathbb{Q}$ has the $\omega_{1}$-approximation property over
$V[G_{\gamma,N}][G^{N}_{\alpha}]$, and $z_{q}$ is
$(M[G_{\gamma,N}][G^{N}_{\alpha}],\mathbb{Q})$-generic.

###### Proof.

It is enough to assume that $\alpha\notin N$ and that $\mathbb{Q}$ is
nontrivial, thus it is the Baumgartner forcing for the specialization of a
tree $T$ of size and height $\omega_{1}$ without cofinal branches in
$V[G^{N}_{\alpha}]$. 5.45 implies that in
$V[G^{N}_{\alpha}][G_{\gamma,N}]=V[G_{\gamma,N}][G^{N}_{\alpha}]$, $T$ is
still a tree of size and height $\omega_{1}$ which has no cofinal branches.
Thus $\mathbb{Q}$ is, in $V[G_{\gamma,N}][G^{N}_{\alpha}]$, a c.c.c forcing
with the $\omega_{1}$-approximation property. Therefore, $z_{q}$ is also
$(M[G_{\gamma,N}][G^{N}_{\alpha}],\mathbb{Q})$-generic. ∎

Let $H$ be a $V[G_{\gamma,N}][G^{N}_{\alpha}]$-generic filter on $\mathbb{Q}$
containing $z_{q}$. By 5.46 and Lemma 2.15, the function $g$ is not guessed in

$M^{*}\coloneqq M[G_{\gamma,N}][G^{N}_{\alpha}][H].$

We consider an assignment $x\mapsto(q_{x},g_{x})$ on $[\mu]^{\omega}$ in
$M^{*}$ such that:

  1. (1)

$q_{x}\in\mathbb{Q}^{\kappa}_{\beta}\\!\upharpoonright\\!N$,

  2. (2)

$(\mathcal{M}_{q_{x}},d_{q_{x}})\leq(\mathcal{M}_{p},d_{p})\\!\upharpoonright\\!M$
in $\mathbb{P}^{\kappa}_{\beta}$,

  3. (3)

$M\cap{\rm dom}(w_{q})\subseteq{\rm dom}(w_{q_{x}})$,

  4. (4)

$q_{x}\\!\upharpoonright\\!\alpha\in G^{N}_{\alpha}$,

  5. (5)

$q_{x}\\!\upharpoonright\\!N\in G_{\beta,N}$,

  6. (6)

$z_{q_{x}}\in H$, and

  7. (7)

$x\subseteq{\rm dom}(g_{x})$ and
$q_{x}\Vdash``g_{x}\\!\upharpoonright\\!x=\dot{f}\\!\upharpoonright\\!x"$.

Note that $x\mapsto(q,g)$ witnesses the above assignment. Thus by elmentarity
and the Axiom of Choice, such an assignment exists in $M^{*}$.

By Lemma 2.17, there is a set $B\in M^{*}$ cofinal in $[\mu]^{\omega}$ such
that for every $x\in B\cap M^{*}$, $g_{x}\nsubseteq g$, but then it implies
that for such $x$, $q_{x}$ and $q$ are incompatible. We shall show that for
every $x\in B\cap M^{*}$, $q_{x}$ and $q$ are compatible. Fix such an $x$.
Then $q_{x}\in M^{*}$ by Lemmas 5.41 and 5.43, and hence $q_{x}\in M$. Let
$r\in G^{N}_{\alpha}$ be a common extension of
$q_{x}\\!\upharpoonright\\!\alpha$ and $q\\!\upharpoonright\\!\alpha$, and let
also $s\in G_{\beta,N}$ be a common extension of $q_{x}\\!\upharpoonright\\!N$
and $q\\!\upharpoonright\\!N$. Let us also put $\hat{q_{0}}=q_{x}\land q$
where the amalgamation is computed in
$\mathbb{Q}^{\kappa}_{\beta}\\!\upharpoonright\\!N$. Set

$d=\\{\xi\in{\rm dom}(w_{\hat{q}_{0}}):\alpha\leq\xi\notin N\\}.$

Let

$\hat{q}=(\mathcal{M}_{\hat{q}_{0}},d_{\hat{q}_{0}},w_{\hat{q}_{0}}\\!\upharpoonright\\!d).$

Notice that $\hat{q}\in\mathbb{Q}^{\kappa}_{\beta}\\!\upharpoonright\\!N$, and
that

$(r,s)\leq(\hat{q}\\!\upharpoonright\\!\alpha,\hat{q}\\!\upharpoonright\\!N).$

We now apply Lemma 5.41 to $(r,s)$ and $\hat{q}$ to find some condition
$t\in\mathbb{Q}^{\kappa}_{\beta}\\!\upharpoonright\\!N/G_{\alpha,N}$ as in the
proof of Lemma 5.41 with $t\leq r,s,\hat{q}$ such that
$t\\!\upharpoonright\\!N\in G_{\beta,N}$. Notice that $t\leq q_{x},q$ in
$\mathbb{Q}^{\kappa}_{\beta}\\!\upharpoonright\\!N$, but $t$ is in
$\mathbb{R}^{N}_{\beta}$, and hence $q$ and $q_{x}$ are compatible in
$\mathbb{R}^{N}_{\beta}$. ∎

##  6\. The proof of Theorem 3.11

Suppose $\kappa<\lambda$ are supercompact cardinals. Consider the structure
$V_{\lambda}=(V_{\lambda},\in,\kappa,U)$, where $U$ is a suitable bookkeeping
function that enumerates all $\mathbb{Q}^{\kappa}_{\lambda}$-terms in
$V_{\lambda}$ which are forced to be trees of size and height $\omega_{1}$
without cofinal branches. Let
$G_{\lambda}\subseteq\mathbb{Q}^{\kappa}_{\lambda}$ be a $V$-generic filter.
Since $\mathbb{Q}^{\kappa}_{\lambda}$ is $\lambda$-c.c, every tree of size and
height $\omega_{1}$ which has no cofinal branches is special in
$V[G_{\lambda}]$. Therefore, in order to show that ${\rm
SGM}^{+}(\omega_{3},\omega_{1})$ holds in $V[G_{\lambda}]$, it is enough to
demonstrate that ${\rm GM}^{+}(\omega_{3},\omega_{1})$ holds in
$V[G_{\lambda}]$. In fact we show that the $\omega_{1}$-guessing models of
size $\omega_{1}$ witnessing ${\rm GM}^{+}(\omega_{3},\omega_{1})$ are
internally club.

######  Lemma 6.1.

Let $\alpha\in E$. Suppose that $N\in\mathcal{M}_{G_{\lambda}}$ is a Magidor
model with $\alpha\in N$. Then $N[G_{\alpha}]$ is an $\omega_{1}$-guessing
model in $V[G_{\lambda}]$.

###### Proof.

Note that the projection map $N\mapsto N\\!\upharpoonright\\!\alpha$ is an
isomorphism and is the identity on $\mathbb{P}^{\kappa}_{\alpha}\cap N$. Thus,
$N[G_{\alpha}]$ and $(N\\!\upharpoonright\\!\alpha)[G_{\alpha}]$ are
isomorphic as well. Therefore, by replacing $N$ with
$N\\!\upharpoonright\\!\alpha$ we may assume that it is an $\alpha$-model. Let
$\overline{N}$ be the transitive collapse of $N$, and let $\pi$ be the
collapse map. For convenience, let us write $\bar{\kappa}$ for $\kappa_{N}$.
Then $\overline{N}=V_{\bar{\gamma}}$, for some $\bar{\gamma}$ with ${\rm
cof}(\bar{\gamma})\geq\bar{\kappa}$ and $\pi(\kappa)=\bar{\kappa}$. Let
$\bar{\alpha}=\pi(\alpha)$. Since $\alpha\in N$ and $N$ is an $\alpha$-model
we have

$\mathbb{P}_{N}=\mathbb{P}^{\kappa}_{\lambda}\cap
N=\mathbb{P}^{\kappa}_{\alpha}\cap N.$

Let
$\mathbb{P}^{\bar{\kappa}}_{\bar{\alpha}}=\pi[\mathbb{P}^{\kappa}_{\alpha}\cap
N]$, and let $p\in G_{\alpha}$ be such that $N\in\mathcal{M}_{p}$. By
Corollary 5.26 $p$ is $(N,\mathbb{P}^{\kappa}_{\alpha})$-strongly generic.
Hence $G_{\alpha,N}=G_{\alpha}\cap N$ is $V$-generic over
$\mathbb{P}^{\kappa}_{\alpha}\cap N$. It follows that

$G^{\bar{\kappa}}_{\bar{\alpha}}=\pi[G_{\alpha,N}]$

is $V$-generic over $\mathbb{P}^{\bar{\kappa}}_{\bar{\alpha}}$. Note that

$\overline{N[G_{\alpha,N}]}=V_{\bar{\gamma}}[G^{\bar{\kappa}}_{\bar{\alpha}}]=V_{\bar{\gamma}}^{V[G_{\alpha,N}]},$

which belongs to $V[G_{\alpha,N}]$. Clearly, the pair
$(\overline{N[G_{\alpha,N}]},V[G_{\alpha,N}])$ has the
$\omega_{1}$-approximation property. On the other hand, by Lemma 5.44, the
quotient forcing $\mathbb{R}^{N}_{\alpha}$ has the $\omega_{1}$-approximation
property, and thus the pair $(V[G_{\alpha,N}],V[G_{\lambda}])$ has the
$\omega_{1}$-approximation property. By Theorem 2.3, the pair

$(\overline{N[G_{\alpha,N}]},V[G_{\lambda}])$

has the $\omega_{1}$-approximation property. Now Theorem 2.4 implies that
$N[G_{\alpha,N}]$ is an $\omega_{1}$-guessing model in $V[G_{\lambda}]$. ∎

One can show with a similar proof that if $\mu>\lambda$ and $N\prec V_{\mu}$
is a $\kappa$-Magidor model containing all the relevant parameters. Then
$N[G_{\lambda}]$ is an $\omega_{1}$-guessing model in $V[G_{\lambda}]$. We now
prove that for all $\mu>\lambda$ the set of strongly $\omega_{1}$-guessing
models is stationary in ${\mathcal{P}}_{\omega_{3}}(V_{\mu}[G_{\lambda}])$.

######  Lemma 6.2.

Suppose that $\mu>\lambda$ and $N\prec V_{\mu}$ is a $\lambda$-Magidor model
containing all the relevant parameters. Then $N[G_{\lambda}]$ is a strongly
$\omega_{1}$-guessing model.

###### Proof.

Since $N$ is a $\lambda$-Magidor model, its transitive collapse $\overline{N}$
equals $V_{\bar{\gamma}}$, for some $\bar{\gamma}<\lambda$. Let
$\bar{\lambda}=N\cap\lambda$, which is in $E$. Note that ${\rm
cof}(\bar{\lambda})\geq\kappa$, and hence the transitive collapse
$\overline{N[G_{\lambda}]}$ of $N[G_{\lambda}]$ equals
$V_{\bar{\gamma}}[G_{\bar{\lambda}}]$. On the other hand, by Corollary 5.35,
the pair $(V[G_{\bar{\lambda}}],V[G_{\lambda}])$ has the
$\omega_{1}$-approximation property. Therefore,
$V_{\bar{\gamma}}[G_{\bar{\lambda}}]$ and hence also $N[G_{\lambda}]$ remains
an $\omega_{1}$-guessing model in $V[G_{\lambda}]$. To see that
$V_{\bar{\gamma}}[G_{\bar{\lambda}}]$ is a strongly $\omega_{1}$-guessing
model, fix some $\delta\in E$ with $\delta>\bar{\gamma}$ and ${\rm
cof}(\delta)<\kappa$. Note that if $M\in\mathcal{M}_{G}^{\delta}$ is a Magidor
model with $\bar{\lambda}\in M$, then by Lemma 6.1 $M[G_{\bar{\lambda}}]$ is
an $\omega_{1}$-guessing model. Moreover, if
$M\in\mathcal{M}_{G^{\kappa}_{\lambda}}^{\delta}$ is a limit of such Magidor
models then by Proposition 5.14,

$M\cap V_{\bar{\gamma}}=\bigcup\\{Q\cap V_{\delta}:Q\in_{\delta}M\mbox{ and
}Q\in\mathcal{M}_{G^{\kappa}_{\lambda}}^{\delta}\\}.$

Hence if we let $\mathcal{G}$ be the collection of the models $(M\cap
V_{\bar{\gamma}})[G_{\bar{\lambda}}]$, for Magidor models
$M\in\mathcal{M}_{G^{\kappa}_{\lambda}}^{\delta}$ with $\bar{\lambda}\in M$,
then $\mathcal{G}$ is an $\in$-increasing chain of length $\omega_{2}$ which
is continuous at $\omega_{1}$-limits and whose union is
$V_{\bar{\gamma}}[G_{\bar{\lambda}}]$. Note that every model of size
$\omega_{1}$ n this chain is a continuous union of an I.C. sequence.
Therefore, $N[G_{\lambda}]$ is a strongly $\omega_{1}$-guessing model in
$V[G_{\lambda}]$, as required. ∎

The following corollary is immediate, and concludes the proof of Theorem 3.11.

######  Corollary 6.3.

The principle ${\rm SGM}^{+}(\omega_{3},\omega_{1})$ holds in
$V[G_{\lambda}]$.

###### .

∎

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