# On fresh sets in iterations of Prikry type forcing notions

Moti Gitik111 We are grateful to participants of TAU Set Theory seminar and in
particular to Menachem Magidor and to Omer Ben-Neria for their comments and
remarks. The work of the first author was partially supported by ISF grant No.
882/22. and Eyal Kaplan

###### Abstract

We examine the existence (and mostly non-existence) of fresh sets in commonly
used iterations of Prikry type forcing notions. Results of [4] are
generalized. As an application, a question of a referee of [9] is answered. In
addition stationary sets preservation is addressed.

##  1  Introduction

We continue here the series of papers [4], [5],[8], [6] on iterations of
Prikry type forcing notions and concentrate on existence or non-existence of
fresh subsets.

Our framework throughout the paper will be as follows:

  1. 1.

GCH,

  2. 2.

$\kappa$ is an inaccessible cardinal,

  3. 3.

${\langle}P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa{\rangle}$
be either Magidor (full) or non-stationary or Easton support iterations of
Prikry type forcing notions.

  4. 4.

For every $p\in P_{\kappa}$ and $\gamma<\kappa$, let $p(\gamma)$ denotes the
$\gamma-$th coordinate of $p$,
i.e.
$p={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}(\gamma)\mid\gamma<\kappa{\rangle}$.
${\rm supp}(p)=\\{\gamma<\kappa\mid
p\upharpoonright\gamma\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}(\gamma)\not=0_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\gamma}}\\}$.

  5. 5.

For every $\beta<\kappa$, ${\langle}Q_{\beta},\leq^{*}_{Q_{\beta}}{\rangle}$
is forced to be $|\beta|-$closed.
Note that for a singular $\beta$ this implies $|\beta|^{+}-$closure.

  6. 6.

For every $\beta<\kappa$,
$|\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}|<\kappa$.

  7. 7.

If the Easton support is used, then for every $p\in P_{\kappa}$ and every
inaccessible $\alpha\leq\kappa$, $\alpha>|{\rm supp}(p)\cap\alpha|$, provided
that for every $\beta<\alpha$, $|P_{\beta}|<\alpha$.

  8. 8.

If the non-stationary support is used, then for every $p\in P_{\kappa}$ and
every inaccessible $\alpha\leq\kappa$, ${\rm supp}(p)\cap\alpha$ is non-
stationary in $\alpha$, provided that for every $\beta<\alpha$,
$|P_{\beta}|<\alpha$.

###### Definition 1.1

(J. Hamkins) A set $A$ of ordinals is called _a fresh set_ if all its initial
segments are in the ground model, but $A$ is not there.

Hamkins [7] proved that if a forcing has a gap at $\kappa$,222 $P$ has a gap
at $\kappa$, if
$P=R*\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}$ such that
$|R|<\kappa$ and
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}$ is
$\kappa+1-$strategically closed. then no fresh subsets of $\kappa$ are added.
We proved in [4] results on non-existence of fresh sets in iterations of
certain Prikry type forcing notions $P_{\kappa}$, as above. Here we would like
to generalize this results.
In addition, results regarding preservation of stationary sets when forcing
with each of the supports.

The paper is organized as follows. In Section 2 we deal with preservations of
stationarity. Later sections address fresh subsets. Section 3 deal with non-
stationary support iterations. Sections 4 and 5 address Easton and full
supports respectively.

##  2 On preservation of stationarity in Prikry type extensions

Let $\kappa$ be an inaccessible cardinal and let $S$ be a stationary subset of
$\kappa$. Suppose that
${\langle}P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa{\rangle}$
is an Easton or a full (Magidor) or a non-stationary support iteration of
Prikry type forcing notions as in the introduction. Let $G_{\kappa}$ be a
generic subset of $P_{\kappa}$.
We would like to address here a question _whether $S$ remains stationary in
$V[G_{\kappa}]$_.

###### Remark 2.1

  1. 1.

Note that we can assume that $\kappa$ is a regular cardinal in $V$, since
otherwise it is possible to replace it by $\operatorname{cof}(\kappa)$.

  2. 2.

Our main interest will be in situations where $\kappa$ is an inaccessible
cardinal in $V[G]$.

  3. 3.

If $\kappa$ is a Mahlo cardinal, then Easton support iterations satisfy
$\kappa-$c.c. and so preserve stationary subsets of $\kappa$.

  4. 4.

If a full support iteration of Prikry forcings is used, then the set of former
measurables which changed their cofinality will be non-stationary, as
witnessed by the regressive function which maps each one of them to the first
element in its Prikry sequence.

###### Theorem 2.2

Assume GCH. Let $\kappa$ be an inaccessible cardinal. Let $P=P_{\kappa}$ be as
in the introduction. Let $G\subseteq P_{\kappa}$ be generic over $V$. Then
$\kappa$ is inaccessible in $V\left[G\right]$, and–

  1. 1.

If $S\subseteq\kappa$ is stationary in $V$ and consists of singulars, then $S$
remains stationary in $V\left[G\right]$.

  2. 2.

If the Easton or nonstationary support is used, then $P_{\kappa}$ preserves
stationary subsets of $\kappa$.

  3. 3.

If the full support is used, $S\subseteq\kappa$ is stationary in $V$, and–

    1. (a)

For every $\alpha\in S$,
$\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha},\leq^{*}\rangle$
is $\left|\alpha\right|^{+}$-complete.

    2. (b)

For every $\alpha<\kappa$,
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}$ has
the property that for every $p,q,r\in Q_{\alpha}$, if $p,q\geq^{*}r$, then
there is $t\in Q_{\alpha}$ such that $t\geq^{*}p,q$.

Then $S$ remains stationary in $V[G]$.

The inaccessibility of $\kappa$ in $V\left[G\right]$ is proved in 2.6. Point 1
of the Theorem is proved in 2.7. Point 2 of the Theorem is proved in Theorems
2.9 and 2.11. Point 3 of the Theorem is proved in 2.10.

Let us start with the following lemma which is a weak form of a strong Prikry
condition:

###### Lemma 2.3

Let $D\subseteq P_{\kappa}$ be a dense open and let $p\in P_{\kappa}$. Then
there are $\alpha<\kappa$ and $q\geq^{*}p$ such that for every $r\in
P_{\alpha},r\geq_{P_{\alpha}}q\upharpoonright\alpha$ there is
$r^{\prime}\geq_{P_{\alpha}}r$ such that
$r^{\prime}{}^{\frown}q\setminus\alpha\in D$.

Proof. Let $D$ and $p={\langle}p_{\gamma}\mid\gamma<\kappa{\rangle}$ be as in
the statement of the lemma.
We prove the lemma for the full support iteration. The arguments for the
Easton and for the non-stationary support iterations are very similar, only
coordinates in supports should be considered.

Suppose that the conclusion of the lemma fails.
We define by recursion, for each $\beta<\kappa$, a condition–

$p(\beta)={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}_{\gamma}\mid\gamma<\beta{\rangle}^{\frown}{\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}_{\gamma}\mid\beta\leq\gamma<\kappa{\rangle}$

so that $p(\beta)\upharpoonright\beta\Vdash\neg\sigma_{\beta}$, where

$\sigma_{\beta}\equiv\exists t\in
P_{\kappa}\setminus\beta,t\geq^{*}p\setminus\beta\exists
r\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\beta}\quad
r^{\frown}t\in D.$

For the first stage, we have–

$\sigma_{1}\equiv\exists t\in P_{\kappa}\setminus 1,t\geq^{*}p\setminus
1\exists
r\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{1}\quad
r^{\frown}t\in D.$

$Q_{0}$ satisfies the Prikry condition, so there is $p_{0}^{*}\geq^{*}p_{0}$
which decides $\sigma_{1}$. If $p_{0}^{*}\Vdash\sigma_{1}$, then
$p_{0}^{*}{}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}$
will be as desired. So, assume that $p_{0}^{*}\Vdash\neg\sigma_{1}$, and
continue.

The successor step is similar to the first stage above.

For limit steps, suppose that $\beta$ is a limit ordinal. Let us show that–

$p(\beta)={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}_{\gamma}\mid\gamma<\beta{\rangle}^{\frown}{\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}_{\gamma}\mid\beta\leq\gamma<\kappa{\rangle}$

is as desired, i.e. $p(\beta)\upharpoonright\beta\Vdash\neg\sigma_{\beta}$.
Suppose otherwise, then there is
$r={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{\gamma}\mid\gamma<\beta{\rangle}\in
P_{\beta}$ such that $r\geq p(\beta)\upharpoonright\beta$ and
$r\Vdash\sigma_{\beta}$. Extend it, if necessary, so that for some
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}$

$r\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}\geq^{*}p\setminus\beta\text{
and
}r^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}\in
D.$

By the definition of order on $\mathcal{P}_{\beta}$, there is
$\beta^{*}<\beta$ such that for every $\gamma$, $\beta^{*}\leq\gamma<\beta$,

$r\\!\\!\upharpoonright\\!\\!\gamma\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{\gamma}\geq^{*}_{\gamma}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}_{\gamma}.$

Consider a $P_{\beta^{*}}$-name

$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}^{\prime}={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{\gamma}\mid\beta^{*}\leq\gamma<\beta{\rangle}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}.$

Then

$r\upharpoonright\beta^{*}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t^{\prime}}}}\geq^{*}p\setminus\beta^{*}\text{
and
}r\upharpoonright\beta^{*}{}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{t}}}^{\prime}\in
D.$

But $r\upharpoonright\beta^{*}\geq
p(\beta^{*})\upharpoonright\beta^{*}\Vdash\neg\sigma_{\beta^{*}}$.
Contradiction.

This completes the construction.
Consider
$p(\alpha)={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}_{\gamma}\mid\gamma<\kappa{\rangle}$.
Pick some $r\geq p(\kappa)$ in $D$. Now we obtain a contradiction as in the
limit stage above.
$\square$

Assuming that $\kappa$ is measurable in the ground model, a stronger version
of Lemma 2.3 can be proved:

###### Lemma 2.4

Assume that $\kappa$ is measurable in $V$ and $U$ is a normal measure on
$\kappa$. Let $D\subseteq P_{\kappa}$ be a dense open and let $p\in
P_{\kappa}$. Then there are $\alpha<\kappa$ and $q\geq^{*}p$, such that
$q\upharpoonright\alpha=p\upharpoonright\alpha$, and for every $r\in
P_{\alpha},r\geq_{P_{\alpha}}p\upharpoonright\alpha$ there is
$r^{\prime}\geq_{P_{\alpha}}r$ such that
$r^{\prime}{}^{\frown}q\setminus\alpha\in D$. Moreover, for every $X\in U$,
$\alpha,q$ above can be chosen such that $\alpha\in X$.

Proof. Let $\sigma_{\beta}$ be as in lemma 2.3. If there exists $\beta\in X$
such that $p\restriction\beta\Vdash\sigma_{\beta}$, we are done.

Assume otherwise. For every $\beta\in X$ there exists $r_{\beta}\geq
p\restriction\beta$ such that $r_{\beta}\Vdash\neg\sigma_{\beta}$. For each
such limit $\beta$, there exists $\beta^{\prime}<\beta$ such that
$r_{\beta}\restriction\beta^{\prime}\Vdash
r_{\beta}\setminus\beta^{\prime}\geq^{*}p\restriction\left[\beta^{\prime},\beta\right)$.
The function $\beta\mapsto\beta^{\prime}$ is regressive, and thus there exist
a set $A\in U$ and $\beta^{*}<\kappa$ such that for every $\beta\in A$,
$r_{\beta}\restriction\beta^{*}\Vdash
r_{\beta}\setminus\beta^{*}\geq^{*}p\restriction\left[\beta^{*},\beta\right)$.
Since $\left|P_{\beta^{*}}\right|<\kappa$, we can shrink $A\in U$ further, and
assume that there exists $r^{*}\in P_{\beta^{*}}$, such that, for every
$\beta\in A$, $r_{\beta}\restriction\beta^{*}=r^{*}$.

Then $r^{*}$ has the following property: for every $\beta\in A$ there exists
$s(\beta)\geq^{*}p\restriction\left[\beta^{*},\beta\right)$ such that
${r^{*}}^{\frown}s(\beta)\Vdash\neg\sigma_{\beta}$. Now apply ineffability: we
can find $A^{*}\subseteq A$, $A^{*}\in U$, and a $P_{\beta^{*}}$-name for a
condition $s^{*}=\left[\beta\mapsto s(\beta)\right]_{U}\in
P\setminus\beta^{*}$, such that $r^{*}\Vdash
s^{*}\geq^{*}p\setminus\beta^{*}$, and, for every $\beta\in A^{*}$,

${r^{*}}^{\frown}s^{*}\restriction\left[\beta^{*},\beta\right)={r^{*}}^{\frown}s(\beta)\Vdash\neg\sigma_{\beta}.$

Finally, pick some $q\geq{r^{*}}^{\frown}s^{*}$, $q\in D$. Let $\beta\in
A^{*}\setminus\beta^{*}+1$ be such that–

$q\restriction\beta\Vdash q\setminus\beta\geq^{*}s^{*}\setminus\beta$

and in particular,

$q\restriction\beta\Vdash q\setminus\beta\geq^{*}p\setminus\beta,\mbox{ and
}{q\restriction\beta}^{\frown}q\setminus\beta\in D$

and thus $q\restriction\beta\Vdash\sigma_{\beta}$; however, since $\beta\in
A^{*}$,

$q\restriction\beta\geq{r^{*}}^{\frown}s^{*}\restriction{\left[\beta^{*},\beta\right)}\Vdash\neg\sigma_{\beta}$

which is a contradiction.
$\square$

###### Lemma 2.5

Let $p\in P_{\kappa}$ be a condition, and assume that
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$ a
$P_{\kappa}$-name for an ordinal. Then there exists $q\geq^{*}p$ and a set of
ordinals $A\in V$ with $\left|A\right|<\kappa$ such that
$q\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}\in
A$.

Proof. Apply lemma 2.3 on the dense open set $D$ of conditions which decide
the value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$. Then
there exists $q\geq^{*}p$ and $\alpha<\kappa$ such that, for every $r\geq
q\restriction\alpha$ there exists $r^{\prime}\geq r$ such that
${r^{\prime}}^{\frown}q\setminus\alpha\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$.
Let $A\in V$ be the set of all possible values of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$ as
decided by some extension of $q$. We argue that $\left|A\right|<\kappa$.

Assume that $q^{\prime}\geq q$ decides the value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$. Denote
$r=q^{\prime}\restriction\alpha$. Then there exists $r^{\prime}\geq r$ such
that
${r^{\prime}}^{\frown}q\setminus\alpha\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$.
So both conditions $q^{\prime}$, ${r^{\prime}}^{\frown}q\setminus\alpha$
decide the value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$; but
those conditions are compatible, since $r^{\prime}\geq
p^{\prime}\restriction\alpha$, and $q^{\prime}\setminus\alpha\geq
q\setminus\alpha$. This shows that every element of $A$ can be realized as the
decided value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}$ by a
condition of the form $r^{\frown}q\setminus\alpha$ for some $r\in P_{\alpha}$.
But the cardinality of the set of such conditions is strictly below $\kappa$,
since $\left|P_{\alpha}\right|<\kappa$, by the assumption on the cardinality
of the forcings
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}$ for
$\beta<\kappa$.
$\square$

###### Corollary 2.6

Let $G_{\kappa}\subseteq P_{\kappa}$. Then $\kappa$ remains inaccessible in
$V[G_{\kappa}]$.

Proof. We concentrate on the proof that $\kappa$ remains a regular cardinal
after forcing with $P_{\kappa}$, since it’s routine to verify that it remains
strong limit.

Assume that $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$
is a $P_{\kappa}$-name for a function from some ordinal $\tau<\kappa$ to
$\kappa$. Let $p\in P_{\kappa}$ be a condition which forces this. We argue
that there exists $p^{*}\geq p$ and some $\mu^{*}<\kappa$, such that
$p^{*}\Vdash{\rm
rng}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}})\subseteq\mu^{*}$.

Let $G_{\tau+1}\subseteq P_{\tau+1}$ be an arbitrary generic extension
containing $p\restriction\tau+1$. We prove that, in
$V\left[G_{\tau+1}\right]$, there exist $q\geq^{*}p\setminus\tau+1$ and
$\mu<\kappa$ such that $q\Vdash{\rm
rng}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}})\subseteq\mu$.
Once we prove that, we are done: let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}$,
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}$ be
$P_{\tau+1}$-names which are forced by $p\restriction\tau+1$ to have the above
properties. Let $\mu^{*}<\kappa$ be an upper bound on the set of possible
values of $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}$,
as forced by extensions of $p\restriction\tau+1$. Since
$\left|P_{\tau+1}\right|<\kappa$, this set is bounded in $\kappa$, and thus
there exists such an upper bound below $\kappa$. Then
$p^{*}={p\restriction\tau+1}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}\Vdash{\rm
rng}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}})\subseteq\mu^{*}$,
as desired.

Work in $V\left[G_{\tau+1}\right]$. Apply lemma 2.5 over and over to construct
a $\leq^{*}$-increasing sequence of conditions $\langle
p_{\xi}\colon\xi\leq\tau\rangle$ in $P_{\kappa}\setminus\tau+1$, such that,
for each $\xi<\tau$ there exists some $\mu_{\xi}<\kappa$ such that
$p_{\xi}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}(\xi)<\mu_{\xi}$.
Note that in limit steps (including the last step) we may take upper bound,
since the direct extension order of $P_{\kappa}\setminus\tau+1$ is more than
$\tau$-closed. Finally, $q=p_{\tau}$ forces that the image of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ is bounded by
$\mu=\bigcup_{\xi<\tau}\mu_{\xi}<\kappa$.
$\square$

###### Theorem 2.7

Let $S\subseteq\kappa$ be a stationary set consisting of singulars. Let
$G_{\kappa}\subseteq P_{\kappa}$. Then $S$ remains stationary in
$V[G_{\kappa}]$.

Proof. Let $C\subseteq\kappa$ be a club in $V\left[G\right]$. Let $p\in G$ be
condition which forces this.
Work in $V$. Pick an elementary submodel $M\preceq H_{\chi}$ such that:

  1. 1.

$|M|=\delta<\kappa$,

  2. 2.

$M\cap\kappa=\delta$,

  3. 3.

$\delta\in S$ (in particular, $\delta$ is singular),

  4. 4.

${}^{\operatorname{cof}(\delta)>}M\subseteq M$,

  5. 5.

$\kappa,P_{\kappa},S,\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}},p\in
M$.

Pick a cofinal sequence in $\delta$, ${\langle}\delta_{i}\mid
i<\operatorname{cof}(\delta){\rangle}$.

Apply lemma 2.5. Construct (in $V$) a $\leq^{*}$-increasing sequence of
conditions in the forcing $P_{\kappa}\setminus\operatorname{cof}(\delta)$,
$\langle p_{\xi}\colon\xi\leq\operatorname{cof}(\delta)\rangle$, such that
each condition $p_{\xi}$ belongs to $M$.

We first construct $p_{0}\geq^{*}p$ in $M$, such that
$p_{0}\restriction\operatorname{cof}(\delta)+1=p\restriction\operatorname{cof}(\delta)+1$,
and, using Lemma 2.5,
$p_{0}\restriction\operatorname{cof}(\delta)+1\Vdash\exists\alpha<\kappa,\
p_{0}\setminus\operatorname{cof}{\delta}+1\Vdash\min\left(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\delta_{0}\right)<\alpha$.
Let $\alpha_{0}$ be the least upper bound of the set of all possible values
for the $P_{\operatorname{cof}(\delta)+1}$-name
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\alpha}}}$. Then
$p_{0}\Vdash\min\left(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\delta_{0}\right)<\alpha_{0}$.
Note that by elementarity, $\alpha_{0}<\delta$.

Assuming that $i<\operatorname{cof}(\delta)$ and $p_{i},\alpha_{i}$ have been
defined and both are in $M$, and let us define $p_{i+1},\alpha_{i+1}$. Let
$p_{i+1}\geq^{*}p_{i}$ in $M$ be such that
$p_{i+1}\restriction\operatorname{cof}(\delta)+1=p\restriction\operatorname{cof}(\delta)+1$,
and there exists $\alpha_{i+1}<\kappa$,
$p_{i+1}\Vdash\min\left(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\max\\{\delta_{i+1},\alpha_{i}\\}\right)<\alpha_{i+1}$.
Take $\alpha_{i+1}$ to be minimal with this property. Then
$\alpha_{i+1}<\delta$.

For the limit step, assume that $j<\operatorname{cof}(\delta)$ is limit and
$\langle p_{i}\colon i<j\rangle$, $\langle\alpha_{i}\colon i<j\rangle$ have
been constructed. Use the fact that $M$ is closed under
$<\operatorname{cof}{\delta}$-sequences to find an upper bound $q\in M$ of
$\langle p_{i}\colon i<j\rangle$, such that
$q\restriction\operatorname{cof}(\delta)+1=p\restriction\operatorname{cof}(\delta)+1$.
We used here the fact that $P_{\kappa}\setminus\operatorname{cof}(\delta)+1$
is $\mbox{cof}(\delta)$-closed. Finally, let $p_{j}\geq^{*}q$ be chosen in $M$
such that
$p_{j}\restriction\operatorname{cof}(\delta)+1=p\restriction\operatorname{cof}(\delta)+1$
and, for some $\alpha_{j}<\kappa$,
$p_{j}\Vdash\min\left(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\max\\{\delta_{j},\sup\\{\alpha_{i}\colon
i<j\\}\\}\right)<\alpha_{j}$. We used here again the fact that $M$ is closed
under $<\operatorname{cof}(\delta)$-sequences and thus
$\langle\alpha_{i}\colon i<j\rangle\in N$. Let $\alpha_{j}$ be the minimal
with the above property. Then $\alpha_{j}\in M$.

This concludes the inductive construction. In the final limit step, take, in
$V$, an upper bound $q^{*}$ for all the conditions ${\langle}p_{i}\colon
i<\operatorname{cof}(\delta){\rangle}$. Then
$q^{*}\Vdash\delta=\bigcup_{i<\operatorname{cof}(\delta)}{\alpha_{i}}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}$,
since $C$ is forced by $p$ to be closed. Thus $\delta\in C\cap S$, as desired.
$\square$

###### Remark 2.8

Basically the same argument works for $\kappa$ replaced by $\kappa^{+}$.

Now let us try to extend the theorem to $S$’s which consists of regular
cardinals. We deal with three supports separately.
First point out the following:

###### Theorem 2.9

Suppose that the Easton support is used in $P_{\kappa}$. Assume that
$S\subseteq\kappa$ is stationary. Then $S$ remains stationary in
$V\left[G\right]$.

Proof. If $\kappa$ is a Mahlo cardinal, the Easton support iteration is
$\kappa$-c.c. and thus preserves stationary subsets of $\kappa$. Thus, we can
assume that $\kappa$ is not Mahlo. In this case, there exists a club
$C\subseteq\kappa$ of singular cardinals. Thus, by shrinking $S$ we can assume
that it consists of singulars. Then, Theorem 2.7 applies.
$\square$

Turn to the full support.

###### Theorem 2.10

Suppose that $P_{\kappa}$ is the full support iteration.
Assume that for every $\beta<\kappa$, for every
$p,q,r\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}$,
if $p,q\geq^{*}r$, then there is
$t\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}$
such that $t\geq^{*}p,q$.
Let $S\subseteq\kappa$ be a stationary such that, for every $\alpha\in S$,
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}$ is
$|\alpha|^{+}-$complete.
Then $S$ remains stationary in $V\left[G\right]$.

Proof. Suppose otherwise. Pick some $p\in P_{\kappa}$ and a name
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}$ such that

$p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\text{
is a club in }\kappa\text{ and
}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\cap
S=\emptyset.$

Pick now $M\preceq H_{\chi}$ and $\delta\in S$, as in Theorem 2.7.
It is enough to find a condition $r\geq p$ which forces
$``\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\text{ is
unbounded in }\delta$” in order to derive a contradiction.
Suppose that there is no such $r$. Let $G_{\delta}\subseteq P_{\delta}$ be a
generic with $p\upharpoonright\delta\in G_{\delta}$. Then , in
$V[G_{\delta}]$,

$p\setminus\delta\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}_{G_{\delta}}\cap\delta\text{
is bounded in }\delta.$

We have $\delta\in S$, so, by the assumption of the theorem,
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\delta}$ is
$\delta^{+}-$complete, and hence, $P_{\kappa}/G_{\delta}$ is
$\delta^{+}-$complete. Then there are $p^{\prime}\in
P_{\kappa}/G_{\delta},p^{\prime}\geq^{*}p\setminus\delta$ and $\rho<\delta$
such that

$p^{\prime}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}_{G_{\delta}}\cap\delta\subseteq\rho.$

Pick some $t\in G_{\delta},t\geq p\upharpoonright\delta$ such that

$t^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{\prime}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}_{G_{\delta}}\cap\delta\subseteq\rho.$

Since $t\geq p\upharpoonright\delta$, there exists $\gamma<\delta$ such that
$t=t\upharpoonright\gamma^{\frown}t\setminus\gamma$ and
$t\setminus\gamma\geq^{*}p\upharpoonright[\gamma,\delta)$. In particular,
$t\restriction\gamma\Vdash{t\setminus\gamma}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{\prime}\geq^{*}p\setminus\gamma$.
Now work in $M$ above the condition $p$. Note that $\gamma,\rho$ above are
below $\delta$ and thus are in $M$. Also
$p,\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\in M$, and
recall that

$p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\mbox{
is a club in }\kappa$

Let $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}\in M$
be a $P_{\kappa}$-name such that
$p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}=\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\check{\rho}+1)$.
By Lemma 2.5, we can find (in $M$) a $P_{\gamma}$-name for a condition
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}_{1}\geq^{*}p\setminus\gamma$
such that $p\upharpoonright\gamma\Vdash\exists\mu<\kappa\
\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}_{1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}<\mu$.
Let $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}\in M$ be
a $P_{\gamma}$-name such that
$p\upharpoonright\gamma\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}_{1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}<\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}$.
In $M$, let $\mu^{*}$ be the supremum of all possible values of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}$, as forced
by extensions of $p\upharpoonright\gamma$. Then $\mu^{*}<\delta$ (since
$\mu^{*}\in M\cap\kappa$). By elementarity,
${p\upharpoonright\gamma}^{\frown}q_{1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\zeta}}}<\mu^{*}$
holds in $V$ as well.

Finally, let $p^{*}\in P_{\kappa}$ be a condition such that
$q^{*}\upharpoonright\gamma=t\upharpoonright\gamma$ and $p^{*}\setminus\gamma$
direct extends both the conditions
${t\setminus\gamma}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{\prime},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}_{1}$.
Note that, since
${t\setminus\gamma}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{\prime}$
and $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{q}}}_{1}$
direct extend $p\setminus\gamma$, the extra assumption of the theorem allows
us to construct such $p^{*}$.

Then on the one hand, $p^{*}$ forces that
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\cap\delta\subseteq\rho$;
on the other hand, it forces that
$\min\left(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\rho+1\right)\leq\mu^{*}<\delta$.
A contradiction.
$\square$

Turn now to the non-stationary support.

###### Theorem 2.11

Assume that the non-stationary support is used. Let $S\subseteq\kappa$ be
stationary. Then $S$ is stationary in $V[G_{\kappa}]$.

Proof. Let $\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}$ be
a $P_{\kappa}-$name and $p\in P_{\kappa}$,
$p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\text{
is a club disjoint to }S$.

We construct:

  1. 1.

$\leq^{*}$-increasing sequence of conditions $\langle p_{i}\colon
i<\kappa\rangle$,

  2. 2.

decreasing sequence of clubs $\langle C_{i}\colon i<\kappa\rangle$, each is
disjoint to the support of $p_{i}$,

  3. 3.

increasing continuous sequences of ordinals $\langle\nu_{i}\colon
i<\kappa\rangle$, $\langle\alpha_{i}\colon i<\kappa\rangle$.

The sequence of conditions will be a fusion sequence, in the sense that, for
every $i<j$, $p_{i}\leq^{*}p_{j}$ and
$p_{j}\restriction\nu_{i}=p_{i}\restriction\nu_{i}$.

Let $p_{0}=p$ and $\alpha_{0}=0$. Pick a club $C_{0}$ disjoint to ${\rm
supp}(p_{0})$. Let $\nu_{0}=\min(C_{0})$.
Let $G_{\nu_{0}}\subseteq P_{\nu_{0}}$ be a generic with
$p_{0}\upharpoonright\nu_{0}\in G_{\nu_{0}}$. Apply Lemma 2.5, inside
$V[G_{\nu_{0}}]$, and find $p^{\prime}_{1}\in
P_{\kappa}/G_{\nu_{0}},p_{1}^{\prime}\geq^{*}p_{0}\setminus\nu_{0}$ and
$A_{1}^{\prime}\subseteq\kappa,|A_{1}^{\prime}|<\kappa$ such that
$p_{1}^{\prime}\Vdash\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}})\in
A_{1}^{\prime}$.
Now back to $V$, we have $|P_{\nu_{0}}|<\kappa$, hence there is
$A_{1},|A_{1}|<\kappa$ such that
$p_{1}=p_{0}\upharpoonright\nu_{0}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}_{1}^{\prime}\Vdash\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}})\in
A_{1}$.
Set $\alpha_{1}=\sup(A_{1})$.

Next, we pick a club $C_{1}\subseteq C_{0}$ disjoint to ${\rm supp}(p_{1})$.
Let $\nu_{1}=\min(C_{1})\setminus\max(\nu_{0},\alpha_{1})$.
Let $G_{\nu_{1}}\subseteq P_{\nu_{1}}$ be a generic with
$p_{1}\upharpoonright\nu_{1}\in G_{\nu_{1}}$. Apply Lemma 2.5, inside
$V[G_{\nu_{1}}]$, and find $p^{\prime}_{2}\in
P_{\kappa}/G_{\nu_{1}},p_{2}^{\prime}\geq^{*}p_{1}\setminus\nu_{1}$ and
$A_{2}^{\prime}\subseteq\kappa,|A_{2}^{\prime}|<\kappa$ such that
$p_{2}^{\prime}\Vdash\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\nu_{1}+1)\in
A_{1}^{\prime}$.
Now back to $V$, we have $|P_{\nu_{1}}|<\kappa$, hence there is
$A_{2},|A_{2}|<\kappa$ such that
$p_{2}=p_{1}\upharpoonright\nu_{1}{}^{\frown}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}_{2}^{\prime}\Vdash\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\nu_{1}+1)\in
A_{2}$.
Set $\alpha_{2}=\sup(A_{2})$.

We do the same at each successor stage $i<\kappa$ of the construction.
Suppose now that $i<\kappa$ is limit. Set
$\alpha_{i}=\nu_{i}=\bigcup_{j<i}\nu_{j}=\bigcup_{j<i}\alpha_{j}$ and let
$p_{i}$ be the coordinateswise union of $p_{j},j<i$. Let
$C_{i}=\bigcap_{j<i}C_{j}$. Pick $\nu_{i+1}=\min(C_{i})\setminus\nu_{i}+1$.
Continue as above.

Finally, let $p_{\kappa}$ be the coordinateswise union of $p_{i},i<\kappa$. It
is in $P_{\kappa}$ since for every $i<j$, $p_{i}\leq^{*}p_{j}$ and
$p_{j}\restriction\nu_{i}=p_{i}\restriction\nu_{i}$ and
$\Delta_{i<\kappa}C_{i}$ is disjoint to its support.
Now,

$p_{\kappa}\Vdash\\{\alpha_{i}\mid i\text{ limit
}\\}\subseteq\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}.$

But $\\{\alpha_{i}\mid i\text{ limit }\\}$ is club in $V$, and so,
$S\cap\\{\alpha_{i}\mid i\text{ limit }\\}\not=\emptyset$. Contradiction.
$\square$

A similar argument can be using elementary submodel shows preservation of
stationary subsets of $\kappa^{+}$ with non-stationary support iterations.

Note $|P_{\kappa}|=\kappa$, in the Easton support case, and so, $\kappa^{+}$
is preserved. By [1], $\kappa^{+}$ is preserved in non-stationary support
iterations. Let us consider the full support iterations. Note that the Magidor
iteration of Prikry forcing notions [10] satisfies $\kappa^{+}-$c.c.. A bit
more general:

###### Proposition 2.12

Suppose that for every $\beta<\kappa$, for every
$s,t,r\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}$,
if
$s\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}t,r$,
then there is
$e\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta},e\geq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}t,r$.
Then $P_{\kappa}$ satisfies $\kappa^{+}-$c.c.

In general it turns out that $\kappa^{+}$ may be collapsed with full support
iteration.

###### Proposition 2.13

Suppose that $\kappa$ is a measurable cardinal. Let $P_{\kappa}$ be the full
support iteration of $Col(\alpha,\alpha^{+})=\\{f\mid
f\in{}^{\xi}\alpha^{+},\xi<\alpha\\}$, for every regular $\alpha<\kappa$. Then
$\kappa^{+}$ is collapsed in $V^{P_{\kappa}}$.

Proof. Let $U$ be a normal measure over $\kappa$.
We start with the following claim:

###### Claim 1

Let
$p={\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}(\beta)\mid\beta<\kappa{\rangle}\in
P_{\kappa}$. Then there are $A^{*}\in U$, $\tau^{*}<\kappa$ and $p^{*}\geq p$
such that $p^{*}\upharpoonright\beta\Vdash{\rm
dom}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}(\beta))=\tau^{*}$,
for every $\beta\in A^{*}$.

Proof. For every regular $\beta<\kappa$ there are $s_{\beta}\in
P_{\beta},s_{\beta}\geq p\upharpoonright\beta$ and $\tau_{\beta}^{0}<\beta$
such that $s_{\beta}\Vdash{\rm
dom}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}(\beta))=\tau_{\beta}^{0}$.
Find $A_{0}^{\prime}\in U$ and $p_{0}\in P_{\kappa}$ such that
$p_{0}\upharpoonright\beta=s_{\beta},$ for every $\beta\in A_{0}^{\prime}$.
For example, take $p_{0}=[\beta\mapsto s_{\beta}]_{U}$.
Then we consider a regressive function $\beta\mapsto\tau_{\beta}^{0}$ on
$A_{0}^{\prime}$. Find $A_{0}\subseteq A_{0}^{\prime},A_{0}\in U$ and
$\tau^{0}<\kappa$ such that $\tau_{\beta}^{0}=\tau^{0},$ for every $\beta\in
A_{0}$.
Repeat the process with $p_{0}$ replacing $p$ and find $A_{1}\subseteq
A_{0},A_{1}\in U$, $p_{1}$ and $\tau^{1}$ such that
$p_{1}\upharpoonright\beta\Vdash{\rm
dom}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}_{0}(\beta))=\tau^{1}$.
Continue by induction. Let $A^{*}=\bigcap_{n<\omega}A_{n}$ and $p^{*}$ be the
coordinatewise union of $p_{n}$’s. Set $\tau^{*}=\bigcup_{n<\omega}\tau^{n}$.
Then $p^{*}\upharpoonright\beta\Vdash{\rm
dom}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{p}}}^{*}(\beta))=\tau^{*}$,
for every $\beta\in A^{*}$, will be as desired.
$\square$ of the claim.

For every $\tau<\kappa$, define a maximal antichain $A_{\tau}$ in
$P_{\kappa}$. Proceed as follows.
Let us pick functions ${\langle}h_{\gamma}\mid\gamma<\kappa^{+}{\rangle}$ such
that ${\rm dom}(h_{\gamma})\in U$, for every $\alpha\in{\rm
dom}(h_{\gamma}),h_{\gamma}(\alpha)<\alpha^{+}$ and $[h_{\gamma}]_{U}=\gamma$,
for example $\kappa^{+}-$canonical functions will do the job.
Fix $\tau<\kappa$. Let $A_{\tau}$ be a maximal antichain in $P_{\kappa}$ of
cardinality $\kappa^{+}$ which consists of $p\in P_{\kappa}$ such that:

(*)_if for some $B\in U$ and $\gamma<\kappa^{+}$, the condition
$t_{B\gamma}={\langle}{t}_{B\gamma}(\alpha)\mid\alpha<\kappa{\rangle}$ is
compatible with $p$, then, for some $B^{\prime}\in U$, $p\geq
t_{B^{\prime}\gamma}$,_
where, for $E\in U$, ${t}_{E\gamma}(\alpha)=0_{\alpha},$ unless $\alpha\in
E\cap{\rm dom}(h_{\gamma})\setminus\tau+1$, and if $\alpha\in E\cap{\rm
dom}(h_{\gamma})\setminus\tau+1$, then
${t}_{E\gamma}(\alpha)=\\{(\tau,h_{\gamma}(\alpha))\\}$, i.e. the value of the
generic function for $\alpha$ at $\tau$ is $h_{\gamma}(\alpha)$.

Let ${\langle}p^{\tau}_{i}\mid i<\kappa^{+}{\rangle}$ be an enumeration of
$A_{\tau}$.

Let $G\subseteq P_{\kappa}$ be a generic. Define $F:\kappa\to(\kappa^{+})^{V}$
by setting $F(\tau)=i$ iff $p^{\tau}_{i}\in G$.
We claim that such $F$ is onto.
Suppose otherwise. Pick some $p\in G$ and $\eta<\kappa^{+}$ such that
$p\Vdash{\rm
rng}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{F}}})\subseteq\eta$.
Apply Claim 1. Let $A^{*}\in U$, $\tau^{*}<\kappa$ and $p^{*}\geq p$ be as in
the conclusion of the claim.
Now for every $\gamma<\kappa^{+}$, we can extend $p^{*}$ to a condition
$p^{\gamma}$ by adding a pair $(\tau^{*},h_{\gamma}(\alpha))$ to
$p^{*}(\alpha)$, for every $\alpha\in A^{*}\cap{\rm
dom}(h_{\gamma})\setminus\gamma+1$. Note that if $\gamma\not=\gamma^{\prime}$,
then $p^{\gamma},p^{\gamma^{\prime}}$ are incompatible. So, the set
$\\{p^{\gamma}\mid\gamma<\kappa^{+}\\}$ consists of $\kappa^{+}-$many
incompatible conditions.
Then, each of $p^{\gamma}$’s must be compatible with a member of
$A_{\tau^{*}}$ with index below $\eta$. Hence, there is $i^{*}<\eta$ such that
$p^{\tau^{*}}_{i^{*}}$ is compatible with $\kappa^{+}-$many $p^{\gamma}$’s.
Pick two of them $\gamma\not=\gamma^{\prime}$.
By (*), then $p^{\tau^{*}}_{i^{*}}\geq
t_{B\gamma},t_{B^{\prime}\gamma^{\prime}}$, for some $B,B^{\prime}\in U$.
However, $\gamma\not=\gamma^{\prime}$ implies that there is $\alpha\in B\cap
B^{\prime}$, such that $h_{\gamma}(\alpha)\not=h_{\gamma^{\prime}}(\alpha)$,
and this is impossible due to the compatibility.
Contradiction.
$\square$.

We conjecture that the measurability assumption can be much weaken. However,
the following positive result can be proved:

###### Proposition 2.14

Suppose that there is a club $C\subseteq\kappa$ such that for every $\alpha\in
C$,
${\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha},\leq^{*}_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}}{\rangle}$
is forced to be $\alpha^{+}-$closed. Then $P_{\kappa}$ preserves $\kappa^{+}$.
In particular, if $\kappa$ is not a Mahlo cardinal, then $P_{\kappa}$
preserves $\kappa^{+}$.

Proof. Let $p\in P_{\kappa}$ and
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ be a name
such that

$p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}:\kappa\to\kappa^{+}.$

Fix a club $C$ such that for every $\alpha\in C$,
${\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha},\leq^{*}_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}}{\rangle}$
is forced to be $\alpha^{+}-$closed. Assume also that for every $\alpha\in C,$
for every $\beta<\alpha,|P_{\beta}|<\alpha$. Let ${\langle}\alpha_{i}\mid
i<\kappa{\rangle}$ be an increasing continuous enumeration of $C$. Apply Lemma
2.5 and find
$p_{0}\geq^{*}p,p_{0}\upharpoonright\alpha_{0}=p\upharpoonright\alpha_{0}$ and
$\eta_{0}<\kappa^{+}$ such that
$p_{0}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}(0)<\eta_{0}$.
Continue by induction and define a $\leq^{*}-$increasing sequence
${\langle}p_{i}\mid i<\kappa{\rangle}$ and sequence ${\langle}\eta_{i}\mid
i<\kappa{\rangle}$ of ordinals below $\kappa^{+}$ such that

  1. 1.

$p_{i}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}(i)<\eta_{i}$,

  2. 2.

for every $i<j$,
$p_{j}\upharpoonright\alpha_{i}=p_{i}\upharpoonright\alpha_{i}$,

There is no problem at limit stages $i$, since
${\langle}P_{\kappa}\setminus\alpha_{i},\leq^{*}{\rangle}$ is
$\alpha_{i}^{+}-$closed since $\alpha_{i}\in C$.

The second item insures that there is $p^{*}\in P_{\kappa}$ such that
$p^{*}\geq^{*}p_{i},$ for every $i<\kappa$. Then

$p^{*}\Vdash{\rm
rng}(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}})\subseteq\bigcup_{i<\kappa}\eta_{i}.$

$\square$

##  3  Non-stationary support iterations

We assume GCH throughout as before. Let $\kappa$ be an inaccessible cardinal.

Let $\langle
P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\colon\alpha\leq\kappa,\beta<\kappa\rangle$
be a non-stationary support iteration of Prikry type forcings, with the
properties stated in the introduction.

Let $I$ be a stationary subset of $\kappa$ which consists of singular
cardinals $\mu$ such that for every $\gamma<\mu,|P_{\gamma}|<\mu$.
Assume that-
_If $\alpha\in I$, then $\Vdash_{P_{\alpha}}\langle
P_{\kappa}\setminus{\alpha},\leq^{*}\rangle\mbox{ is
}\alpha^{++}\mbox{-closed.}$_

Note that for a singular cardinal $\alpha$, we have
$\Vdash_{P_{\alpha}}\langle P_{\kappa}\setminus{\alpha},\leq^{*}\rangle\mbox{
is }\alpha^{+}\mbox{-closed.}$ We will need a slightly more closure. A typical
situation is where $\kappa$ is Mahlo, $Q_{\alpha}$ is trivial at every
accessible $\alpha$ and each forcing
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\alpha}$ has
cardinality below the least inaccessible above $\alpha$.

###### Theorem 3.1

$P_{\kappa}$ does not add fresh unbounded subsets to $\kappa$.

Proof. Recall the following fusion lemma for the non-stationary support
iteration of Prikry forcings333The fusion property for non-stationary support
iterations is due to Friedman and Magidor [2]. A version suitable for
iterations of Prikry type forcings appeared in [1]. The proof is basically
given in lemma 3.3 in [4].:

###### Lemma 3.2

Let $p\in P_{\kappa}$. For every $\beta<\kappa$, let $F(\beta)$ be a
$P_{\beta}$-name for a $\leq^{*}$-dense open subset of $P\setminus\beta$ above
$p\setminus\beta$, and assume that this is forced by $p\restriction{\beta}$.
Then there exist $p^{*}\geq^{*}p$ and a club $C\subseteq\kappa$ such that for
every singular $\beta\in C$, $p^{*}\restriction{\beta}\Vdash
p^{*}\setminus\beta\in F(\beta).$

Let $G\subseteq P_{\kappa}$ be generic over $V$, and assume for contradiction
that there exists a function $f\in 2^{\kappa}$ which is the characteristic
function of a fresh subset of $\kappa$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ be a
$P_{\kappa}$-name for it, and assume that this is forced by some condition in
$G$. For simplicity, assume that this is the weakest condition.

Let $\zeta\in\kappa\cap I$ be the least ordinal for which a new subset is
added in the extension from $V$ to $V\left[G\right]$. Such $\zeta$ exists,
since the forcings $Q_{\alpha}$ (for $\alpha<\kappa$) have cardinality below
$\kappa$, and at least one of them is non-trivial.

Note that $\zeta\in\kappa\cap I$,
${\langle}P_{\kappa}\setminus\zeta,\leq^{*}{\rangle}$ is $\zeta^{++}$-closed.
Pick a condition $q\in P_{\zeta}$ which forces that a new subset is added to
$\zeta$. For simplicity, assume that the weakest condition in $P_{\zeta}$
forces this (else, work above a condition in $P_{\kappa}$ whose restriction to
$P_{\zeta}$ equals $q$).

We divide into two cases:

Case 1.444 It basically repeats those of 4.11, [4]. There exists
$\mu\in\left(\zeta,\kappa\right)\cap I$ and a condition $p^{*}\in P_{\mu}$
which forces that the following property holds:

$\displaystyle\exists p\in G_{\mu}\ \exists s\in P\setminus\mu\forall
r\geq^{*}s\ \exists\xi<\kappa\ \exists r_{0},r_{1}\geq^{*}r,$ $\displaystyle
V\vDash\left({p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi},\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\right)\mbox{,
and the decisions are different.}$

(here, $G_{\mu}$ denotes the canonical name for the generic set for
$P_{\mu}$). By extending $p^{*}$, we can decide the value of $p$ in the
statement above, and thus assume that $p^{*}\geq p$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}$ be a
$P_{\mu}$-name for $s$ from the above property, and assume that this is forced
by $p^{*}$.

Let us apply the same methods as in the main lemma in [7]. We construct, in
$V$, a binary tree of conditions, $\langle\langle
p^{*},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\sigma}\rangle\colon\sigma\in{}^{\mu>}2\rangle$
and a tree of functions $\langle b_{\sigma}\colon\sigma\in{}^{\mu>}2\rangle$
such that
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\emptyset}=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}$,
and for every $\sigma\in{}^{\mu>}2$:

  1. 1.

$\forall i<2,\ \langle
p^{*},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\sigma^{\frown}\langle
i\rangle}\rangle\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{lh\left(b_{\sigma^{\frown}\langle
i\rangle}\right)}=b_{\sigma^{\frown}\langle i\rangle}$.

  2. 2.

$b_{\sigma^{\frown}\langle 0\rangle}\perp b_{\sigma^{\frown}\langle
1\rangle}$.

  3. 3.

If $\mbox{lh}(\sigma)$ is limit, then $p^{*}$ forces that
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\sigma}$ is
an upper bound, with respect to the direct extension order, of
$\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\sigma\restriction{\xi}}\colon\xi<\mbox{lh}(\sigma)\rangle$.

  4. 4.

$b_{\sigma}$ is an end extension of $b_{\sigma\restriction{\xi}}$ for every
$\xi<\mbox{lh}(\sigma)$.

Now assume that $g\subseteq P_{\mu}$ is generic over $V$ with $p^{*}\in g$. In
$V\left[g\right]$, let $h\in 2^{<\mu}$ be the characteristic function of a new
subset of $\mu$ (such a new subset exists because $\mu$ is above $\zeta$). $h$
defines a branch through the binary tree, $\langle\langle
p^{*},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{h\restriction{\xi}}\rangle\colon\xi<\mu\rangle$.
Since
$\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{h\restriction{\xi}}\colon\xi<\mu\rangle$
form the sequence of moves of the first player, when the second plays
according to the strategy $\tau$, there exists an upper bound $s^{*}\in
P\setminus\mu$, which extends all the conditions in the sequence. Thus, there
exists an upper bound for the branch, of the form $\langle
p^{*},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}^{*}\rangle$.
It forces that–

$b=\bigcup_{\xi<\mu}b_{h\restriction{\xi}}$

is an initial segment of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$. We argue
that this must be a strict initial segment of $f$. Indeed, otherwise,
$\left(\mbox{cf}(\kappa)\right)^{V\left[G_{\mu}\right]}\leq\mu$. But, since
$\kappa$ is inaccessible, $G_{\mu}\subseteq P_{\mu}$ is a forcing whose
cardinality is strictly below $\kappa$, so it preserves cofinalities greater
of equal to $\kappa$.

Therefore, $b$ is a strict initial segment of $f$, and thus $b\in V$.
Therefore, $h$ can be defined, in $V$, using the binary tree and the set $b$.
This is a contradiction to the choice of $h$.

Case 2. For every $\mu\in\left(\zeta,\kappa\right)\cap I$, every condition in
$P_{\mu}$ forces that–

$\displaystyle\forall
p\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\mu}\
\forall s\in P_{\kappa}\setminus\mu\exists r\geq^{*}s\ \forall\xi<\kappa\
\forall r_{0},r_{1}\geq^{*}r,$ $\displaystyle V\vDash\mbox{ If
}{p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
and }\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
then the decisions are the same.}$

Define for every $\mu<\kappa$

$\displaystyle e\left(\mu\right)=\\{$ $\displaystyle r\in
P_{\kappa}\setminus\mu\colon\forall p\in G_{\mu}\ \forall\xi<\kappa\ \forall
r_{0},r_{1}\geq^{*}r,$ $\displaystyle V\vDash\mbox{ If
}{p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
and }\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
then the decisions are the same.}\\}$

###### Claim 2

Suppose that $\mu\in\left(\zeta,\kappa\right)\cap I$. Then $e(\mu)$ is a dense
open in ${\langle}P_{\kappa}\setminus\mu,\leq^{*}{\rangle}$.555 It is the only
place in the proof where $\mu^{++}-$completeness is used.

Proof. Just note that $\left|G_{\mu}\right|\leq\mu^{+}$ and
${\langle}P_{\kappa}\setminus\mu,\leq^{*}{\rangle}$ is $\mu^{++}-$closed.
$\square$ of the claim.

Given $\mu\in\left(\zeta,\kappa\right)\cap I$ as above, the following set is
also forced to be $\leq^{*}$-dense open in $P\setminus\mu$:

$\displaystyle d\left(\mu\right)=\\{$ $\displaystyle r\in
P\setminus\mu\colon\exists g\in 2^{\mu},\
r\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction\mu=g\\}.$

The $\leq^{*}$-density of $d(\mu)\subseteq P\setminus\mu$ follows as well from
the fact that the direct extension order of $P\setminus\mu$ is more than
$\mu$-closed.

We can now apply the standard fusion argument 3.2. There exists $p\in
P_{\kappa}$ and a club $C\subseteq\kappa$ such that $\min(C)>\zeta$, and, for
every $\mu\in C\cap I$,

$p\restriction{\mu}\Vdash p\setminus\mu\in d(\mu)\cap e(\mu).$

For each $\mu\in C\cap I$, there exists a condition in $G$ of the form
${q_{\mu}}^{\frown}p\setminus\mu$, where $q_{\mu}\in P_{\mu}$, which decides
the value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{\mu}$.
The reason is that $p\restriction{\mu}$ forces that $p\setminus\mu$ is in
$d(\mu)$, and thus the value of
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{\mu}$
is decided by the forcing $P_{\mu}$.

For each such $\mu$, $q_{\mu}$ is an extension of $p\restriction{\mu}$, and
thus there exists a finite set $b_{\mu}\subseteq\mu$ such that at every
$\delta\in\mu\setminus b_{\mu}$, $q_{\mu}(\delta)$ direct extends $p(\delta)$
(as forced by $q_{\mu}\restriction{\delta}$).

The function $\mu\mapsto\max b_{\mu}$ is a regressive function in
$V\left[G\right]$, and its domain is the set $C\cap I$. $C\cap I$ is
stationary in $V$ since $I$ is assumed to be stationary. By theorem 2.2, it is
also stationary in $V\left[G\right]$. Since $\kappa$ is still regular in
$V\left[G\right]$, we can find an unbounded subset $S\subseteq\kappa$ and an
ordinal $\mu^{*}<\kappa$ such that for every $\mu\in S$,
$b_{\mu}\subseteq\mu^{*}$. By increasing $\mu^{*}$, we can assume that it
belongs to $C\cap I$.

Now, shrink $S$ further to stabilize the function $\mu\mapsto
q_{\mu}\restriction{\mu^{*}}$. This is possible since $S$ is unbounded in
$\kappa$, and $q_{\mu}\restriction{\mu^{*}}$ is a condition in $P_{\mu^{*}}$
which has a small cardinality (and, again, $\kappa$ is inaccessible in
$V\left[G\right]$).

So we can assume that there exists a condition $q^{*}\in P_{\mu^{*}}$, such
that for every $\mu\in S$, there exists some direct extension $r_{\mu}\in
P\setminus\mu^{*}$ of $p\setminus\mu^{*}$, such that–

${q^{*}}^{\frown}r_{\mu}\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{\mu}$

and ${q^{*}}^{\frown}r_{\mu}\in G$.

Recall that $\mu^{*}\in C\cap I$ and the condition $p$ obtained by fusion
above also satisfies that $p\restriction{\mu^{*}}\Vdash p\setminus\mu^{*}\in
e(\mu^{*})$. Since $q^{*}\geq p\restriction{\mu^{*}}$, the condition $q^{*}$
forces that, for every $\xi<\kappa$, any pair of direct extensions of
$p\setminus{\mu^{*}}$ which decide
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{\xi}$,
decide this initial segment the same way.

It follows that ${q^{*}}^{\frown}p\setminus\mu^{*}$ decides
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ entirely, and
forces it to be the following function of $V$:

$\displaystyle h=\bigcup_{\mu\in\kappa\setminus\mu^{*}}\\{g\in 2^{\mu}\colon$
$\displaystyle\mbox{there exists a }P_{\mu^{*}}\mbox{-name for an
extension}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}\geq^{*}p\setminus\mu^{*},$
$\displaystyle\mbox{such that
}{q^{*}}^{\frown}r\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction{\mu}=g\\}$

which is a contradiction. $\square$

###### Remark 3.3

Lemma 3.2 implies that $V[G]\vDash 2^{\kappa}=\kappa^{+}$. Indeed, assume that
$A$ is a subset of $\kappa$ in $V[G]$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}$ be a
$P_{\kappa}$-name for it. For every singular $\beta<\kappa$, define in
$V^{P_{\beta}}$ the set–

$F(\beta)=\\{q\in P_{\kappa}\setminus\beta\colon\exists
A_{\beta}\subseteq\beta,\
q\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\beta=A_{\beta}\\}.$

Note that $\beta$ is singular and thus $F(\beta)$ is $\leq^{*}$-dense open.
Let $p^{*}\in G$ and $C\subseteq\kappa$ be such that for every singular
$\beta\in C$, $p^{*}\upharpoonright\beta\Vdash p^{*}\setminus\beta\in
F(\beta)$. Then there exists a $P_{\beta}$-name
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}_{\beta}$ for a
subset of $\beta$, such that
$p^{*}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\beta=(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}_{\beta})_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}\upharpoonright
P_{\beta}}$. Then
$A=(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}})_{G}$ can
be computed in $V[G]$ from the sequence
${\langle}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}_{\beta}\colon\beta\in
C{\rangle}$. By using canonical names for bounded subsets of $\kappa$, and by
GCH in $V$, there are at most $\kappa^{+}$ such sequences. So there are at
most $\kappa^{+}-$many subset of $\kappa$ in $V[G]$.

The situation with higher cardinals was clarified in [4]. The following was
shown basically in [4], 4.11:

###### Lemma 3.4

$P_{\kappa}$ does not add fresh unbounded subsets to $\kappa^{+}$, or to any
cardinal $\lambda$ of $V$ with $\mbox{cf}\left(\lambda\right)>\kappa$.

Proof. The proof is a variation of the proof of theorem 3.1, and it basically
appears in [4]. Assume that
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ is a
$P_{\kappa}$-name for the characteristic function of a fresh unbounded subset
of $\lambda$. Divide into cases as in the proof of theorem 3.1. Case $1$
remains the same. Case $2$ is simplified, since the sets $d(\mu)$ is no longer
required. Indeed, in the notations of the proof of theorem 3.1, assume that
$p\in P_{\kappa}$ is a condition and $C\subseteq\kappa$ is a club, such that
for every $\mu\in C\cap I$,

$p\restriction\mu\Vdash p\setminus\mu\in e(\mu)$

Now, work in $V\left[G\right]$. For each $\xi<\lambda$, let $q_{\xi}\in
P_{\kappa}$ be an extension of $p$ which decides
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction\xi$.
Let $b_{\xi}\subseteq\kappa$ be a finite set such that, for every $\alpha\in
b_{\xi}$, $q_{\xi}\restriction_{\alpha}\Vdash
q_{\xi}(\alpha)\geq^{*}p(\alpha)$. Let $\mu_{\xi}<\kappa$ be an upper bound on
$b_{\xi}$. Since $\mbox{cf}(\lambda)>\kappa$ in in $V$, the same holds true in
$V\left[G\right]$ as well (by the same proof as in corollary 2.6). Thus, there
exists $\mu^{*}\in C\cap I$ such that, for an unbounded $S\subseteq\lambda$,
$\mu_{\xi}<\mu^{*}$. By shrinking $S$, we can assume that, for some $q^{*}\in
P_{\mu^{*}}$, $q_{\xi}\restriction_{\mu^{*}}=q^{*}$. Then $q^{*}$ satisfies
that, for every $\xi\in S$, there exists a direct extension $r_{\xi}\in
P\setminus\mu^{*}$ of $p\setminus\mu^{*}$, such that
${q^{*}}^{\frown}r_{\xi}\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\restriction\xi$.
Now, as in the proof of case $2$ in theorem 3.1, the condition
${q^{*}}^{\frown}p\setminus\mu^{*}$ forces
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ to be the
following function of $V$:

$\displaystyle h=\bigcup_{\xi<\kappa^{+}}\\{g\in 2^{\mu}\colon$
$\displaystyle\mbox{there exists a }P_{\mu^{*}}\mbox{-name for an
extension}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}\geq^{*}p\setminus\mu^{*},$
$\displaystyle\mbox{such that
}{q^{*}}^{\frown}r\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}\upharpoonright{\xi}=g\\}$

which is a contradiction.
$\square$

We finish with an application for iterations of Prikry type forcings with the
nonstationary support. The referee of [9] asked if such an iteration, below a
cardinal $\kappa$, can add new measurable cardinals below $\kappa$. We show
that the answer is negative666The answer for the same question in the full or
Easton support is known to be negative, see [3]..

###### Theorem 3.5

Assume GCH and let $\kappa$ be an inaccessible cardinal.
Let $P_{\kappa}$ be a non-stationary support iteration of Prikry-type forcings
satisfying the conditions from the beginning of the section. Assume:

$(*)$ For every Mahlo cardinal $\alpha<\kappa$ which is not measurable in $V$,
${\langle}P_{\kappa}\setminus\alpha,\leq^{*}{\rangle}$ is $\alpha^{+}-$closed.
Let $\lambda$ be a cardinal such that
$\forall\tau<\lambda(|P_{\tau}|<\lambda)$. Let $G\subseteq P_{\kappa}$ be
generic over $V$. Then If $\lambda$ is measurable in $V[G]$, it was already
measurable in $V$.
Furthermore, if the assumption $(*)$ is strengthened to–

$(**)$ For every Mahlo cardinal $\alpha<\kappa$,
${\langle}P_{\kappa}\setminus\alpha,\leq^{*}{\rangle}$ is $\alpha^{+}-$closed.
then $\lambda$ is measurable in $V[G]$ if and only if it is measurable in $V$.

Proof. Assume first that $(**)$ holds and $\lambda$ is a measurable cardinal
in $V$. If $\lambda>\kappa$, the Levy-Solovay Theorem [11] shows that
$\lambda$ is measurable in $V[G]$. If $\lambda\leq\kappa$, standard arguments
show that $\lambda$ remains measurable in $V\left[G\upharpoonright
P_{\lambda}\right]$ (see [5]). If $\lambda<\kappa$, the forcing
${\langle}P_{\kappa}\setminus\lambda,\leq^{*}{\rangle}$ is
$\lambda^{+}$-closed, so $P_{\kappa}\setminus\lambda$ does not add subsets to
$\lambda$ and $\lambda$ remains measurable in $V[G]$. Thus, let us concentrate
on the other direction assuming $(*)$.

We first recall Lemma $2.1$ in [4]: given a forcing notion which does not add
new fresh unbounded subsets to cardinals of $V$ in the interval
$\left[\kappa,\left(2^{\kappa}\right)^{V}\right]=\left[\kappa,\kappa^{+}\right]$,
every $\kappa$-complete ultrafilter in the generic extension extends a
$\kappa$-complete ultrafilter from $V$. Assume now that $\alpha$ is measurable
in $V\left[G\right]$, and let $W\in V\left[G\right]$ be a nontrivial
$\kappa$-complete ultrafilter on $\alpha$. If $\alpha\geq\kappa$, then by the
results in this section, $P_{\kappa}$ does not add fresh unbounded subsets to
$\alpha$, $\alpha^{+}$, and thus $W\cap V\in V$ by Lemma $2.1$ in [4]. Thus,
assume that $\alpha<\kappa$, and assume that $\alpha$ is not measurable in
$V$. So, $\alpha$ is a Mahlo cardinal in $V$. If the forcing
${\langle}P_{\kappa}\setminus\alpha,\leq^{*}{\rangle}$ is
$\alpha^{++}-$closed, then $W\in V\left[G\restriction P_{\alpha}\right]$.
However we assumed only that
${\langle}P_{\kappa}\setminus\alpha,\leq^{*}{\rangle}$ is $\alpha^{+}-$closed.
In this case $W$ need not be in $V\left[G\restriction P_{\alpha}\right]$.
Proceed then as follows. Work in $V\left[G\restriction P_{\alpha}\right]$. Let
${\langle}A_{i}\mid i<\alpha^{+}{\rangle}$ be an enumeration of all subsets of
$\alpha$ (such enumeration exists by applying remark 3.3 on $P_{\alpha}$).
Define a $\leq^{*}-$ increasing sequence of conditions ${\langle}p_{i}\mid
i<\alpha{\rangle}$ in $V\left[G\restriction P_{\alpha}\right]$ such that for
every $i<\alpha$,
$p_{i}||A_{i}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{W}}}$.
Set $W^{\prime}=\\{A_{i}\mid i<\alpha,p_{i}\Vdash
A_{i}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{W}}}\\}$.
Then $W^{\prime}$ will be an $\alpha-$complete ultrafilter over $\alpha$ in
$V\left[G\restriction P_{\alpha}\right]$.

Apply now 3.1 and 3.4 to $P_{\alpha}$. It follows that no fresh subsets are
added to $\alpha,\alpha^{+}$. Now, by Lemma $2.1$ from [4], $W^{\prime}\cap
V\in V$ is a nontrivial $\alpha$-complete ultrafilter over $\alpha$ in $V$. A
contradiction.
$\square$

###### Remark 3.6

  1. 1.

The closure assumptions made on
${\langle}P_{\kappa}\setminus\alpha,\leq^{*}{\rangle}$ are needed.
For example, start with $V=L[U]$, where $U$ is a normal ultrafilter over
$\alpha$. Iterate $Cohen(\beta),\beta<\alpha$. Let $V$ be this model. Then
$\alpha$ is not a measurable in $V$. Force with $Cohen(\alpha)$ over $V$. Then
$\alpha$ will be a measurable in the extension. Here we take $Q_{\beta}$ to be
trivial for every $\beta<\alpha$ and $Q_{\alpha}=Cohen(\alpha)$.

  2. 2.

Also, the assumption $\forall\tau<\lambda(|P_{\tau}|<\lambda)$ is necessary.
Just use the previous example (with $\lambda=\alpha$). $Q_{0}=Cohen(\alpha)$
resurrects measurability of $\alpha$.

  3. 3.

Note that a measurable cardinal in $V$ need not be such in $V[G]$ without
assuming $(*)$. Just use the Prikry forcing or the iteration of such forcings.

##  4 On fresh sets in the Easton support iterations

We start with an easier case of the Easton support iterations with $\kappa$
being a Mahlo cardinal. $\kappa-$c.c. of the forcing will be used to show the
following:

###### Theorem 4.1

Suppose that $\kappa$ is a Mahlo cardinal and $P_{\kappa}$ is an Easton
support iteration. Then no fresh sets are added to $\kappa$.

Proof. Suppose otherwise. Work in $V$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}$ be a name of
such subset.

Define a tree $T$ of possibilities as follows. Fix an increasing enumeration
${\langle}\kappa_{\xi}\mid\xi<\kappa{\rangle}$ of all inaccessible cardinals
below $\kappa$.

For every $\xi<\kappa$, let

$Lev_{\xi}(T)=\\{x\subseteq\kappa_{\xi}\mid\exists p\in P_{\kappa}\quad
p\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\kappa_{\xi}=x\\}.$

Let $x\in Lev_{\alpha}(T),y\in Lev_{\beta}(T)$. Set $x>_{T}y$ iff
$\alpha>\beta$ and $x\cap\kappa_{\beta}=y$.

Then ${\langle}T,<_{T}{\rangle}$ is a $\kappa-$tree, since $\kappa$ is an
inaccessible.

###### Lemma 4.2

${\langle}T,<_{T}{\rangle}$ has a $\kappa-$branch.

Proof. Let ${\langle}x_{\gamma}\mid\gamma<\kappa{\rangle}$ be an enumeration
of $T$.
There is a club $C\subseteq\kappa$ such that for every $\gamma,\delta\in C$
the following hold:

  1. 1.

$\kappa_{\gamma}=\gamma$,

  2. 2.

the level of $x_{\gamma}\geq\gamma$,

  3. 3.

if $\gamma<\delta$, then the level of $x_{\gamma}<\delta$.

For every $\gamma\in C$, pick $p_{\gamma}\in P_{\kappa}$ such that
$p_{\gamma}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\kappa_{\xi_{\gamma}}=x_{\gamma}$,
where $\xi_{\gamma}\geq\gamma$ denotes the level of $x_{\gamma}$.

Now, $\kappa$ is a Mahlo cardinal and an Easton support was used, hence there
is a stationary $S\subseteq C$ such that for every $\gamma,\delta\in S$,
$p_{\gamma}$ and $p_{\delta}$ are compatible.
Take any two $\gamma<\delta$ in $S$. Then
$x_{\delta}\cap\xi_{\gamma}=x_{\gamma}$ due to the compatibility of
$p_{\gamma}$ and $p_{\delta}$.
So, $\\{x_{\gamma}\mid\gamma\in S\\}$ is a $\kappa-$branch.
$\square$

Let now $b=\\{x_{i}\mid i<\kappa\\}$ be a maximal $\kappa-$branch in $T$. For
every $i<\kappa$ fix $p_{i}\in P_{\kappa}$ which witnesses that $x_{i}\in T$.
By the assumption made, $\bigcup_{i<\kappa}x_{i}\not=A$, since this union is
in $V$.
Then for every $i<\kappa$, there is $i^{\prime}\geq i$ such that
$x_{i^{\prime}}$ is a splitting point of $T$.
Denote by $y_{i^{\prime}}$ an immediate successor of $x_{i^{\prime}}$ which is
not in $b$. Let $q_{i^{\prime}}$ be a condition which witnesses that
$y_{i^{\prime}}\in T$.
Let $C\subseteq\kappa$ be a club such that for every $i_{1},i_{2}\in
C,i_{1}<i_{2}$, we have $i_{1}^{\prime}<i_{2}$.

The next lemma provides the desired contradiction, since $P_{\kappa}$
satisfies $\kappa-$c.c.

###### Lemma 4.3

The conditions $\\{q_{i^{\prime}}\mid i\in C\\}$ are pairwise incompatible.

Proof. Let $i_{1}<i_{2}$ be in $C$. Then

$q_{i_{1}^{\prime}}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\kappa_{i_{1}^{\prime}}\not=x_{i_{1}^{\prime}+1}.$

However,

$q_{i_{2}^{\prime}}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\kappa_{i_{1}^{\prime}}=x_{i_{1}^{\prime}+1},$

since $y_{i_{2}^{\prime}}>_{T}x_{i_{2}}>_{T}x_{i_{1}^{\prime}+1}$. This is
possible onle when $q_{i_{1}^{\prime}}$ and $q_{i_{2}^{\prime}}$ are
incompatible.
$\square$

Let us give an example of the Easton support iteration $P_{\kappa}$ which adds
a fresh subset, however we give up here the assumption that
$|\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}|<\kappa$.

Let ${\langle}\kappa_{\beta}\mid\beta<\kappa{\rangle}$ be an increasing
sequence of measurable cardinals above an inaccessible $\kappa$.

Let
${\langle}P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa{\rangle}$
be an Easton support iterations of the Prikry forcings, i.e. for each
$\beta<\kappa$, $Q_{\beta}$ is the Prikry forcing with a normal ultrafilter
over $\kappa_{\beta}$.

Let $G_{\kappa}$ be a generic subset of $P_{\kappa}$. For every
$\beta<\kappa$, let $b_{\beta}$ be the Prikry sequence added by $G_{\kappa}$
to $\kappa_{\beta}$.

###### Lemma 4.4

The set

$A=\\{\alpha<\kappa\mid\text{ the first element of the sequence
}b_{\kappa_{\alpha}}\text{ is }0\\}$

is a fresh subset of $\kappa$.

Proof. Every initial segment of $A$ is in $V$ due to the support used. On the
other hand $A\not\in V$, since every condition in the forcing $P_{\kappa}$
should be bounded in $\kappa$, and so it can be extended to one which forces
$b_{\kappa_{\alpha}}(0)=0$ or to one forcing $b_{\kappa_{\alpha}}(0)\not=0$.
$\square$

Let us turn now to a general case, i.e. we assume only that $\kappa$ is an
inaccessible.

Our aim will be to prove the following :

###### Theorem 4.5

Let $\kappa$ be an inaccessible cardinal and
${\langle}P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa{\rangle}$
be the Easton support iteration of Prikry type forcing notions. Let $I$ be a
stationary subset of $\kappa$ which consists of singular cardinals $\mu$ such
that for every $\gamma<\mu,|P_{\gamma}|<\mu$. Suppose that for every
$\alpha\in I$, $\Vdash_{P_{\alpha}}\langle
P_{\kappa}\setminus{\alpha},\leq^{*}\rangle\mbox{ is
}\alpha^{++}\mbox{-closed.}$
Let $G_{\kappa}\subseteq P_{\kappa}$ be a generic. Then, in $V[G_{\kappa}]$,
there is no fresh subsets of $\kappa$.

###### Remark 4.6

Similar results were proved in [4] for cardinals above $\kappa$. The proof
there is based on the fact that $|P_{\kappa}|=\kappa$ and it is much easier.

Proof. Let $G_{\kappa}$ be a generic subset of $P_{\kappa}$.

We would like to show that there is no fresh subset of $\kappa$ in
$V[G_{\kappa}]$.

Suppose otherwise. Work in $V$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}$ be a name of
such subset and let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ be a name of
the characteristic function of $A$. Fix some $p\in G_{\kappa}$ which forces
this.

Let $\zeta<\kappa$ be an ordinal for which a new subset is added in the
extension from $V$ to $V\left[G\right]$. Such $\zeta$ exists, since the
forcings $Q_{\alpha}$ (for $\alpha<\kappa$) have cardinality below $\kappa$,
and at least one of them is non-trivial. By increasing if necessary, we can
assume that $\zeta$ is a singular cardinal. Then
${\langle}P_{\kappa}\setminus\zeta,\leq^{*}{\rangle}$ is more than
$\zeta$-closed. Thus, there exists a condition $q\in G_{\kappa}\upharpoonright
P_{\zeta}$ which forces that a new subset is added to $\zeta$. For simplicity,
assume that $p\upharpoonright\zeta$ forces this.

Given a condition $r\in P_{\kappa}$, let us denote by $r({\gamma})$ its
$\gamma-$th coordinate,
i.e. $r={\langle}r(\gamma)\mid\gamma<\kappa{\rangle}$.

We divide into two cases as in Theorem 3.1.

Case 1. There exists $\mu\in\left(\zeta,\kappa\right)\cap I$ and a condition
$p^{*}\in P_{\mu}$ which forces that the following property holds:

$\displaystyle\exists p\in G_{\mu}\ \exists s\in P\setminus\mu\forall
r\geq^{*}s\ \exists\xi<\kappa\ \exists r_{0},r_{1}\geq^{*}r,$ $\displaystyle
V\vDash\left({p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi},\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\right)\mbox{,
and the decisions are different.}$

Case 2. For every $\mu\in\left(\zeta,\kappa\right)\cap I$, every condition in
$P_{\mu}$ forces that–

$\displaystyle\forall
p\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\mu}\
\forall s\in P_{\kappa}\setminus\mu\exists r\geq^{*}s\ \forall\xi<\kappa\
\forall r_{0},r_{1}\geq^{*}r,$ $\displaystyle V\vDash\mbox{ If
}{p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
and }\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
then the decisions are the same.}$

The treatment of the first case is exactly as in Theorem 3.1. Let us deal with
the second case.

As in 3.1, we define for every $\mu<\kappa$

$\displaystyle e\left(\mu\right)=\\{$ $\displaystyle r\in
P_{\kappa}\setminus\mu\colon\forall p\in G_{\mu}\ \forall\xi<\kappa\ \forall
r_{0},r_{1}\geq^{*}r,$ $\displaystyle V\vDash\mbox{ If
}{p}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
and }\
{p}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
then the decisions are the same.}\\}$

By Claim 1 of 3.1, it is $\leq^{*}-$dense open subset of
$P_{\kappa}\setminus\mu$, for every $\mu\in(\zeta,\kappa)\cap I$. Again, here
is the only place where $\mu^{++}-$closure of the direct order on
$P_{\kappa}\setminus\mu$ is used.

Given a generic $G_{\kappa}\subseteq P_{\kappa},p\in G_{\kappa}$, define in
$V[G_{\kappa}]$,

$S=\\{\xi<\kappa\mid\xi\text{ ia limit ordinal and }\exists t\geq^{*}p,t\in
G_{\kappa}\text{ such that
}t\upharpoonright\xi\Vdash_{P_{\xi}}t\setminus\xi||_{P_{\kappa}\setminus{\xi}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi)\\}.$

For every $\xi\in S$, fix some $t^{\xi}\in G_{\kappa}$ such that
$t^{\xi}\geq^{*}p$ and
$t^{\xi}\upharpoonright\xi\Vdash_{P_{\xi}}t^{\xi}\setminus\xi||_{P_{\kappa}\setminus{\xi}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi$.
So, there is $p^{\xi}\in G\upharpoonright P_{\xi},p^{\xi}\geq
t^{\xi}\upharpoonright\xi,(p^{\xi})^{\frown}t^{\xi}\setminus\xi\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi=a_{\xi}$,
for some $a_{\xi}\in V$. Then there is a finite $b^{\xi}\subseteq\xi$ such
that $p^{\xi}\setminus b^{\xi}\geq^{*}p\upharpoonright(\xi\setminus b^{\xi})$.

Suppose for a moment that $S$ is stationary in $V[G_{\kappa}]$.
Then we can find a stationary subset $S^{\prime}$ of $S$ and a finite $b$ such
that for every $\xi\in S^{\prime}$, $b^{\xi}=b$.
Now we can freeze $p^{\xi}\upharpoonright\max(b)$. Denote $\max(b)$ by
$\mu^{*}$.

Let $\mu\in I$ be a cardinal above $\mu^{*}$.
Consider the set $e(\mu)$ defined above. It is $\leq^{*}-$dense open subset of
$P_{\kappa}\setminus\mu$, in $V[G_{\mu}]$ above $p\setminus\mu$. In particular
there is $r\in P_{\kappa}\setminus\mu,r\geq^{*}p\setminus\mu$ such that
for every $p^{\prime}\in G_{\mu}$, for every $\xi<\kappa$ and for every
$r_{0},r_{1}\geq^{*}r$

$V\models\text{ If
}p^{\prime}{}^{\frown}r_{0}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi\text{
and
}p^{\prime}{}^{\frown}r_{1}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi\text{
then the decisions are the same}.$

Recall that $t^{\xi}\geq^{*}p$ and
$p^{\xi}\setminus\mu^{*}\geq^{*}p\upharpoonright(\mu^{*},\xi)$, for every
$\xi\in S^{\prime}$. If we were able to conclude from this that
$p^{\xi}\setminus\mu\geq^{*}r$, then it will imply that $A\in V$. However it
need not be the case since the support of $r$ may be bigger than those of
$p\setminus\mu$ and incompatibility may occur on coordinates outside of ${\rm
supp}(p)$.

Let us argue that it is possible to overcome this obstacle.
Work in $V$. Set $p_{0}=p$. Use Lemma 2.5 to find $q\geq^{*}p$ and
$\mu_{0}\geq\sup({\rm supp}(p)),\mu_{0}\in I$ such that
$q\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}^{*}\leq\mu_{0}$.
Then, we use the density of $e(\mu_{0})$ to find
$r_{0}\geq^{*}q\setminus\mu_{0}$ such that

$q\upharpoonright\mu_{0}\Vdash_{P_{\mu_{0}}}r_{0}\in e(\mu_{0}).$

Set $p_{1}=q\upharpoonright\mu_{0}{}^{\frown}r_{0}$.
Next, we run the argument above with $p=p_{0}$ replaced by $p_{1}$. Again,
using Lemma 2.5 find $q_{1}\geq^{*}p_{1}$ and $\mu_{1}\geq\sup({\rm
supp}(p_{1})),\mu_{0}+1$,$\mu_{1}\in I$ such that
$q_{1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{\mu}}}^{*}\leq\mu_{1}$,
where $\mu^{*}$ is now defined using $p_{1}$ instead of $p$.
Then, use the density of $e(\mu_{1})$ to find
$r_{1}\geq^{*}q_{1}\setminus\mu_{1}$ such that

$q_{1}\upharpoonright\mu_{1}\Vdash_{P_{\mu_{1}}}r_{1}\in e(\mu_{1}).$

Set $p_{2}=q\upharpoonright\mu_{1}{}^{\frown}r_{1}$.
Continue by induction and define $p_{n},q_{n},r_{n},\mu_{n}$, for every
$n<\omega$.
Finally set $\mu_{\omega}=\bigcup_{n<\omega}\mu_{n}$ and
$p_{\omega}=\bigcup_{n<\omega}p_{n}=q_{\omega}=\bigcup_{n<\omega}q_{n}$. Then,
for every $n<\omega$,

$p_{\omega}\upharpoonright\mu_{n}\Vdash p_{\omega}\setminus\mu_{n}\in
e(\mu_{n}),$

since $p_{\omega}\setminus\mu_{n}\geq^{*}r_{n}\in e(\mu_{n})$ and $e(\mu_{n})$
is dense open. Also, $\sup({\rm supp}(p_{\omega}))=\mu_{\omega}$.
Pick now a generic $G\subseteq P_{\kappa}$ with $p_{\omega}\in G$. Let
$S_{p_{\omega}}$ be defined as $S$ above only with $p_{\omega}$ replacing $p$.
Assuming its stationarity, define $q^{\xi}\geq^{*}p_{\omega},p^{\xi}\in
G\upharpoonright P_{\xi}$ for $\xi\in S^{\prime}$ exactly as above. Then there
will be a stationary $S^{\prime}\subseteq S$ and $n^{*}<\omega$ such that
$p^{\xi}\upharpoonright(\mu_{n^{*}},\xi)\geq^{*}p_{\omega}\upharpoonright(\mu_{n^{*}},\xi)$,
since $\sup({\rm supp}(p_{\omega}))=\mu_{\omega}$ and a non-direct extension
is used at finitely many places only.
Shrink $S^{\prime}$ further to $S^{\prime\prime}$ and stabilise the value of
the function $\xi\mapsto p^{\xi}\upharpoonright\mu_{n^{*}}^{+}$.
Finally, we use that $p_{\omega}\setminus\mu_{n^{*}}\in e(\mu_{n^{*}})$.

Hence, the following lemma will complete the proof. We prove it for the
initial $p$, but the same argument works for $S_{p_{\omega}}$ or any $S_{x}$
with $x\geq p$.

###### Lemma 4.7

$S$ is stationary in $V[G_{\kappa}]$.

Proof. The argument will be similar to those of 2.2.
Suppose otherwise. Let $C\subseteq\kappa$ be a club disjoint from $S$. Assume
that $p\in G_{\kappa}$ forces this, otherwise replace it by a stronger
condition doing this.
Work in $V$. Pick an elementary submodel $M\preceq H_{\chi}$ such that

  1. 1.

$|M|=\delta<\kappa$,

  2. 2.

$M\cap\kappa=\delta$,

  3. 3.

$\operatorname{cof}(\delta)<\delta$,

  4. 4.

${}^{\operatorname{cof}(\delta)>}M\subseteq M$,

  5. 5.

$\kappa,P_{\kappa},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}},p\in
M$.

Pick a cofinal in $\delta$ sequence ${\langle}\delta_{i}\mid
i<\operatorname{cof}(\delta){\rangle}$ consisting of singulars and with
$\delta_{0}>\operatorname{cof}(\delta)$.

Let $G_{\operatorname{cof}(\delta)+1}\subseteq
P_{\operatorname{cof}(\delta)+1}$ be a generic with
$p\upharpoonright{\operatorname{cof}(\delta)+1}\in
G_{\operatorname{cof}(\delta)+1}$.
Work in $V[G_{\operatorname{cof}(\delta)+1}]$.

Consider

$D^{0}=\\{r\geq p\mid\exists c<\kappa\quad r\Vdash
c=\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\delta_{0})\\}.$

Clearly, $D^{0}$ is a dense open and it belongs to $M$.
Apply Lemma 2.3. Then there will be $\alpha_{0}<\kappa$ and
$q_{0}\geq^{*}p,q\upharpoonright{\operatorname{cof}(\delta)+1}=p\upharpoonright{\operatorname{cof}(\delta)+1}$,
in $M$ such that for every $r\in
P_{\alpha_{0}},r\geq_{P_{\alpha_{0}}}q_{0}\upharpoonright\alpha_{0}$ there is
$r^{\prime}\geq_{P_{\alpha_{0}}}r$ such that
$r^{\prime}{}^{\frown}q_{0}\setminus\alpha_{0}\in D_{0}$. So, for every such
$r^{\prime}$ there is $c(r^{\prime})<\kappa$ such that

$r^{\prime}{}^{\frown}q_{0}\setminus\alpha_{0}\Vdash
c(r^{\prime})=\min(\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}\setminus\delta_{0}).$

Note that $P_{\alpha_{0}}\subseteq M$. Hence all $c(r^{\prime})$’s are in $M$.
Also $|P_{\alpha_{0}}|<\kappa$. Hence their sup is below $\kappa$, and then,
by elementarity, in $M$. Denote it by $c_{0}^{*}$.

Define, for every $\tau<\kappa$,

$D_{\tau}=\\{r\in P\mid r\geq^{*}p\text{ and
}\exists\xi\in\kappa\setminus\tau\quad
r\upharpoonright\xi\Vdash_{P_{\xi}}(r\setminus\xi)||_{P_{>\xi}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi)\\}.$

###### Claim 3

$D_{\tau}$ is $\leq^{*}-$dence open above $p$.

Proof. Set $\tau_{0}=\tau+1$. Consider

$D(\tau_{0})=\\{r\in P_{\kappa}\mid r\perp p\text{ or }(r\geq p\text{ and
}r\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\tau_{0})\\}.$

By Lemma 2.3, there are $\alpha_{0}$ and $q\geq^{*}p$ such that

$q\upharpoonright\alpha_{0}\Vdash\exists
b\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\alpha_{0}}\quad
b^{\frown}q\setminus\alpha_{0}\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\tau_{0}.$

If $\alpha_{0}\leq\tau_{0}$ then we are done. Suppose that
$\alpha_{0}>\tau_{0}$. Consider $D(\alpha_{0})$ and again, using Lemma 2.3,
pick $\alpha_{1}$ and $q_{1}\geq^{*}q$ such that

$q_{1}\upharpoonright\alpha_{1}\Vdash\exists
b\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\alpha_{1}}\quad
b^{\frown}q_{1}\setminus\alpha_{1}\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha_{0}.$

If $\alpha_{1}\leq\alpha_{0}$, then $q_{1}\in D(\alpha_{0})$ and we are done.
If $\alpha_{1}>\alpha_{0}$, then continue and define in the same fashion
$\alpha_{2},q_{2}$ etc.
Suppose that the process continues infinitely many steps. Then we will have

$\alpha_{0}<\alpha_{1}<...<\alpha_{i}<...\text{ and
}q_{0}\leq^{*}q_{1}\leq^{*}...\leq^{*}q_{i}\leq^{*}...,i<\omega.$

Let $\alpha^{*}=\bigcup_{i<\omega}\alpha_{i}$ and $q^{*}\geq^{*}q_{i}$, for
every $i<\omega$. Let $G\subseteq P_{\kappa}$ be a generic with $q^{*}\in G$.
For every $i<\omega$, let $G_{\alpha_{i}}=G\upharpoonright P_{\alpha_{i}}$ and
$G^{*}=G\upharpoonright P_{\alpha^{*}}$.
Now, for every $i<\omega$, there are $r_{i+1}\in G_{\alpha_{i+1}}$ and
$a_{i}\in V$ such that

$r_{i+1}{}^{\frown}q_{i+1}\setminus\alpha_{i+1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha_{i}=a_{i}.$

Then,

$r_{i+1}{}^{\frown}q^{*}\setminus\alpha_{i+1}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha_{i}=a_{i}.$

Set $a=\bigcup_{i<\omega}a_{i}$. Then, in $V[G_{\alpha^{*}}]$,
$q^{*}\setminus\alpha^{*}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha^{*}=a$,
since of $r_{i}$’s is in $G_{\alpha^{*}}$. So, there is $r\in G_{\alpha^{*}}$
such that

$r^{\frown}q^{*}\setminus\alpha^{*}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha^{*}=a.$

Hence,

$r\Vdash_{P_{\alpha^{*}}}(q^{*}\setminus\alpha^{*}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha^{*}).$

The only requirement on $G_{\alpha^{*}}$ is that
$q^{*}\upharpoonright\alpha^{*}\in G_{\alpha^{*}}$. Hence,

$q^{*}\upharpoonright\alpha^{*}\Vdash
q^{*}\setminus\alpha^{*}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\alpha^{*}.$

So, $q^{*}\in D_{\tau}$.
$\square$ of the claim.

Consider $D_{c_{0}^{*}}$.

It is in $M$, as well. So, inside $M$, we can pick $\xi_{0}\geq
c_{0}^{*},\xi_{0}\in I$ and $t_{0}\geq^{*}p$ such that

$t_{0}\upharpoonright\xi_{0}\Vdash_{P_{\xi_{0}}}t_{0}\setminus\xi_{0}||_{P_{\kappa}\setminus{{\xi_{0}}}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi_{0})\\}.$

We continue the same process only with $c_{0}$ replaced by
$\max(\xi_{0},\delta_{1})$ and $p$ by $q_{0}$.

At limit stages of the construction and at the stage
$\operatorname{cof}(\delta)$ itself we would like to put ${\langle}q_{i}\mid
i<\operatorname{cof}(\delta){\rangle}$ into a single condition $q^{*}$. This
is possible since $\delta_{0}>\operatorname{cof}(\delta)$, and so, we have
enough completeness to proceed.

We have
$p\upharpoonright{\operatorname{cof}(\delta)+1}^{\frown}q^{*}\Vdash\delta\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{C}}}$,
since $C$ is forced to be closed by $p$.

Let $G\subseteq P_{\kappa}$ be a generic with $q^{*}\in G$. Set
$G_{\delta}=G\upharpoonright P_{\delta}$. Then for every
$i<\operatorname{cof}(\delta)$, $q_{i}\upharpoonright\delta\in G_{\delta}$.
Hence, there are $r_{i}\in G_{\xi_{i}},r_{i}\geq q_{i}\upharpoonright\xi_{i}$
and $a_{i}\in V$ such that

$r_{i}{}^{\frown}q_{i}\setminus\xi_{i}\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi_{i}=a_{i}.$

Set $a=\bigcup_{i<\operatorname{cof}(\delta)}a_{i}$. Then, in V[G],
$A\cap\delta=a$, since each $r_{i}{}^{\frown}q_{i}\setminus\xi_{i}$ is in $G$.
Remember that the only requirement on a generic set $G$ was that $q^{*}$
belongs to it. Hence, back in $V$,

$q^{*}\upharpoonright\delta\Vdash_{P_{\delta}}q^{*}\setminus\delta||_{P_{>\delta}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\delta.$

So, $\delta\in S$ and as it was shown above $\delta\in C$ as well.
Contradiction.
$\square$

##  5  No fresh subsets of $\kappa$ in the full support

Our aim will be to prove the following :

###### Theorem 5.1

Let $\kappa$ be an inaccessible cardinal and
${\langle}P_{\alpha},\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa{\rangle}$
be the full iteration of Prikry type forcing notions. Suppose that for every
$\beta<\kappa$, for every
$x,y,z\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}$,
if
$z\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}x,y$
and $x,y$ are compatible according to
$\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}$,
then they are compatible according to
$\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}$,
i.e. there is
$e\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta},e\geq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}x,y$.777
Note that if $\leq=\leq^{*}$, then this holds trivially. Prikry, Magidor,
Radin forcings, their supercompact versions, etc., have this property.
Actually, any reasonable forcing of this type has this property.

Let $G_{\kappa}\subseteq P_{\kappa}$ be a generic. Then, in $V[G_{\kappa}]$,
there is no fresh subsets of $\kappa$.

###### Remark 5.2

More restrictive results were proved in [4] for cardinals above $\kappa$. The
present proof can be easily modified for higher cardinals.

Proof. Let $G_{\kappa}$ be a generic subset of $P_{\kappa}$.

We would like to show that there is no fresh subset of $\kappa$ in
$V[G_{\kappa}]$.

Suppose otherwise. Work in $V$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}$ be a name of
such subset and let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}$ be a name of
the characteristic function of $A$. Fix some $p\in G_{\kappa}$ which forces
this.

Let $\zeta<\kappa$ be an ordinal for which a new subset is added in the
extension from $V$ to $V\left[G\right]$. Such $\zeta$ exists, since the
forcings $Q_{\alpha}$ (for $\alpha<\kappa$) have cardinality below $\kappa$,
and at least one of them is non-trivial. By increasing if necessary, we can
assume that $\zeta$ is a singular cardinal. Then
${\langle}P_{\kappa}\setminus\zeta,\leq^{*}{\rangle}$ is more than
$\zeta$-closed. Thus, there exists a condition $q\in G_{\kappa}\upharpoonright
P_{\zeta}$ which forces that a new subset is added to $\zeta$. For simplicity,
assume that $p\upharpoonright\zeta$ forces this.

Let $I$ be a subset of $\kappa$ which consists of singular cardinals $\tau$
such that for every $\rho<\tau,|P_{\rho}|<\tau$.

As before, we divide into two cases:

Case 1. There exists $\mu\in\left(\zeta,\kappa\right)\cap I$ and a condition
$p^{*}\in P_{\mu},p^{*}\geq p\upharpoonright\mu$ which forces that the
following property holds:

$\displaystyle\exists
p^{\prime}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\mu}\
\exists s\in P_{\kappa}\setminus\mu,s\geq p\setminus\mu\forall
r\geq^{*}s,r({\mu})=s({\mu}),r({\mu^{+}})=s({\mu^{+}})\ \exists\xi<\kappa\
\exists r_{0},r_{1}\geq^{*}r,$ $\displaystyle
r_{0}({\mu})=r_{1}({\mu})=r({\mu}),r_{0}(\mu^{+})=r_{1}({\mu^{+}})=r({\mu^{+}}),$
$\displaystyle
V\vDash\left({p^{\prime}}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi},\
{p^{\prime}}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\right)\mbox{,
and the decisions are different,}$

where $G_{\mu}$ denotes the canonical name for the generic set for $P_{\mu}$.
By extending $p^{*}$, if necessary, we can decide the value of $p^{\prime}$ in
the statement above, and thus assume that $p^{*}\geq p^{\prime}$. Let
$\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}$ be a
$P_{\mu}$-name for $s$ from the above property, and assume that this is forced
by $p^{*}$.

Note that here we do not assume $\mu^{++}-$completence of the direct order.
Additional requirements are included in order to compensate this. Still, the
treatment of this case repeats completely Case 1 of 3.1.

Case 2. For every $\mu\in\left(\zeta,\kappa\right)\cap I$, every condition in
$P_{\mu}$ stronger than $p\upharpoonright\mu$ forces ( and so,
$p\upharpoonright\mu$ forces) that–

$\displaystyle\forall
p^{\prime}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\mu}\
\forall s\in P\setminus\mu,s\geq p\setminus\mu\exists
r\geq^{*}s,r({\mu})=s({\mu}),r({\mu^{+}})=s({\mu^{+}})\ \forall\xi<\kappa\
\forall r_{0},r_{1}\geq^{*}r,$ $\displaystyle
r_{0}({\mu})=r_{1}({\mu})=r({\mu}),r_{0}({\mu^{+}})=r_{1}({\mu^{+}})=r({\mu^{+}}),$
$\displaystyle V\vDash\mbox{ If
}{p^{\prime}}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}}\cap{\xi}\mbox{
and }\
{p^{\prime}}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}}\cap{\xi}\mbox{
then the decisions are the same.}$

For every $\mu\in\left(\zeta,\kappa\right)\cap I$, define (in $V[G_{\mu}]$)

$\displaystyle e^{\prime}\left(\mu\right)=\\{$ $\displaystyle r\in
P_{\kappa}\setminus\mu\colon\forall p^{\prime}\in G_{\mu}\ \forall\xi<\kappa\
\forall r_{0},r_{1}\geq^{*}r,$ $\displaystyle
r_{0}({\mu})=r_{1}({\mu})=r({\mu}),r_{0}({\mu^{+}})=r_{1}({\mu^{+}})=r({\mu^{+}}),$
$\displaystyle V\vDash\mbox{ If
}{p^{\prime}}^{\frown}r_{0}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
and }\
{p^{\prime}}^{\frown}r_{1}\parallel{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{f}}}}\restriction{\xi}\mbox{
then the decisions are the same.}\\}$

###### Claim 4

For every $\mu\in\left(\zeta,\kappa\right)\cap I$,

$e^{\prime}(\mu)$ is $\leq^{*}$-dense in $P\setminus\mu$ (above
$p\setminus\mu$).888 It need not be open.

Proof. Note that $\left|G_{\mu}\right|\leq\mu^{+}$, the forcing
${\langle}P_{\kappa}\setminus\mu^{++}{\rangle}$ is $\mu^{++}-$closed and the
coordinates $\mu,\mu^{+}$ do not change. Hence, for every given $r^{\prime}\in
P_{\kappa}\setminus\mu$, we can construct a $\leq^{*}-$increasing sequence of
a length $|G_{\mu}|$ of conditions stronger than $r^{\prime}$, which takes
care of each $p^{\prime}\in G_{\mu}$. Then its upper bound will be in
$e(\mu)$.
$\square$ of the claim.

Given a generic $G_{\kappa}\subseteq P_{\kappa},p\in G_{\kappa}$, define in
$V[G_{\kappa}]$,

$S=\\{\xi<\kappa\mid\xi\text{ ia limit ordinal and }\exists t\geq^{*}p,t\in
G_{\kappa}\text{ such that
}t\upharpoonright\xi\Vdash_{P_{\xi}}t\setminus\xi||_{P_{\kappa}\setminus{\xi}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi)\\}.$

For every $\xi\in S$, fix some $t^{\xi}\in G_{\kappa}$ such that
$t^{\xi}\geq^{*}p$ and
$t^{\xi}\upharpoonright\xi\Vdash_{P_{\xi}}t^{\xi}\setminus\xi||_{P_{\kappa}\setminus{\xi}}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi$.
So, there is $p^{\xi}\in G\upharpoonright P_{\xi},p^{\xi}\geq
t^{\xi}\upharpoonright\xi,(p^{\xi})^{\frown}t^{\xi}\setminus\xi\Vdash\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi=a_{\xi}$,
for some $a_{\xi}\in V$. Then there is a finite $b^{\xi}\subseteq\xi$ such
that $p^{\xi}\setminus b^{\xi}\geq^{*}p\upharpoonright(\xi\setminus b^{\xi})$.

The argument of Lemma 4.7 applies without changes in the present situation,
and shows that $S$ is stationary (in $V[G_{\kappa}]$).

Then we can find a stationary subset $S^{\prime}$ of $S$ and a finite $b$ such
that for every $\xi\in S^{\prime}$, $b^{\xi}=b$.
Now we can freeze $p^{\xi}\upharpoonright\max(b)^{++}$.
If the following holds,

$(\aleph)$: _For every $\beta<\kappa$, for every
$s,t,r\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}$,
if
$s\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}t,r$,
then there is
$e\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta},e\geq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}t,r$_.999
For example if $P_{\kappa}$ is the Magidor iteration of Prikry forcings. Also
note that we do not to split into Cases 1,2 in this type of situation.

then it is easy to conclude that $A\in V$, which is impossible.
Deal with the general situation. Let $\mu\in I$ be a cardinal
$\geq\max(b)$.Freeze $p^{\xi}\upharpoonright\mu^{++}+1$.
Then for every $\xi,\rho\in S^{\prime}$,

$p^{\xi}(\mu)=p^{\rho}(\mu)\text{ and }p^{\xi}(\mu^{+})=p^{\rho}(\mu^{+}).$

There is $s\in G_{\kappa},s\geq p,s\upharpoonright\mu^{++}\geq
p^{\xi}\upharpoonright\mu^{++},\xi\in S^{\prime}$, such that

$s\Vdash\forall\xi\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{S}}}^{\prime}\exists
t^{\xi}\geq^{*}p,t^{\xi}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\kappa}\quad\exists
p^{\xi}\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{G}}}_{\kappa}\upharpoonright
P_{\xi}\quad b_{\xi}=b\text{ which has the properties above}.$

Consider the set $e^{\prime}(\mu)$ defined above. By the claim, it is
$\leq^{*}-$dense subset of $P_{\kappa}\setminus\mu$, in $V[G_{\mu}]$ above
$p\setminus\mu$. In particular there is $r\in
P_{\kappa}\setminus\mu,r\geq^{*}s\setminus\mu$ such that
for every $p^{\prime}\in G_{\mu}$, for every $\xi<\kappa$ and for every
$r_{0},r_{1}\geq^{*}r$

$V\models\text{ If
}p^{\prime}{}^{\frown}r_{0}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi\text{
and
}p^{\prime}{}^{\frown}r_{1}||\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{A}}}\cap\xi\text{
then the decisions are the same}.$

Pick now a generic $G^{\prime}\subseteq P_{\kappa}$ with
$G_{\mu}=G^{\prime}\upharpoonright P_{\mu}$ and
$s\upharpoonright\mu^{\frown}r\in G^{\prime}$.

We assumed the following,

$(\beth)$: _For every $\beta<\kappa$, for every
$x,y,z\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}$,
if
$z\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}x,y$
and $x,y$ are compatible according to
$\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}$,
then they are compatible according to
$\leq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}^{*}}$,
i.e. there is
$e\in\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta},e\geq_{\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{Q}}}_{\beta}}^{*}x,y$._

Now, $P_{\kappa}$ is a full support iteration. So, $(\beth)$ implies -

$(\gimel)$: _For every $\beta<\kappa$, for every $u,v,w\in P_{\kappa}$, if
$w\leq_{P_{\kappa}}^{*}u,v$ and $u,v$ are compatible according to
$\leq_{P_{\kappa}}$, then they are compatible according to
$\leq_{P_{\kappa}}^{*}$.101010 It need not be the case for Easton or non-
stationary support, since then the support of $w$ may be strictly smaller than
those of $u$, $v$, and $u,v$ may disagree on a common coordinate outside the
support of $w$. _

It follows, in this case, that $A\in V$. Contradiction.
$\square$

## References

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