# Forcing Axioms and Definabilty of the Nonstationary Ideal on $\omega_{1}$

Stefan Hoffelner111The first and the third author were supported by funded by
the Deutsche Forschungsgemeinschaft (DFG German Research Foundation) under
Germanys Excellence Strategy EXC 2044 390685587, Mathematics Münster:
Dynamics-Geometry-Structure., Paul Larson222The second author was supported in
part by NSF research grants DMS-1201494 and DMS-1764320., Ralf Schindler and
Liuzhen Wu

(5.6.2022)

###### Abstract

We show that under $\mathsf{BMM}$ and “there exists a Woodin cardinal$"$, the
nonstationary ideal on $\omega_{1}$ can not be defined by a $\Sigma_{1}$
formula with parameter $A\subset\omega_{1}$. We show that the same conclusion
holds under the assumption of Woodin’s $(\ast)$-axiom. We further show that
there are universes where $\mathsf{BPFA}$ holds and $\hbox{NS}_{\omega_{1}}$
is $\Sigma_{1}(\omega_{1})$-definable. Last we show that if the canonical
inner model with one Woodin cardinal $M_{1}$ exists, there is a universe where
$\hbox{NS}_{\omega_{1}}$ is saturated, $\Sigma_{1}(\omega_{1})$-definable and
$\mathsf{MA_{\omega_{1}}}$ holds.

##  1 Introduction

This article deals with the possibility of a (boldface)
$\bf{\Delta_{1}}$-definition (over $H(\omega_{2})$) of the nonstationary ideal
on $\omega_{1}$ in the presence of various forcing axioms. As we shall see,
stronger assumptions rule out the existence of such
$\bf{\Sigma}_{1}$-definitions, whereas weaker assumptions are consistent with
such $\bf{\Sigma}_{1}$-definitions, even in the presence of
$\hbox{NS}_{\omega_{1}}$ being saturated.

The main results are as follows.

######  Theorem 1.1.

Assume $\mathsf{BMM}$ and that there exists a Woodin cardinal. Then for no
$\Sigma_{1}$-formula $\varphi(v_{0},v_{1})$ and no parameter
$A\subset\omega_{1}$ does it hold that

$\forall S\in P(\omega_{1})(S\text{ is stationary
}\Leftrightarrow\varphi(S,A)).$

######  Theorem 1.2.

Assume Woodin’s axiom $(\ast)$, then there is no $A\subset\omega_{1}$ and no
$\Sigma_{1}$-formula $\varphi(-,A)$ in the language of set theory such that

$\forall S\in P(\omega_{1})\,(S\text{ is stationary
}\Leftrightarrow\varphi(S,A)).$

In contrast to these two impossibility results we also obtain two theorems
which show that under weaker assumptions, $\Sigma_{1}$ definitions of
$\hbox{NS}_{\omega_{1}}$ are possible.

######  Theorem 1.3.

There is a universe in which $\mathsf{BPFA}$ holds and
$\hbox{NS}_{\omega_{1}}$ is $\Sigma_{1}(\omega_{1})$-definable.

######  Theorem 1.4.

Assume that the canonical inner model with one Woodin cardinal $M_{1}$ exists.
Then there is a generic extension of $M_{1}$ where $\hbox{NS}_{\omega_{1}}$ is
saturated and $\Sigma_{1}(\omega_{1})$-definable and
$\mathsf{MA_{\omega_{1}}}$ holds.

The paper is organized as follows. We will prove the theorems in the order
stated above, thus we start with the two impossibility results, then follow up
with the two possibility results. The methods and techniques which are used in
this article are quite varied and we will provide only very little preliminary
definitions, instead assuming the reader knows the basics of the stationary
tower forcing (see [7] for an extensive account) and $\mathbb{P}_{\text{max}}$
(see [6] or [14]), as well as the coding technique of A. Caicedo and B.
Velickovic ([1]).

##  2 Impossibility Results

This section collects two results which show that strong assumptions entail
the impossibility of a boldface $\Sigma_{1}$-definition of
$\hbox{NS}_{\omega_{1}}$. We assume that the reader is familiar with Woodin’s
stationary tower and with $\mathbb{P}_{\text{max}}$-forcing.

###  2.1 Impossibility under $\mathsf{BMM}+$“There exists a Woodin Cardinal$"$

The goal of this section is to prove that under $\mathsf{BMM}+$“there exists a
Woodin cardinal$"$, no $\Sigma_{1}$ formula (boldface) can define stationary
subsets of $\omega_{1}$ correctly.

######  Theorem 2.1.

Assume $\mathsf{BMM}$ and the existence of a Woodin cardinal $\delta$. Then
for no parameter $A\subset\omega_{1}$ and for no $\Sigma_{1}$-formula
$\varphi(-,A)$ in the language of set theory, does $\varphi$ define the
stationary subsets of $\omega_{1}$ i.e, we do not have that

$\forall T\in P(\omega_{1})(T\text{ is stationary
}\Leftrightarrow\varphi(T,A)).$

###### Proof.

Assume for a contradiction that there is a $\Sigma_{1}$\- formula $\varphi$
and a set $A\subset\omega_{1}$ such that $\forall T\in P(\omega_{1})(T$ is
stationary iff $\varphi(T,A)$). Let $\delta$ be our Woodin cardinal.

Let

$S_{0}=\\{X\prec H(\omega_{2})\,:\,|X|=\aleph_{1}\land X\text{ is
transitive}\\}$

and let $g$ be $\mathbb{P}_{<\delta}$-generic over $V$ where
$\mathbb{P}_{<\delta}$ is the associated full stationary tower. Let us pick a
generic filter $g$ which contains $S_{0}$, which is possible by the
stationarity of $S_{0}$. As usual we can form the generic elementary embedding
in the universe $V[g]$:

$j:V\rightarrow M\subset V[g]$

for a transitive inner model $M$ of $V[g]$. Membership in the generic filter
$g$ for the stationary tower forcing can be characterized using $j$, namely we
have that

$\forall a(a\in g\leftrightarrow j"\bigcup a\in j(a))$

thus $S_{0}\in g$ yields $j"H(\omega_{2})^{V}\in j(S_{0})$. In particular
$j"H(\omega_{2})^{V}$ is transitive, and as $H(\omega_{2})^{V}$ is the
transitive collapse of $j"H(\omega_{2})^{V}$, we obtain that
$j"H(\omega_{2})^{V}=H(\omega_{2})^{V}$ and that the critical point $crit(j)$
of the elementary embedding $j$ must be $\geq\omega_{2}^{V}$. As
$H(\omega_{2})^{V}\in j(S_{0})$, $|H(\omega_{2})^{V}|=\aleph_{1}$ in $M$, so
$crit(j)=\omega_{2}^{V}$.

We have $P(\omega_{1})\cap V\in M$. Indeed by $\mathsf{BMM}$,
$2^{\aleph_{1}}=\aleph_{2}$, so let $f:\omega_{2}\rightarrow P(\omega_{1})\cap
V$ be a bijection with $f\in V$. Then $j(f)\upharpoonright\omega_{2}^{V}=f$
and $j(f)\in M$ so $P(\omega_{1})\cap V=ran(f)\in M$.

It is a theorem of A. D. Taylor (see [12]) that $\mathsf{MA_{\omega_{1}}}$
implies that $\hbox{NS}_{\omega_{1}}$ is not $\omega_{1}$-dense. As
$P(\omega_{1})\cap V$ has size $|\omega_{2}^{V}|=\aleph_{1}$ in $M$, there is
a stationary set $D\subset\omega_{1}$ in $M$ such that $T\backslash D$ is
stationary for every stationary $T\in P(\omega_{1})\cap V$. By Theorem 2.5.8
of [7], $M^{<\delta}\cap V[g]\subset M$, which gives us that $V[g]$ is a
stationary set preserving extension of $V$. Further it is still true in $V[g]$
that $D$ is stationary and $T\backslash D$ is stationary for all
$T\notin\hbox{NS}_{\omega_{1}}^{V}$.

In the next step we use the ordinary club shooting forcing
$\mathbb{P}_{\omega_{1}\backslash D}$ over $V[g]$ to shoot a club through the
complement of $D$. The forcing will not destroy any stationary subsets from
$V\cap P(\omega_{1})$:

######  Fact 2.2.

If $h$ denotes a generic filter for $\mathbb{P}_{\omega_{1}\backslash D}$ over
$V[g]$, then if $T\in P(\omega_{1})\cap V$ is stationary in $V$ then it will
remain stationary in $V[g,h]$.

###### Proof.

Fix a stationary (in $V$) $T\in P(\omega_{1})\cap V$ and let
$p\in\mathbb{P}_{\omega_{1}\backslash D}$ be a condition and $\tau$ be a name
in $V[g]^{\mathbb{P}_{\omega_{1}\backslash D}}$ such that $p\Vdash\tau$ is a
club in $\omega_{1}$. We shall find a $q^{\prime}<p$ such that
$q^{\prime}\Vdash\tau\cap T\neq\emptyset$.

As $T\backslash D$ is stationary in $V[g]$, we fix a sufficiently large
regular $\theta$ and may pick a countable $X\prec H(\theta)^{V[g]}$ such that
$p,\mathbb{P}_{\omega_{1}\backslash D}$ and $\tau$ are elments of $X$ and
which satisfies $\alpha=X\cap\omega_{1}\in T\backslash D$. It is
straightforward to construct a condition $q\in\mathbb{P}_{\omega_{1}\backslash
D}$, $q<p$ with domain $\alpha$ such that for every dense
$D\subset\mathbb{P}_{\omega_{1}\backslash D}$, $D\in X$ there is a
$\xi<\alpha$ such that $q\upharpoonright\xi\in D\cap X$. Finally
$q^{\prime}:=q\cup\\{(\alpha,\alpha)\\}$ is as desired.

∎

So $V[g,h]$ is a stationary set preserving extension of $V$. But now by our
hypothesis and by elementarity of $j:V\rightarrow M$ we get that

$M\models\varphi(D,A)$

and hence

$V[g]\models\varphi(D,A)$

as $\varphi$ is $\Sigma_{1}$, and consequentially

$V[g,h]\models\varphi(D,A).$

In $V[g,h]$ the set $D$ is nonstationary, thus

$V[g,h]\models\exists D(D\text{ is nonstationary }\land\varphi(D,A)).$

This statement is $\Sigma_{1}$ with parameter $A\subset\omega_{1}$ in the
language of set theory, and as $\mathsf{BMM}$ is assumed to hold true in $V$
we conclude that

$V\models\exists D(D\text{ is nonstationary }\land\varphi(D,A))$

which is a contradiction. ∎

###  2.2 Impossibility under $(\ast)$

Our next goal is to derive the same conclusion from Woodin’s
$(\ast)$-principle. Recall that the $(\ast)$-principle states that

  * •

$AD$ holds in $L(\mathbb{R})$ and

  * •

$L(P(\omega_{1}))$ is a $\mathbb{P}_{\text{max}}$-generic extension of
$L(\mathbb{R})$.

It has been shown very recently by the third author and D. Aspero that
$\mathsf{MM^{++}}$ implies $(\ast)$, solving a long-standing open question.
Its proof paved the way for a third impossibility result, namely that under
$\mathsf{MM}$, there is no $A\subset\omega_{1}$ and no $\Sigma_{1}$-formula
which defines stationarity. The proof is due to the third author and Xiuyuan
Sun and will appear soon (see [10]).

######  Theorem 2.3.

Assume $(\ast)$, then there is no $A\subset\omega_{1}$ and no
$\Sigma_{1}$-formula $\varphi(-,A)$ in the language of set theory such that

$\forall T\in P(\omega_{1})\,(T\text{ is stationary
}\Leftrightarrow\varphi(T,A)).$

###### Proof.

Let $V=L(\mathbb{R})[g]$, where $g$ is $\mathbb{P}_{\text{max}}$-generic over
$L(\mathbb{R})$. Suppose for a contradiction that there is a
$\mathbb{P}_{\text{max}}$-name for a subset of $\omega_{1}$, $\dot{A}$, a
$\Sigma_{1}$-formula $\varphi(-,\dot{A})$ and a
$\mathbb{P}_{\text{max}}$-condition $p=(p,J,b)$ such that

$p\Vdash\forall S(S\text{ is stationary iff }\varphi(S,\dot{A}))$

By [8], the conclusion of the theorem is true for every $A\in
P(\omega_{1})^{L(\mathbb{R})}$, so we may assume that $\dot{A}^{g}=A\in
P(\omega_{1})^{V}\backslash L(\mathbb{R})$. It is well-known that any such
$A\subset\omega_{1}$ gives rise to a $\mathbb{P}_{\text{max}}$-generic filter
$g_{A}:=\\{(p,I,a)\in\mathbb{P}_{\text{max}}\,:\,$ $p$ generically iterates to
some $p_{\omega_{1}}=(p_{\omega_{1}},\hbox{NS}_{\omega_{1}}^{V},A)\\}$ and
$L(\mathbb{R})[g_{A}]=L(\mathbb{R})[g]$. Hence we assume without loss of
generality that the $\dot{A}$ above is the name for such a distinguished
subset of $\omega_{1}$ which gives rise to the
$\mathbb{P}_{\text{max}}$-generic filter.

Claim Let $q<p$, $q=(q,NS^{q}_{\omega_{1}},c)$ such that $i:p\rightarrow
p^{\ast}=(p^{\ast},J,c)$ witnesses $q<_{\mathbb{P}_{\text{max}}}p$ then

$q\Vdash\exists T\in P(\omega_{1})\backslash NS^{q}_{\omega_{1}}(\forall
S\in(J^{+})^{p^{\ast}}(S\backslash T\notin NS_{\omega_{1}}^{q})).$

###### proof of the Claim.

Assume that $q<p$ as witnessed by $i:p\rightarrow p^{\ast}$, then
$|p|=\aleph_{1}$ in $q$ and $q\models\mathsf{MA_{\omega_{1}}}$. Hence by the
result of Taylor, there is no $\omega_{1}$-dense, normal ideal in $q$. In
particular, there must be a positive set
$T\in(\hbox{NS}_{\omega_{1}}^{+})^{q}$ such that $S\backslash T$ is an element
of $(\hbox{NS}_{\omega_{1}}^{+})^{q}$ for all $S\in
p^{\ast},S\in(\hbox{NS}_{\omega_{1}}^{+})^{q}$.

∎

Now let $q$ be as in the last Claim and for $g$ our
$\mathbb{P}_{\text{max}}$-generic over $L(\mathbb{R})$ we let $q\in g$ which
we can assume by density. By the choice of $p$ and $q<p$, $q\in g$ we have
that

$V\models S\subset\omega_{1}\text{ is stationary
}\Leftrightarrow\varphi(S,A).$

Let $j:q\rightarrow q^{\ast}=(q^{\ast},K,A)$ be the unique iteration of $q$ of
length $\omega_{1}$ given by $g$ which sends $c$ to $A$. By standard
$\mathbb{P}_{\text{max}}$-arguments, $K=NS_{\omega_{1}}^{V}\cap q^{\ast}$. By
elementarity

$q^{\ast}\models j(T)\in P(\omega_{1})\backslash K(\forall S\in
j((J^{+})^{p^{\ast}}(S\backslash j(T)\notin K))$

and as $\varphi$ is $\Sigma_{1}$,

$q^{\ast}\models\varphi(j(T),A).$

In other words $j(T)$ is stationary in $V$ and for all $S\in
j((J^{+})^{p^{\ast}})$, $S\backslash j(T)$ is stationary in $V$. Hence

$\displaystyle(H(\omega_{2}^{V}),\hbox{NS}_{\omega_{1}}^{V},A)\models$
$\displaystyle\,j(T)\notin\hbox{NS}_{\omega_{1}}\land\varphi(j(T),A)$
$\displaystyle\land S\backslash j(T)\notin\hbox{NS}_{\omega_{1}}\text{ for all
}S\in j((J^{+})^{p^{\ast}}.$

Now let $L$ denote the ideal generated from
$\hbox{NS}_{\omega_{1}}^{V}\cup\\{j(T)\\}$, i.e. $X\in L$ iff $(X\backslash
j(T))\cap C=\emptyset$ for some club $C$. It is easy to see that $L$ is a
normal ideal and $j((J^{+})^{p^{\ast}})\subset L^{+}$, so $j(J)=L\cap
j(p^{\ast})$.

Now we take a countable elementary substructure of $(H(\omega_{2}^{V}),L,A)$
and transitively collapse it. This results in a $\mathbb{P}_{\text{max}}$
condition $(r,\bar{L},a)$ (see [14], Lemma 3.14) which gives us:

Claim: There is a condition $r=(r,\bar{L},a)<_{\mathbb{P}_{\text{max}}}p$ such
that if $j:p\rightarrow p^{\ast}$ witnesses that
$r<_{\mathbb{P}_{\text{max}}}p$, then

$r\Vdash\exists T\in\bar{L}\,(\varphi(T,a))$

But now if we consider the generic iteration of $r$ given by $g$,

$k:(r,\bar{L},a)\rightarrow(r^{\ast},\hbox{NS}_{\omega_{1}}\cap r^{\ast},A)$

of length $\omega_{1}$, then $r^{\ast}\models\varphi(k(T),A)$ and so
$V\models\varphi(k(T),A)$. Yet $k(T)$ is nonstationary in
$V=L(\mathbb{R})[g]$. But $r<_{\mathbb{P}_{\text{max}}}p$ which is a
contradiction. ∎

##  3 Possibility Results

###  3.1 $\mathsf{BPFA}$ and the $\Pi_{1}$-definabilty of
$\hbox{NS}_{\omega_{1}}$

Goal of this section is to show that $\mathsf{BPFA}$ is consistent with a
$\Sigma_{1}$-definition of $\hbox{NS}_{\omega_{1}}^{+}$. Its proof relies on a
new coding technique which exploits mutually stationary sets.

####  3.1.1 Mutually stationary preserving forcing

######  Definition 3.1.

Let $K$ be a collection of regular cardinals with bounded below $\kappa$, and
suppose that we have $S_{\eta}\subseteq\eta$ for each $\eta\in K$. Then the
collection of sets $\\{S_{\eta}\mid\eta\in K\\}$ is _mutually stationary_ if
and only if for all algebras $\mathcal{A}$ on $\kappa$, there is an
$N\prec\mathcal{A}$ such that

$\text{for all }\eta\in K\cap N,\ \sup(N\cap\eta)\in S_{\eta}.$

Foreman-Magidor ([2]) show that every sequence $\vec{S}$ with
$S_{\eta}\subset\eta\cap Cof(\omega)$ is mutually stationary. Let
$\mathcal{T}_{\vec{S}}$ be the collection of all countable $N$ such that for
all $\eta_{i}\in N$, $\sup(N\cap\eta_{i})\in S_{i}.$

######  Theorem 3.2 (Foreman-Magidor).

Let $\langle\eta_{i}\mid i<j\rangle$ be an increasing sequence of regular
cardinals. Let $\vec{S}=\langle S_{i}\mid i<j\rangle$ be a sequence of
stationary sets such that $S_{i}\subseteq\eta_{i}\cap\textrm{Cof}(\omega)$. If
$\theta$ is a regular cardinal greater than all $\eta_{i}$ and $\mathcal{A}$
is a algebra on $\theta$, then there is a $N\prec\mathcal{A}$ belongs to the
class $\mathcal{T}_{\vec{S}}$. In particular, $\vec{S}$ is mutually
stationary.

From now on, we assume all stationary subsets of ordinals discussed in this
section are concentrated on countable cofinality. The corresponding notion for
being club in this context is that of an unbounded set which is closed under
$\omega$-sequences.

######  Definition 3.3.

Suppose $\vec{S}=\\{S_{\eta}\mid\eta\in K\\}$ is mutually stationary and that
for every $\eta\in K$, $S_{\eta}$ is stationary, co-stationary in $\eta$
$\cap$ Cof $({\omega})$. We say a forcing poset $\mathbb{P}$ is
$\vec{S}$-preserving if the following holds: Suppose
$\theta>2^{|\mathbb{P}^{+}|}$ is regular. Suppose $M$ is a countable
elementary submodel of $H(\theta)$ with $\\{\mathbb{P},\vec{S}\\}\subset M$
and $M\in\mathcal{T}_{\vec{S}}$. Suppose $p\in\mathbb{P}\cap M$. Then there
exists a $(M,\mathbb{P})$-generic condition $q$ extending $p$.

Remark

  1. 1.

Any proper forcing is $\vec{S}$-preserving.

  2. 2.

When $K=\\{\omega_{1}\\}$ and $\vec{S}=S\subset\omega_{1}$, the definition of
$\vec{S}$-preserving is identical to the usual definition of $S$-proper
forcing.

  3. 3.

Let $\vec{S}$ be such that each $S_{\eta}\in\vec{S}$ is stationary, co-
stationary in $\eta\cap Cof(\omega)$. Then an example of a non-proper,
$\vec{S}$ preserving forcing is the forcing poset $Club(S_{\eta})$, i.e, the
forcing which adds an unbounded subset to $S_{\eta}$ which is closed under
$\omega$-sequences, via countable approximations.

The iteration theorems for countable support iterated proper forcing can be
generalized to $\vec{S}$-preserving forcing.

######  Lemma 3.4.

If $\langle P_{i},\dot{Q}_{i}\mid i<\alpha\rangle$ is a countable support
iterated forcing system and for each $i<\alpha$,
$\Vdash_{P_{i}}$“$\dot{Q}_{i}$ is $\vec{S}$-preserving” then $P_{\alpha}$ is
$\vec{S}$-preserving.

###### Proof.

(Sketch, following the proof of [11], Theorem 3.2 ) We will only need to show
by induction on $j\leq\alpha$ that for any $N\in\mathcal{T}_{\vec{S}}$, if
$j\in N$, then:

(*)N For all $i<j$, $i\in N$ and for all $p\in N\cap P_{j}$ and $q\in P_{i}$
is $(N,O_{i})$ generic condition extending $p\upharpoonright i$, there is an
$r\in P_{j}$ such that $r$ is $(N,P_{j})$-generic condition extending $p$ and
$r\upharpoonright i=q$.

The statement (*)N is identical to the statement (*) in the proof of [11],
Theorem 3.2. It can be checked that the original proof also works here. ∎

In our proof of the main theorem of this section, we will use forcings which
have a specific form, so they get a name.

######  Definition 3.5.

Let $\kappa$ be an inaccessible cardinal. Let $\vec{S}=\langle S_{i}\mid
i<\kappa\rangle$ be mutually stationary. We say a forcing poset $\mathbb{P}$
is a $\vec{S}$-coding if $\delta\leq\kappa$ and
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\mid\alpha<\delta\rangle$
satisfies the following:

  * •

$\mathbb{P}$ is a countable support iteration

  * •

For any $\alpha<\delta$, one of the followings holds:

    1. 1.

Assume that $\alpha$ is inaccessible and $\mathbb{P}_{\alpha}$ is forcing
equivalent to a forcing of size less or equal than $\alpha$.333We say two
forcing $P$ and $Q$ are equivalent if their Boolean completions $B(P)$ and
$B(Q)$ are isomorphism. Assume that in $V^{\mathbb{P}_{\alpha}}$, $\langle
B_{\beta}\mid\beta<2^{\alpha}\rangle$ is an enumeration of $V_{\alpha}$. Then
$\dot{Q}_{\alpha}$ is allowed to be

$\prod_{j\in B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2j})\times\prod_{j\notin
B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2j+1})$

using countable support.

    2. 2.

In all other cases, we have that
$\Vdash_{\mathbb{P}_{\alpha}}\dot{Q}_{\alpha}$ is proper.

Let $\eta$ be an regular cardinal, we say $\mathbb{P}$ is an $\eta$-$\vec{S}$
coding if (1) is replaced by

  * (1’)

$\alpha\geq\eta$ is inaccessible and $\mathbb{P}_{\alpha}$ is forcing
equivalent to a forcing of size less or equal than $\alpha$. In
$V^{\mathbb{P}_{\alpha}}$, $\langle B_{\beta}\mid\beta<2^{\alpha}\rangle$ is
an enumeration of $V_{\alpha}$. Then $\dot{Q}_{\alpha}$ is allowed to be

$\prod_{j\in B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2j})\times\prod_{j\notin
B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2j+1})$

By Lemma 3.4, if $\mathbb{P}$ is a $\vec{S}$-coding forcing, then $\mathbb{P}$
is $\vec{S}$-preserving. It is also clear that an iteration is a
$\vec{S}$-coding forcing if and only all its initial segments are
$\vec{S}$-coding forcing.

######  Lemma 3.6.

Suppose that $\vec{S}$ is stationary, co-stationary. Suppose $\mathbb{P}$ is
an $\vec{S}$ coding forcing. Then for any $i\in\kappa$, the followings are
equivalent:

  * (a)

$\Vdash_{\mathbb{P}}S_{i}$ contains an $\omega$-club.

  * (b)

there are $\beta,\alpha,j$ and a set $B_{\beta}\subset V_{\alpha}$ such that
$j\in B_{\beta}$ if $\beta\cdot(\alpha+1)+2j=i$ and $i$ is even and $j\notin
B_{\beta}$ if $\beta\cdot(\alpha+1)+2j+1=i$ and $i$ is odd.

###### Proof.

((b) $\rightarrow$ (a)) Follow from the definition of the forcing.

((a) $\rightarrow$ (b)) Fix an $i$ and assume without loss of generality that
$i$ is even. Write $i=\alpha\cdot({\beta+1})+2j$ and suppose for a
contradiction that $j$ is not an element of $B_{\beta}$. By the definition of
an $\vec{S}$ coding forcing, we must have added a club through
$S_{\alpha(\beta+1)+2j+1}$ instead. Let $\vec{T}$ be the sequence $\langle
T_{k}\mid k<\kappa\rangle$, where $T_{k}=S_{k}$ if $k\neq i$ and
$T_{i}=\eta_{i}\setminus S_{i}$. It follows from Theorem 3.2 again that
$\vec{T}$ is mutually stationary. We will prove that $\mathbb{P}$ is
$\vec{T}$-preserving to derive a contradiction. Indeed we shall see that
$\vec{T}$-perservation implies that $\eta_{i}\setminus S_{i}$ must remain
stationary after forcing with $\mathbb{P}$, yet $\mathbb{P}\Vdash``S_{i}$
contains an $\omega$-club$"$ which is impossible.

To see that $\vec{T}$ preserving forcings preserve the stationarity of every
$S_{\eta_{i}}\in\vec{T}$, we only need to note that for any name $\dot{C}$ of
a subset of $\eta_{i}$ which is unbounded and $\omega$-closed, and any
countable elementary substructure $N$ which contains $\dot{C}$ and for which
$\sup(N\cap\eta_{i})\in S_{\eta_{i}}$, any $(N,\mathbb{P})$-generic condition
$q$, $q$ forces $\dot{C}\cap(S_{\eta_{i}})\neq\emptyset$.

Next we show by induction that each $Q_{\beta}$ is forced to be
$\vec{T}$-preserving. Work in $V[G_{\beta}]$. If $\dot{Q}_{\beta}/G_{\beta}$
is proper, then it is also $\vec{T}$-preserving. Otherwise, (1) holds. Now
$\dot{Q}_{\beta}/G_{\beta}$ is a countable support product of shooting club
forcing. Fix any $N\in\mathcal{T}_{T}$ which is a countable substructure of
$H(\theta)^{V[G_{\beta}]}$. For any $p\in N\cap\dot{Q}_{\beta}$, we can
construct a countable decreasing sequence of conditions $\langle p_{i}\mid
i<\omega\rangle$ meeting all dense set in $N$. Define $q$ by setting $q(i)$ to
be the closure of $\bigcup_{n<\omega}p_{n}(i)$ if $i\in N$ and trivial
otherwise. Note any nontrivial $q(i)$ is equal to
$\bigcup_{n<\omega}p_{n}(i)\cup\\{\sup(N\cap\eta_{i})\\}$, where
$\eta_{i}=\sup(S_{i})$ is a regular cardinal. As $N\in\mathcal{T}_{T}$ and no
$\beta,\alpha,j$ witness that $\beta\cdot(\alpha+1)+2j=i$ or
$\beta\cdot(\alpha+1)+2j+1=i$, we have $\sup(N\cap\eta_{i})\in S_{i}$,
whenever $q(i)$ is non-trivial. Hence $q$ is a condition. ∎

The proof also show that $\vec{S}$-coding preserves stationary subset of
$\omega_{1}$ if $\sup(S_{0})>\omega_{1}$. As a Corollary of Lemma 3.6 and the
definition of $\vec{S}$-coding, in any generic extension by $\vec{S}$-coding
and any even $i$, at most one of $S_{i}$ and $S_{i+1}$ contains a club.

######  Lemma 3.7.

Suppose $\mathbb{P}=\langle
P_{\alpha},\dot{Q}_{\alpha}\mid\alpha<\delta\rangle$ is a countable support
iteration. Suppose for any $\alpha>0$, $\dot{Q}_{\alpha}$ is forced to be
$\eta_{\alpha}$-$\vec{S}$ coding of length $l(\alpha)$, where
$\eta_{\alpha}=\max\\{|\mathbb{P}_{\alpha}^{+}|,\Sigma_{\beta<\alpha}l(\beta)\\}$.
Also let $\eta_{0}$ be regular. Then $\mathbb{P}$ is forcing equivalent to a
$\eta_{0}$-$\vec{S}$ coding.

###### Proof.

Let $\epsilon_{\alpha}=\Sigma_{\beta<\alpha}l(\beta)$ for any
$\alpha\leq\delta$. We define a countable support iteration
$\mathbb{P}^{\prime}=\langle
P_{\alpha}^{\prime},\dot{Q}_{\alpha}^{\prime}\mid\alpha<\epsilon_{\delta}\rangle$.
By induction on $\alpha<\delta$, we give the definition of
$P^{\prime}_{\epsilon_{\alpha}}$. We will also verify that
$P^{\prime}_{\epsilon_{\alpha}}$ is forcing equivalent to $P_{\alpha}$ and
$P^{\prime}_{\epsilon_{\alpha}}$ is a $\eta_{0}$-$\vec{S}$ coding.

If $\alpha$ is limit. Then $P^{\prime}_{\epsilon_{\alpha}}$ is the countable
support limit of $P^{\prime}_{\gamma}$ for $\gamma<\epsilon_{\alpha}$. Hence
$P^{\prime}_{\epsilon_{\alpha}}$ is the countable support limit of
$P^{\prime}_{\epsilon_{\gamma}}$ for $\gamma<\alpha$. As
$P^{\prime}_{\epsilon_{\gamma}}$ is forcing equivalent to $P_{\gamma}$.
$P^{\prime}_{\epsilon_{\alpha}}$ is forcing equivalent to the countable
support limit of $P_{\gamma}$ for $\gamma<\alpha$. Hence
$P^{\prime}_{\epsilon_{\gamma}}$ is forcing equivalent to $P_{\alpha}$.

If $\alpha=\beta+1$. Let $\epsilon_{\alpha}=\epsilon_{\beta}+l(\alpha)$. Now
$\dot{Q}_{\beta}$ is a name for a forcing iteration $\langle
P^{\alpha}_{\gamma},\dot{Q}^{\alpha}_{\gamma}\mid\gamma<l(\alpha)\rangle$. Now
as $P_{\beta}$ is forcing equivalent to $P^{\prime}_{\epsilon_{\beta}}$. We
can view $\dot{Q}_{\alpha}$ as a $P^{\prime}_{\epsilon_{\beta}}$ name. By
induction on $\gamma<l(\alpha)$, we can set
$\dot{Q}^{\prime}_{\epsilon_{\beta}+\gamma}$ be
$P^{\prime}_{\epsilon_{\beta}+\gamma}$ name of $\dot{Q}^{\alpha}_{\gamma}$ and
verify that $P^{\prime}_{\epsilon_{\beta}+\gamma}$ is forcing equivalent to
$P^{\prime}_{\epsilon_{\beta}}*P^{\alpha}_{\gamma}$.

It remains to check that $P^{\prime}_{\epsilon_{\alpha}}$ is a
$\eta_{0}$-$\vec{S}$ coding. Fix a $\beta<\epsilon_{\alpha}$, with
$\dot{Q}_{\beta}$ is not forced to be proper. There is $\gamma$ such that
$\beta\in[\epsilon_{\gamma},\epsilon_{\gamma+1})$. $\beta^{\prime}$ is
inaccessible cardinal greater then $\eta_{\gamma}$. Let
$\beta=\epsilon_{\gamma}+\beta^{\prime}$. As $\epsilon_{\gamma}<\eta_{\gamma}$
and $\beta$ is inaccessible. $\beta=\beta^{\prime}$ is inaccessible. Now
$P^{\prime}_{\beta}$ is forcing equivalent to $P_{\gamma}*P^{\gamma}_{\beta}$.
$P_{\gamma}$ is of size less than $\beta$ and $P^{\gamma}_{\beta}$ is forced
to be forcing equivalent to a forcing of size less of equal to $\beta$. Hence
$P^{\prime}_{\beta}$ is also forcing equivalent to a forcing of size less of
equal to $\beta$. The rest of (2) is routine to check. ∎

We are mainly interested in executing $\vec{S}$-coding forcing over $L$. Now
we can define the coding machinery to be used in the latter section.

######  Definition 3.8.

Suppose $\alpha$ is in inaccessible in $L$ and $X\subseteq P(\alpha)$. We say
$\vec{S}$ codes $X\subset\alpha$ if

  1. 1.

For any even $i\in[\alpha,(\alpha^{+})^{L})$, one of $S_{i}$ and $S_{i+1}$
contains a club.

  2. 2.

For any $x\in X$, there is a $\beta<\alpha$ such that

$j\in x\text{ if and only if }S_{\alpha\cdot(\beta+1)+2i}\text{ contains a
club}$

and

$j\notin x\text{ if and only if }S_{\alpha\cdot(\beta+1)+2i+1}\text{ contains
a club}$

Let $\vec{C}=\langle C_{i}\mid i\in[\alpha,(\alpha^{+})^{L})\rangle$ be a club
sequence which witnesses (2), then we say that $\vec{C}$ is a $\vec{S}$ code
for $X$.

A useful fact is the upward absoluteness of the coding between certain pair of
models.

######  Lemma 3.9.

Suppose $N\subset M$ are two ZFC- transitive models. $N\models$ $\vec{C}$ is a
$\vec{S}$ code for $X\subset\alpha$, $\alpha$ is inaccessible in $L^{M}$,
$((\alpha^{+})^{L})^{M}=((\alpha^{+})^{L})^{N}$. Suppose $M\models$ for any
even $i\in[\alpha,(\alpha^{+})^{L})$, at most one of $S_{i}$ and $S_{i+1}$
contains a club. Then $M\models$ $\vec{C}$ is a $\vec{S}$ code for $X$.

###### Proof.

This follows from the definition. Note that being a club is an absolute
property between transitive models. ∎

######  Lemma 3.10.

Suppose $\mathbb{P}$ is a $\vec{S}$-coding over $L$, $\alpha$ is inaccessible,
$X\subseteq P(\alpha)$ and $S$ codes $X$, then $X=P(\alpha)^{L[G_{\alpha}]}$.

###### Proof.

By Lemma 3.6, for any $i$, $S_{i}$ contains a club if there is some stage
$\alpha$ such that $\dot{Q}_{\alpha}$ satisfies (2) in the definition. But
$\dot{Q}_{\alpha}$ forces $S$ codes $P(\alpha)^{L[G_{\alpha}]}$ in
$L[G_{\alpha+1}]$. Then by Lemma 3.9, $S$ codes $P(\alpha)^{L[G_{\alpha}]}$ in
$L[G]$. ∎

####  3.1.2  A model of BPFA and $\Delta_{1}$-definalbility of
$NS_{\omega_{1}}$

######  Theorem 3.11.

Suppose that $V=L$ and $\delta$ is a reflecting cardinal. Then there is a
forcing poset $P$ such that in $L^{P}$, the following statements hold:

  1. 1.

BPFA

  2. 2.

$\omega_{1}=\omega_{1}^{L}$ and $\omega_{2}=\delta$.

  3. 3.

The nonstationary ideal on $\omega_{1}$ is
$\Sigma_{1}(\\{\omega_{1}\\})$-definable over $\langle
H(\omega_{2}),\in\rangle$.

###### Proof.

We first choose a sequence $\langle
S_{\alpha}\mid\alpha\in\mathop{\textrm{Lim}}(\delta)\rangle$ uniformly in
$\alpha<\delta$ satisfying

  * •

$S_{\alpha}\subset\alpha$

  * •

If $\alpha$ is a regular cardinal, then $S_{\alpha}$ is stationary co-
stationary in $\alpha\cap\textrm{Cof}(\omega)$.

The existence of such a sequence $\langle
S_{\alpha}\mid\alpha\in\mathop{\textrm{Lim}}(\delta)\rangle$ follows from the
fact that $\diamondsuit_{\lambda}$ holds in $L$ for any $L$-cardinal $\lambda$
and is a routine construction.

Now we define the forcing poset $P$. The forcing $P=\langle
P_{\alpha},\dot{Q}_{\alpha}\mid\alpha<\delta\rangle$ will be a countable
support iteration of length $\delta$. We require the size of each iterand to
be smaller than $\delta$. As a consequence $P$ satisfies $\delta$-cc. We
demand that $\dot{Q}_{\alpha}$ is trivial unless $\alpha$ is an inaccessible
cardinal and $P_{\alpha}\subset L_{\alpha}$. We split into two cases if
$\alpha$ is inaccessible.

  * •

($\alpha$ is Mahlo) We follow an idea from Goldstern-Shelah [3]. Choose a
$A\subset\omega_{1}$ and a $\Sigma_{1}$ formula $\psi(x,y)$. $A$ and $\psi$ is
chosen in a bookkeeping way so that during the whole iteration, each pair
$(A,\psi)$ will be dealt with unboundedly many times. Since $\delta$ is
reflecting, whenever there is a $\alpha$-$\vec{S}_{>\alpha}$ coding forcing
poset $Q$ which adds a witness to $\psi(x,A)$, there is such a $Q$ of size
less than $\delta$. In this case, let $\dot{Q}_{\alpha}$ be the name of $Q$
and use this forcing at stage $\alpha$. When there is no such $Q$, we set
$\dot{Q}_{\alpha}$ to be trivial forcing.

  * •

($\alpha$ is inaccessible but not Mahlo) $\dot{Q}_{\alpha}$ is trivial unless
$|P_{\alpha}|\leq\alpha$. Work in $L[G_{\alpha}]$, where $G_{\alpha}$ is a
$L$-$P_{\alpha}$ generic filter. $|P_{\alpha}|=\alpha$ is inaccessible. We
choose $\langle B_{\beta}\mid\beta<2^{\alpha}\rangle$ to be an enumeration of
$P(\alpha)^{V[G_{\alpha}]}$. Let $Q$ be the forcing

$\prod_{i\in B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2i})\times\prod_{i\notin
B_{\beta}}Club(S_{\alpha\cdot(\beta+1)+2i+1})$

with countable support. Then we force with $Q$ at stage $\alpha$.

It follows from the definition of $P$ that for any $\alpha<\delta$, the
forcing $\dot{Q}_{\alpha}$ is forced to be a $\alpha-\vec{S}$ coding forcing.
Applying Lemma 3.7, we know that $P$ is a $\vec{S}$-coding forcing. Moreover,
for any $\alpha$ Mahlo, the tail $P/P_{\alpha}$ is a
$\alpha$-$\vec{S}_{>\alpha}$ coding.

Now as $P$ is $\vec{S}$-preserving, $P$ preserves $\omega_{1}$. On the other
hand, $P$ is $\delta$-c.c and $P$ preserves cardinals above $\delta$. By the
definition of the forcing in the inaccessible but not Mahlo case, all
cardinals below $\delta$ are collapsed to $\omega_{1}$. In summary,
$\omega_{1}^{L^{P}}=\omega_{1}^{L}$ and $\omega_{2}^{L^{P}}=\delta$.

######  Lemma 3.12.

$P\Vdash$ $\mathsf{BPFA}$.

###### Proof.

Work in $L[G]$. Let $A\subset\omega_{1}$ and $Q\in L[G]$ be a proper forcing
which adds a witness to the $\Sigma_{1}$-formula $\psi(x,A)$. Now let $\alpha$
be a stage such that $A\in L[G_{\alpha}]$, $\alpha$ is Mahlo and $(A,\psi)$ is
to be dealt with at stage $\alpha$. In $L[G]$, the forcing $P/P_{\alpha}*Q$ is
a $\alpha-\vec{S}_{>\alpha}$ coding forcing adding a witness to $\psi(x,A)$.
Hence, we must be in the first case of the definition of our iteration at
stage $\alpha$ and $Q_{\alpha}$ is a $\alpha-\vec{S}_{>\alpha}$ coding forcing
which adds a witness to $\psi(x,A)$. Thus
$H(\omega_{2})^{L[G_{\alpha+1}]}\models\exists x\psi(x,A)$. By upward
absoluteness, $H(\omega_{2})^{L[G]}\models\exists x\psi(x,A)$. ∎

######  Lemma 3.13.

Work in $L[G]$. Let $S$ be a subset of $\omega_{1}$, then the following are
equivalent:

  * (a)

$S$ is stationary.

  * (b)

there is a $\alpha$ inaccessible in $L$ with $P_{\alpha}\subseteq L_{\alpha}$
and $S$ is stationary in $L[G_{\alpha}]$.

  * (c)

there is a $\alpha$ inaccessible in $L$, there is $\vec{C}\in L[G_{\alpha+1}]$
which is a $\vec{S}$ code for $P(\alpha)^{L[G_{\alpha}]}$, and there is a
transitive model $M$ of $\mathsf{ZF}^{-}$ such that $\vec{C}\in M$ and $M$
thinks that $S$ is stationary.

###### Proof.

((a) $\rightarrow$ (c)) Let $\alpha$ be inaccessible but not Mahlo and
consider stage $\alpha$ of the iteration which we can assume to be such that
$S\in L[G_{\alpha}]$ and $|P_{\alpha}|\leq\alpha$. Now $Q_{\alpha}$ forces the
existence of a $\vec{C}$-sequence which is an $\vec{S}$ code for
$P(\alpha)^{L[G_{\alpha}]}$. Any transitive $M$ which contains $\vec{C}$ is as
desired in (c), moreover $M$ will automatically think that $S$ is stationary
if $S$ is stationary in $L[G]$.

((b) $\rightarrow$ (a)) The tail of forcing $P/P_{\alpha}$ is
$\alpha$-$S_{>\alpha}$ coding. Now the proof of Lemma 3.6 shows $P/P_{\alpha}$
preserves stationarity of $S$.

((c) $\rightarrow$ (b)) If $M$ is as in the assumption,
$P(\alpha)^{L[G_{\alpha}]}$ is a subset of $M$, and if $M$ thinks that $S$ is
stationary, so must $V_{\alpha+1}^{L[G_{\alpha}]}=P(\alpha)^{L[G_{\alpha}]}$,
and hence $L[G_{\alpha}]$. ∎

We now present the $\Sigma_{1}$ definition of stationary subsets of
$\omega_{1}$ over $(H(\omega_{2}),\in,\\{\omega_{1}\\})$. Let $\psi(x)$
describe the following statement: there are objects $A$ and $M$ such that

  * (i)

$A=\langle C_{i}\mid i\in[\alpha,\beta)\rangle$ is a sequence with $C_{i}$ is
an $\omega$-club in $\sup(C_{i}\cap Ord)=\eta_{i}$.

  * (ii)

All $\eta_{i}$ are regular cardinals in $L$. $\eta_{\alpha}=\alpha$ is
inaccessible in $L$.

  * (iii)

$M$ is a transitive ZFC- model with $\omega_{1}=\omega_{1}^{M}$,
$((\alpha^{+})^{L})^{M}=(\alpha^{+})^{L}$, and $x\in M$.

  * (iv)

$M\models x\subset\omega_{1}$ is stationary.

  * (v)

$M\models A$ is a $\vec{S}$ codes for $P(\alpha)$.

It is routine to check (i), (iii), (iv) and (v) are all $\Sigma_{1}$ over
$\langle H(\omega_{2}),\in,\\{\omega_{1}\\}\rangle$. For (ii), as $L[G]\models
BPFA$, we can apply a trick of Todorcevic (cf. Proof of Lemma 4, [13]) to get
$\Sigma_{1}$ formulas $\psi_{0}(x)$ and $\psi_{1}(x)$ such that
$H(\omega_{2})\models\psi_{0}(\beta)$ if and only if $\beta$ is a regular
cardinal in $L$ and $H(\omega_{2})\models\psi_{1}(\beta)$ if and only if
$\beta$ is an inaccessible cardinal in $L$.

Here $\psi_{0}(x)$ is the formula describing the existence of a specialization
function of the tree $T_{x}$, where $T_{x}$ is derived from the canonical
global square sequence in $L$. It is a consequence of $\mathsf{BPFA}$ that $x$
is uncountable regular in $L$ if and only if such a specialization function
exists. Now $\psi_{1}(x)$ is a formula says $\alpha$ is regular and a limit of
regular cardinals in $L$.

It is now clear that $\psi(x)$ is $\Sigma_{1}$ over $\langle
H(\omega_{2}),\in,\\{\omega_{1}\\}\rangle.$ What is left is to show that
$\psi(x)$ indeed characterizes stationarity.

######  Lemma 3.14.

In $L[G]$, for any $x\subseteq\omega_{1}$, $H(\omega_{2})\models\psi(x)$ if
and only if $x$ is stationary.

###### Proof.

If $x\subset\omega_{1}$ is stationary, then, by Lemma 3.13 (c), we do
immediately get a witness $M\in H(\omega_{2})$ and an $A$ so that $\psi(x)$
holds.

Conversely, let $x$, $A$, $M$ be as given by the definition of $\psi(x)$, we
check that $x$ is stationary. But if $\psi(x)$ holds, then $M$ thinks that $A$
is a $\vec{S}$-code for $P(\alpha)$. By item $(ii)$ of $\psi$, this also means
that $A$ is a $\vec{S}$-code for $P(\alpha)$ and by (c) of Lemma 3.13 it is
true that $x$ is indeed stationary. ∎

∎

###  3.2 A model for $\mathsf{MA_{\omega_{1}}}$, $\hbox{NS}_{\omega_{1}}$
being saturated and $\Delta_{1}(\\{\omega_{1}\\})$-definable

In this section we improve an earlier result of [5] namely we show that given
a Woodin cardinal, there is a model such that $\hbox{NS}_{\omega_{1}}$ is
saturated, $\Delta_{1}$-definable with parameters from $H(\omega_{2})$ while
Martin’s Axiom also holds true. If one forces over the canonical inner model
with one Woodin cardinal $M_{1}$, then the construction yields a model where
additionally $\hbox{NS}_{\omega_{1}}$ is definable with $\omega_{1}$ as the
only parameter.

####  3.2.1 Short summary of the main features of the model $W_{1}$

The proof relies heavily on the coding machinery introduced in [5], where it
is shown that, given a Woodin cardinal $\delta$, then there is a universe such
that $\hbox{NS}_{\omega_{1}}$ is saturated and
$\Delta_{1}(\vec{C},\vec{T}^{0})$-definable, where $\vec{C}$ is an arbitrary
ladder system on $\omega_{1}$ and $\vec{T}^{0}$ is an independent sequence
(independence will be defined in a moment below) of Suslin trees of length
$\omega$. As this coding is rather convoluted, we will not define it here in
detail but instead only highlight the most important notions and features of
it. Our notation will be exactly as there.

We shall use Suslin trees on $\omega_{1}$ for creating a
$\bf{\Sigma_{1}}$-definition of stationarity. To facilitate things, the trees
should satisfy a certain property:

######  Definition 3.15.

Let $\vec{T}=(T_{\alpha}\,:\,\alpha<\kappa)$ be a sequence of Suslin trees. We
say that the sequence is an independent family of Suslin trees if for every
finite set $e=\\{e_{0},e_{1},...,e_{n}\\}\subset\kappa$ the product
$T_{e_{0}}\times T_{e_{1}}\times\cdot\cdot\cdot\times T_{e_{n}}$ is a Suslin
tree again.

Independent sequences of Suslin trees can be used to code arbitrary
information using the two well-known and mutually exclusive ways to destroy a
Suslin tree, namely either shooting a branch through the tree or specializing
it. More precisely, given a set $X\subset\omega_{1}$ and an independent
sequence of Suslin trees $\vec{T}=(T_{i}\,:\,i<\omega_{1})$, we can code the
characteristic function of $X$ into $\vec{T}$ via forcing with the finitely
supported product of

$\mathbb{P}_{i}=\begin{cases}T_{i}&\text{ if }i\in X\\\ Sp(T_{i})&\text{ if
}i\notin X\end{cases}$

where $T_{i}$ just denotes the forcing one obtains when forcing with the
Suslin tree $T_{i}$ (which adds a cofinal branch to $T_{i}$) and $Sp(T_{i})$
denotes the forcing which spezializes the tree $T_{i}$. Note that the
independence of $\vec{T}$ buys us that the finitely supported product has the
ccc as well. We will eventually use this mechanism to create a generic
extension of $V$ with a $\Sigma_{1}(\vec{C},\vec{T}^{0})$-definition of being
stationary on $\omega_{1}$.

The construction of such a universe shall be sketched now. We start with a
universe $V$ with one Woodin cardinal $\delta$ with $\diamondsuit$. We fix a
ladder system $\vec{C}$ and an $\omega$-sequence of independent Suslin trees
$\vec{T}^{0}$ and start a first, nicely supported iteration (using Miyamoto’s
nice iterations, see [9]) of length $\delta$ over $V$ which combines Shelah’s
proof of the saturation of $\hbox{NS}_{\omega_{1}}$ from a Woodin cardinal
with the forcings invented by A. Caicedo and B. Velickovic (see [1]) and other
forcings whose purpose is to create a model $W_{0}$ which will have several
features listed below which will turn our to be useful.

In a next step we force over $W_{0}$ with a variant of almost disjoint coding
which is due to L. Harrington (see [4]), which is used to ensure that over the
resulting generic extension of $W_{0}$, denoted by $W_{1}$, there is a
${\Sigma_{1}}(\vec{C},\vec{T}^{0})$-definable $\omega_{2}$-sequence of
$\aleph_{1}$-sized, transitive models which are sufficiently smart to
determine whether a member is a stationary subset of $\omega_{1}$ or a Suslin
tree in $W_{0}$. These sufficiently smart models are called suitable, and the
purpose of suitable models is that they can correctly compute a fixed,
independent sequence $\vec{T}=(T_{\alpha}\,:\,\alpha<\omega_{2})$ of Suslin
trees. Due to this correctness, we can identify whether an $\omega_{1}$-block
of trees is in $\vec{T}$ in a $\Sigma_{1}(\vec{C},\vec{T}^{0})$-way. The
sequence $\vec{T}$ will be used later to code up being stationary in a
$\Sigma_{1}(\vec{T}^{0},\vec{C})$-way.

To summarize the above, starting from an arbitrary $V$ which contains a Woodin
cardinal $\delta$ with $\diamondsuit$, and fixing a ladder system on
$\omega_{1}$ $\vec{C}$ and an $\omega$-sequence of independent Suslin trees
$\vec{T}^{0}$, we create first a generic extension $W_{0}$ and then a further
generic extension $W_{1}$ such that in $W_{1}$ the following holds:

  1. 1.

$\delta=\aleph_{2}$.

  2. 2.

In $W_{1}$, the nonstationary ideal is saturated and the saturation is ccc-
indestructible.

  3. 3.

Every real in $W_{0}$ is coded by a triple of limit ordinals
$(\alpha,\beta,\gamma)$ below $\omega_{2}$ relative to the ladder system
$\vec{C}$ (in the sense of Caicedo-Velickovic, see [5], pp. Theorem 18
$(\ddagger)$).

  4. 4.

Every subset $X\subset\omega_{1}$, $X\in W_{0}$ is coded by a real $r_{X}\in
W_{0}$ relative to the fixed almost disjoint family of reals $F$ we
recursively obtain from our ladder system $\vec{C}$.

  5. 5.

There is an $\omega_{2}$-sequence of independent Suslin trees
$\vec{T}=(T_{i}\,:\,\omega<i<\omega_{2})\in W_{1}$ whose initial segments are
uniformly and correctly definable in suitable models. The set of suitable
models is itself $\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable in $W_{1}$ using
as parameters the ladder system $\vec{C}$, and one $\omega$-block of
independent Suslin tree $\vec{T}^{0}=(T_{n}\,:\,n\in\omega)$. As a
consequence, $\vec{T}$ is $\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable over
$W_{1}$.

  6. 6.

The definition of $\vec{T}$ remains valid in all generic extensions of $W_{1}$
by forcings with the countable chain condition. So $W_{1}$ is a reasonable
candidate for a ground model using coding forcings which themselves have the
ccc.

In a second iteration using $W_{1}$ as the ground model, we force with coding
forcings using Suslin trees that are applied to make $\hbox{NS}_{\omega_{1}}$
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable. We force with a finitely
supported iteration of ccc forcings over $W_{1}$. As a consequence we preserve
the saturation of the nonstationay ideal and the sequence $\vec{T}$ is still
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable. The only forcings which are used
in this second iteration are the Suslin trees from our independent sequence
$\vec{T}^{>0}=(T_{i}\,:\,i>\omega)$, which we either specialize or destroy via
the addition of an $\omega_{1}$-branch. We will use a bookkeeping function and
start to write characteristic functions of every stationary subset of
$\omega_{1}$ into $\omega_{1}$-blocks of $\vec{T}^{>0}$ using either the
spezialization forcing or shooting a branch through elements of
$\vec{T}^{>0}$.

This will eventually yield a universe $W_{\omega_{2}}$ where the nonstationary
ideal remains saturated and where stationary subsets of $\omega_{1}$ can be
characterized as follows:

######  Fact 3.16.

There are $\Sigma_{1}(\vec{C},\vec{T}^{0})$-formulas $\Phi(r)$ and $\Psi(S)$
where the formula $\Phi(r)$ defines a set of reals such that every member is
an almost disjoint code for an $\aleph_{1}$-sized, transitive models which can
be used to compute the sequence $\vec{T}^{>0}$ of Suslin trees correctly
(these models are the suitable models mentioned earlier). The formula
$\Psi(S)$ then defines stationary subsets of $W_{\omega_{2}}$ in the following
way:

  * $\Psi(S)$

if and only if there is an $\aleph_{1}$-sized, transitive model $N$ which
contains $\vec{C}$ and $\vec{T}^{0}$ such that $N$ models that

    * –

There exists a real $x$ such that $\Phi(x)$ holds, i.e. $x$ is a code for a
suitable model $M$.

    * –

There exists an ordinal $\alpha$ in the suitable model $M$ such that
$\vec{T}^{\prime}$ is the $\alpha$-th $\omega_{1}$ block of the definable
sequence of independent Suslin trees as computed in $M$ and $N$ sees a full
pattern on $\vec{T}^{\prime}$.

    * –

$\forall\beta<\omega_{1}$ $(\beta\in S$ if and only if
$\vec{T}^{\prime}(\beta)$ has a branch).

    * –

$\forall\beta<\omega_{1}$ $(\beta\notin S$ if and only if
$\vec{T}^{\prime}(\beta)$ is special).

Note that $\Psi(S)$ is of the form $\exists N(N\models...)$, thus $\Psi$ is a
$\Sigma_{1}$-formula.

####  3.2.2 Forcing over $W_{1}$

We shall show how to modify the just sketched construction to obtain a model
where additionally $\mathsf{MA_{\omega_{1}}}$ holds. As before we will
construct the model $W_{1}$. We will proceed however not coding up stationary
subsets of $\omega_{1}$ as we do in [5] but instead code up a second
$\omega_{2}$-block of generically added Suslin trees first.

In an $\omega_{2}$-length iteration we first use Tennenbaum’s forcing over
$W_{1}$ to add an independent sequence of Suslin trees of length $\omega_{2}$
and use our definable independent sequence $\vec{T}$ to code up the added
Suslin trees. First let us briefly recall the definition of Tennenbaum’s
forcing.

######  Definition 3.17.

Tennenbaum’s forcing $\mathbb{P}_{T}$ consists of conditions which are finite
trees $(T,<_{T})$, $T\subset\omega_{1}$ such that $\alpha<\beta$ if
$\alpha<_{T}\beta$, and $(T_{1},<_{T_{1}})<(T_{2},<_{T_{2}})$ holds if
$T_{2}\subset T_{1}$ and $<_{T_{2}}=<_{T_{1}}\cap(T_{2}\times T_{2})$.

It is well-known that $\mathbb{P}_{T}$ is Knaster and adds generically a
Suslin tree to the ground model.

So we start with the model $W_{1}$ as our ground model. Let $\vec{C}$ be our
fixed ladder system on $\omega_{1}$ and let $\vec{T}^{0}$ be a fixed
independent sequence of Suslin trees of length $\omega$. In $W_{1}$ there is a
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable $\omega_{2}$-sequence of
$\omega_{1}$-blocks of independent Suslin trees
$\vec{T}=(\vec{T}^{\alpha}\,:\,\alpha<\omega_{2})$, where for every
$0\neq\alpha<\omega_{2}$,
$\vec{T}^{\alpha}=(T_{\eta}^{\alpha}\,:\,\eta<\omega_{1})$, and $\vec{T}$
forms an independent sequence of Suslin trees.

Over $W_{1}$ we start a finitely supported iteration
$\mathbb{Q}=((\mathbb{Q}_{\alpha},\dot{\mathbb{R}}_{\alpha})\,:\,\alpha<\omega_{2})$
and let $H_{\alpha}$ denote the generic filter for $\mathbb{Q}_{\alpha}$. For
every $\alpha<\omega_{2}$, using $W_{1}[G_{\alpha}]$ as the ground model,
$\dot{\mathbb{R}}^{G_{\alpha}}_{\alpha}$ is defined to be
$\mathbb{Q}^{1}_{\alpha}\ast\mathbb{Q}^{2}_{\alpha}$, and
$\mathbb{Q}^{1}_{\alpha}$ is Tennenbaum’s $\mathbb{P}_{T}$ and
$\mathbb{Q}_{\alpha}^{2}$ codes up the generically by
$\mathbb{Q}_{\alpha}^{1}$ added tree, called $h_{\alpha}\subset\omega_{1}$ in
the following way:

$\mathbb{Q}_{\alpha}^{2}=\begin{cases}T^{\alpha}_{\eta}&\text{ if }\eta\in
h_{\alpha}\\\ Sp(T^{\alpha}_{\eta})&\text{ if }\eta\notin
h_{\alpha}\end{cases}$

It is immediate that the resulting universe $W_{2}=W_{1}[H_{\omega_{2}}]$ is a
ccc extension of $W_{1}$, thus $\hbox{NS}_{\omega_{1}}$ remains saturated and
as in [H] one can show that the generically added sequence of Suslin trees
$(h_{\alpha}\,:\,\alpha<\omega_{2})$ is an independent,
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable sequence of Suslin trees via the
formula:

######  Fact 3.18.

There is a $\Sigma_{1}(\vec{C},\vec{T}^{0})$-formula $\Psi(h)$ which defines
the generically added Suslin trees $(h_{\alpha}\,:\,\alpha<\omega_{2})$ of
$W_{2}$:

  * $\Psi(h)$

if and only if there is an $\aleph_{1}$-sized, transitive model $N$ which
contains $\vec{C}$ and $\vec{T}^{0}$ such that $N$ models that

    * –

There exists a real $x$ such that $\Phi(x)$ holds, i.e. $x$ is a code for a
suitable model $M$.

    * –

There exists an ordinal $\alpha$ in the suitable model $M$ such that
$\vec{T}^{\prime}$ is the $\alpha$-th $\omega_{1}$ block of the definable
sequence of independent Suslin trees as computed in $M$ and $N$ sees a full
pattern on $\vec{T}^{\prime}$.

    * –

$\forall\beta<\omega_{1}$ $(\beta\in h$ if and only if
$\vec{T}^{\prime}(\beta)$ has a branch).

    * –

$\forall\beta<\omega_{1}$ $(\beta\notin h$ if and only if
$\vec{T}^{\prime}(\beta)$ is special).

Note that $\Psi(h)$ is of the form $\exists N(N\models...)$, thus $\Psi$ is a
$\Sigma_{1}$-formula.

In a second step, we use $W_{2}$ as our ground model and force in an
$\omega_{2}$-length, finitely supported iteration $\mathsf{MA_{\omega_{1}}}$
while simultaneously coding up stationary subsets of $\omega_{1}$ using the
boldface $\Sigma_{1}$-definable sequence of trees
$(h_{\alpha}\,:\,\alpha<\omega_{2})$. We write $\vec{h}^{\alpha}$ for the
$\alpha$-th $\omega_{1}$-block of elements of the sequence
$(h_{\beta}\,:\,\beta<\omega_{2})$ and let $h^{\alpha}_{\beta}$ denote the
$\beta$-th element of the $\alpha$-th $\omega_{1}$-block.

We do the usual forcing to code characteristic functions of stationary subsets
of $\omega_{1}$ into $\vec{h}^{\alpha}$, but additionally we feed in forcings
of size $\aleph_{1}$ with the countable chain condition to produce a model of
$\mathsf{MA_{\omega_{1}}}$. Note that as all the forcings we use have the ccc
and thus we will preserve the saturation of $\hbox{NS}_{\omega_{1}}$.

In order to prevent the forcings we use to get $\mathsf{MA_{\omega_{1}}}$ from
damaging our codes, we will force $\mathsf{MA_{\omega_{1}}}$ in a “diagonal
way$"$. We define a finitely supported iteration
$((\mathbb{R}_{\alpha},\dot{\mathbb{S}}_{\alpha})\,:\,\alpha<\omega_{2})$ of
ccc forcing over $W_{2}$ inductively using a bookkeeping function $F$. We let
$F\in W_{2}$, $F:\omega_{2}\rightarrow\omega_{2}\times\omega_{2}\times 2$ such
that for every $(\alpha,\beta,i)\in\omega_{2}\times\omega_{2}\times 2$,
$F^{-1}(\alpha,\beta,i)$ is an unbounded subset in
$\omega_{2}\times\omega_{2}\times 2$. Assume we are at stage
$\alpha<\omega_{2}$ and we have already defined the iteration
$\mathbb{R}_{\alpha}$ up to $\alpha<\omega_{2}$. We let $I_{\alpha}$ denote
the generic filter for $\mathbb{R}_{\alpha}$. We also assume by induction that
every of the first $\alpha$-many $\omega_{1}$-blocks of trees
$\vec{h}^{\beta}$, $\beta<\alpha$ has already been used for codings, but the
elements of the sequences $(\vec{h}^{\eta}\,:\eta\geq\alpha)$ still form an
independent sequence of Suslin trees in $W_{2}[I_{\alpha}]$. The forcing we
have to use next is determined by the value of $F(\alpha)$.

  1. 1.

If $F(\alpha)=(\beta,\gamma,0)$, then we look at the $\beta$-th stationary
subset $S$ of $\omega_{1}$ in $W_{2}[I_{\gamma}]$ and use the $\alpha$-th
$\omega_{1}$-block of $\vec{h}$,
$\vec{h}^{\alpha}=(h^{\alpha}_{\eta}\,:\,\eta<\omega_{1})$ to code up $S$ i.e.
we will force with
$\dot{\mathbb{S}}^{I_{\alpha}}_{\alpha}:=\prod_{i<\omega_{1}}\mathbb{P}_{i}$
with finite support, where

$\mathbb{P}_{i}=\begin{cases}h^{\alpha}_{\eta}&\text{ if }\eta\in S\\\
Sp(h^{\alpha}_{\eta})&\text{ if }\eta\notin S\end{cases}$

where $h^{\alpha}_{\eta}$ here is considered as a forcing notion when forcing
with the tree and $Sp(h^{\alpha}_{\eta})$ denotes the specialization forcing
for the tree $h^{\alpha}_{\eta}$.

  2. 2.

If $F(\alpha)=(\beta,\gamma,1)$, then we look at the $\beta$-th forcing
$\mathbb{B}$ of size $\aleph_{1}$ in $W_{2}[I_{\gamma}]$ which has the
countable chain condition as seen in the universe $W_{2}[I_{\alpha}]$. We can
consider the iteration $((\mathbb{R}_{\zeta}\,:\,\zeta<\alpha)\ast\mathbb{B})$
which has a dense subforcing of size $\aleph_{1}$ (the dense set is just the
set of conditions in $\mathbb{R}_{\alpha}\ast\mathbb{B}$ which are fully
decided), ccc forcing in $W_{2}$ and can thus be seen as a subset of
$\omega_{1}$ in $W_{2}$. As $W_{2}=W_{1}[H_{\omega_{2}}]$, there is a stage
$\nu<\omega_{2}$ such that (a forcing equialent to)
$((\mathbb{R}_{\zeta}\,:\,\zeta<\alpha)\ast\mathbb{B})\in W_{1}[H_{\nu}]$. Now
if $\nu\leq\alpha$ we let $\dot{\mathbb{S}}^{I_{\alpha}}_{\alpha}$ be
$\mathbb{B}$. Otherwise we force with the trivial forcing.

In the first case of the definiton we will write codes in the sequence
$(\vec{h}^{\alpha}\,:\,\alpha<\omega_{2})$ of blocks of Suslin trees. The
definition of the second case ensures that we will not accidentally write an
unwanted pattern when forcing for $\mathsf{MA_{\omega_{1}}}$.

######  Lemma 3.19.

Assume we are at stage $\alpha$ of our iteration and we are in the nontrivial
part of case 2 of the definition, thus we force with an $\aleph_{1}$-sized
$\dot{\mathbb{S}}^{I_{\alpha}}_{\alpha}=\mathbb{B}$. Then, if $I_{\alpha+1}$
is
$\mathbb{R}_{\alpha+1}=\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha}$-generic
over $W[H_{\omega_{2}}]$, all the Suslin trees in $\vec{h}_{\zeta}$,
$\zeta\geq\alpha$ remain Suslin trees in $W[H_{\omega_{2}}][I_{\alpha+1}]$.

###### Proof.

Assume that we are at stage $\alpha$, thus the model we have produced so far
is $W_{1}[H_{\omega_{2}}][I_{\alpha}]$ and we force with
$\dot{\mathbb{S}}_{\alpha}$ which is a ccc forcing of size $\aleph_{1}$ in
$W_{1}[H_{\nu}][I_{\alpha}]$ for $\nu\leq\alpha$. Consider some block
$\vec{h}^{\zeta}$, $\zeta\geq\alpha$ in the universe
$W_{1}[H_{\omega_{2}}][\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha}]$. The
latter universe is obtained via the iteration

$(\mathbb{Q}_{\zeta}\,:\,\zeta<\omega_{2})\ast\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha}=(\mathbb{Q}_{\zeta}\,:\,\zeta\leq\nu)\ast(\mathbb{Q}_{\zeta}/\mathbb{Q}_{\nu}\,:\zeta>\nu)\ast\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha}$

and the right hand side can be rewritten as

$(\mathbb{Q}_{\zeta}\,:\,\zeta\leq\nu)\ast((\mathbb{Q}_{\zeta}/\mathbb{Q}_{\nu}\,:\zeta>\nu)\times(\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha})).$

As we can switch the order in products, the latter can be written as

$(\mathbb{Q}_{\zeta}\,:\,\zeta\leq\nu)\ast((\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha})\times(\mathbb{Q}_{\zeta}/\mathbb{Q}_{\nu}\,:\zeta>\nu)),$

and consequentially the $W_{1}[H_{\nu}]$-generic filter $H_{\nu,\omega_{2}}$
for the tail
$(\mathbb{Q}_{\zeta}/\mathbb{Q}_{\nu}\,:\,\zeta>\nu,\zeta<\omega_{2})$ remains
generic over the model
$W_{1}[(\mathbb{Q}_{\zeta}\,:\,\zeta\leq\nu)\ast(\mathbb{R}_{\alpha}\ast\dot{\mathbb{S}}_{\alpha})]$.
This means in particular that the generically added Suslin trees
$\vec{h}^{\zeta},\zeta>\nu$ for Tennenbaum’s forcing for adding a Suslin tree
which are elements in $H_{\nu,\omega_{2}}$ remain generic even over the ground
model $W_{1}[H_{\nu}\ast I_{\alpha+1}]$. Now Tennebaum’s forcing is computed
in an absolute way in every universe with the same $\omega_{1}$, and
trivially, every generic filter for it is a Suslin tree. Hence we obtain that
every $h^{\zeta}_{\eta}$, $\zeta>\nu,\eta<\omega_{1}$ is a Suslin tree in
$W_{1}[H_{\nu}\ast I_{\alpha+1}]$, thus every $h^{\zeta}_{\eta}$,
$\zeta\geq\alpha\geq\nu$ is a Suslin tree in $W_{1}[H_{\omega_{2}}\ast
I_{\alpha+1}]$ as claimed. ∎

After $\omega_{2}$-many steps we arrive at $W_{2}[I_{\omega_{2}}]$ which has
the desired properties. The first thing to note is that we are in full control
of the codes which are written into the sequences of blocks of Suslin trees
$(\vec{h}^{\alpha}\,:\,\alpha<\omega_{2})$.

######  Lemma 3.20.

In $W_{2}[I_{\omega_{2}}]$, a set $S\subset\omega_{1}$ is stationary if and
only if there is an $\alpha<\omega_{2}$ such that

  * $\forall\beta<\omega_{1}((\beta\in S\leftrightarrow h^{\alpha}_{\beta}$ has a branch $)$ and $(\beta\notin S\leftrightarrow h^{\alpha}_{\beta}$ is special $))$.

###### Proof.

If $S$ is stationary then the rules of the iteration guarantee that there is
such an $\alpha<\omega_{2}$ with the desired properties. On the other hand if
there is an $\alpha<\omega_{2}$ such that the $\alpha$-th block
$\vec{h}^{\alpha}$ sees a certain 0,1-pattern then by the last Lemma, this
pattern must come from the first case in the definition of our iteration.
Hence $S$ has to be stationary. ∎

######  Theorem 3.21.

In $W_{2}[I_{\omega_{2}}]$ $\mathsf{MA_{\omega_{1}}}$ holds,
$\hbox{NS}_{\omega_{1}}$ is saturated and
$\Delta_{1}(\vec{C},\vec{T}^{0})$-definable.

###### Proof.

That $\hbox{NS}_{\omega_{1}}$ is saturated is clear as $W_{2}[I_{\omega_{2}}]$
is a ccc extension of $W_{0}$ and $\hbox{NS}_{\omega_{1}}$ is saturated in
$W_{0}$ and ccc indestructible. The proof that $\mathsf{MA_{\omega_{1}}}$
holds in $W_{2}[I_{\omega_{2}}]$ is also clear as a standard computation
yields that the continuum is $\aleph_{2}$ in $W_{2}[I_{\omega_{2}}]$. Hence,
it is sufficient to show that $\mathsf{MA_{\omega_{1}}}$ holds for ccc posets
of size $\aleph_{1}$. Let $\mathbb{P}\in W_{2}[I_{\omega_{2}}]$ be such, then
there is a stage $\nu<\omega_{2}$ such that $\mathbb{P}\in W_{2}[I_{\nu}]$.
The rules of the iteration yield that we will consider $\mathbb{P}$
unboundedly often after stage $\nu$. Thus there will be a stage
$\alpha<\omega_{2}$ such that $\mathbb{P}$ is considered by the bookkeeping
$F$ and $\mathbb{R}_{\alpha}\ast\mathbb{P}$ is an element of $W_{1}[H_{\nu}]$
for $\nu\leq\alpha$, and hence we used $\mathbb{P}$ in the iteration
$(\mathbb{R}_{\eta}\,:\,\eta<\omega_{2})$, so $\mathsf{MA_{\omega_{1}}}$
holds.

In order to see that every $S\in W_{2}[I_{\omega_{2}}]$ has a
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definition we exploit the fact that the
trees $(h_{\alpha}\,:\,\alpha<\omega_{2})$ are
$\Sigma_{1}(\vec{C},\vec{T}^{0})$-definable in $W_{1}[I_{\omega_{2}}]$. We
claim that the following $\Sigma_{1}(\vec{C},\vec{T}^{0})$-formula
$\varphi(S)$ defines being stationary in $W_{2}[I_{\omega_{2}}]$:

  * $\varphi(S)$ if and only if there exists a triple of $(M_{1},M_{2},M_{3})$ of transitive models of size $\aleph_{1}$ such that $M_{1}\subset M_{2}\subset M_{3}$, $M_{1}$ is a suitable model and $M_{2}$ sees a full pattern on the trees in some $\omega_{1}$-block $\vec{T}^{\alpha}$. This pattern itself yields an $\omega_{1}$-block of trees $\vec{h}^{\beta}$ and $M_{3}$ sees a full pattern on $\vec{h}^{\beta}$ and this pattern is the characteristic function for $S$.

Note that the formula $\varphi(S)$ is of the form $\exists
M_{1},M_{2},M_{3}\,\sigma(M_{1},M_{2},M_{3},S)$ and $\sigma$ is $\Delta_{1}$
as all the statements in $\sigma$ are of the form $M_{i}\models...$ which is a
$\Delta_{1}$-formula.

By absoluteness and the way we defined our iteration, it is clear that if
$S\subset\omega_{1}$ is stationary in $W_{2}[I_{\omega_{2}}]$, then
$\varphi(S)$ holds.

On the other hand, if $\varphi(S)$ is true and $M_{1},M_{2}$ and $M_{3}$ are
witnesses to the truth of $\varphi(S)$, then, as they see full patterns, their
local patterns must coincide with the patterns in the real world
$W_{2}[I_{\omega_{2}}]$. But the last Lemma ensures that the patterns of
$W_{2}[I_{\omega_{2}}]$ characterize stationarity, so the proof is finished. ∎

As in [5], instead of working in an arbitrary $V$ with a Woodin cardinal, we
can work over the canonical inner model with one Woodin cardinal $M_{1}$. This
has the advantage, that we can replace the two parameters $\vec{C}$ and
$\vec{T}^{0}$ by just $\\{\omega_{1}\\}$. We will not go into any details and
just claim that the above proof can be applied over $M_{1}$, with all the
modifications exactly as in [5]. We therefore obtain the last theorem of this
article.

######  Theorem 3.22.

Let $M_{1}$ be the canonical inner model with one Woodin cardinal. Then there
is a generic extension of $M_{1}$, in which $\hbox{NS}_{\omega_{1}}$ is
saturated, $\Sigma_{1}(\omega_{1})$-definable, and $\mathsf{MA_{\omega_{1}}}$
holds.

####  3.2.3 Open questions

We end with a couple of natural problems which remained open.

######  Question 1.

Assume $\mathsf{PFA}$. Is there a $\Sigma_{1}$-formula and a set
$A\subset\omega_{1}$ such that

$\forall S\in P(\omega_{1})(S\text{ is stationary
}\Leftrightarrow\varphi(S,A))?$

######  Question 2.

Assume the existence of a Woodin cardinal. Is there a universe in which
$\mathsf{BPFA}$ holds, $\hbox{NS}_{\omega_{1}}$ is $\bf{\Delta_{1}}$-definable
over $H(\omega_{2})$ and $\hbox{NS}_{\omega_{1}}$ is saturated?

######  Question 3.

Assume the existence of a reflecting cardinal. Is $\mathsf{BPFA}$ consistent
with the non-existence of a $\Sigma_{1}(\omega_{1})$-definition of
$\hbox{NS}_{\omega_{1}}$?

## References

  * [1] A. Caicedo and B. Velickovic The bounded proper forcing axiom and wellorderings of the reals. Mathematical Research Letters, 13 (2-3) (2006). pp. 393-408
  * [2] M. Foreman and M. Magidor  Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on $P_{\kappa}(\lambda)$. Acta Math. 186 (2001), no. 2, 271–300.
  * [3] M. Goldstern and S. Shelah  The bounded proper forcing axiom. The Journal of Symbolic Logic, Volume 60, Number 1. March 1995. pp.58–73.
  * [4] L. Harrington Long projective wellorderings. Ann. Math. Logic, 12(1):1–24, 1977.
  * [5] S. Hoffelner $\hbox{NS}_{\omega_{1}}$ $\Delta_{1}$-definable and saturated. Journal of Symbolic Logic 86 (1), 2021, pp. 25 - 59.
  * [6] P.B. Larson Forcing over Models of Determinacy. Handbook of Set Theory. Springer, 2010.
  * [7] P.B. Larson The stationary tower. Notes on a course by W. Hugh Woodin. American Mathematical Society University Lecture Series, vol 32, 2004.
  * [8] P. Lücke, R. Schindler and P. Schlicht $\Sigma_{1}(\kappa)$-definable subsets of $H(\kappa^{+})$. Journal of Symbolic Logic, Volume 82, Issue 3, September 2017, pp. 1106-1131.
  * [9] T. Miyamoto On Iterating Semiproper Preorders. Journal of Symbolic Logic 67, pp. 1431-1468, 2002.
  * [10] R. Schindler and X. Sun $\mathsf{MM}$ and the definability of $\hbox{NS}_{\omega_{1}}$. Preprint 2022.
  * [11] S. Shelah  Proper and improper forcing. Second edition. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1998. xlviii+1020 pp
  * [12] A. Taylor Regularity properties of ideals and ultrafilters. Annals of Mathematical Logic, 16, pp.33-55, 1979.
  * [13] S.Todorcevic Localized reflection and fragments of PFA, Set theory (Piscataway, NJ, 1999), volume 58 of DIMACS series in discrete mathematics and theoretical computer science, pp. 135–148. American Mathematics Society, Providence, RI (2002)
  * [14] H. W. Woodin The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. De Gruyter, 2001.