$(H_{\kappa}^{V},\in_{C}^{V})\prec_{1}(V,\in_{C}^{V})$ holds in all models
$(V,\in)$ of $T$;

  * •

$\psi^{H_{\kappa}}$ is forcible over any model of $T$.

We use below this same pattern to analyze what can be said with regard to
$2^{\aleph_{0}}>\aleph_{2}$ by looking at the AMC-spectrum of set theory.

The rest of this section is dedicated to the proof of Thm. 1.12 (or of its
variant in line with Corollary 3.5.4).

######  Definition 3.5.6.

Let $\psi_{\aleph_{1}}(x)$ be the boolean combination of universal formulae
for $\in_{\Delta_{0}}$ whose unique solution is the first uncountable
cardinal363636Previously we noted this formula by $(x\text{ is }\aleph_{1})$
however for the sake of readabillty, we introduce below a constant symbol
$\omega_{1}$ to denote the least uncountable cardinal and we prefer to write
$\psi_{\aleph_{1}}(\omega_{1})$ rather than $(\omega_{1}\text{ is
}\aleph_{1})$, other occasions showing that this new convention is convenient
will arise along this section.. $c:[\omega_{1}]^{2}\to\omega$ is a _ladder
system on $\omega_{1}$_ if for every $\alpha$ limit point of $\omega_{1}$
$c\restriction\left\\{\alpha\right\\}\times\alpha\to\omega$ is surjective and
monotone increasing.

$P:\omega_{1}\to\omega$ is a _partition of $\omega_{1}$ in countably many
stationary sets_ if $P^{-1}[{n}]$ is stationary for all $n\in\omega$ and $P$
is surjective.

######  Fact 3.5.7.

There is a $\Delta_{0}$-formula $\psi_{\mathrm{ladder}}(x,y)$ such that any
solution of $\psi_{\mathrm{ladder}}(x,a)$ in any
$\in_{\Delta_{0}}\cup\left\\{a\right\\}$-model of

$\mathsf{ZF}_{\Delta_{0}}+\psi_{\aleph_{1}}(a)$

defines a ladder system on $\omega_{1}$ according to the model.

Accordingly there is a $\Pi_{1}$-formula $\psi_{\mathrm{countpartstat}}(x,y)$
for $\in_{\Delta_{0}}$ such that any solution of
$\psi_{\mathrm{countpartstat}}(x,a)$ in any model of

$\mathsf{ZF}_{\Delta_{0}}+\psi_{\aleph_{1}}(a)$

defines a partition of $\omega_{1}$ in countably many stationary sets
according to the model.

Furthermore

$\mathsf{ZFC}^{-}_{\Delta_{0}}+\psi_{\aleph_{1}}(a)\models\exists
z\,\psi_{\mathrm{ladder}}(z,a)\wedge\exists
y\,\psi_{\mathrm{countpartstat}}(y,a).$

######  Theorem 3.5.8 (Moore [18]).

Let $T$ be the $\in_{\Delta_{0}}\cup\left\\{\omega_{1},c,A\right\\}$-theory

$\mathsf{ZFC}^{-}_{\Delta_{0}}+\psi_{\aleph_{1}}(\omega_{1})+\psi_{\mathrm{countpartstat}}(A,\omega_{1})+\psi_{\mathrm{ladder}}(c,\omega_{1}).$

There is an
$\in_{\Delta_{0}}$-formula373737$\psi_{\mathrm{Moore}}(r,\alpha,f,C,\omega_{1},c,A)$
states that $f:\omega_{1}\to\alpha$ is a surjection, $C$ is a club subset of
$\omega_{1}$ and for all $\alpha\in C$, if $A(\alpha)=n$, then for eventually
all $\beta\in\alpha\cap C$,
$c(\beta,\alpha)<c(\operatorname{otp}(f[\beta]),\operatorname{otp}(f[\alpha])$
if and only if $r(n)=1$.
$\psi_{\mathrm{Moore}}(x,y,u_{0},u_{1},z_{0},z_{1},z_{2})$ such that in any
$\in_{\Delta_{0}}\cup\left\\{\omega_{1},c,A\right\\}$-model $\mathcal{M}$ of
$T$:

  1. (a)

If

$\mathcal{M}\models\psi_{\mathrm{Moore}}(r,\alpha,f,C,\omega_{1},c,A),$

then

$\mathcal{M}\models(\alpha\text{ is an
ordinal})\wedge(f:\alpha\to\omega_{1}\text{ is an
injection})\wedge(C\subseteq\omega_{1}\text{ is a
club})\wedge(r\subseteq\omega).$

  2. (b)

If

$\mathcal{M}\models r,s\subseteq\omega\wedge\exists
h\,(h:\alpha\to\omega_{1}\text{ is an injection}),$

then

$\mathcal{M}\models\forall
f,g,C,D\,\left[(\psi_{\mathrm{Moore}}(s,\alpha,f,C,\omega_{1},c,A)\wedge\psi_{\mathrm{Moore}}(r,\alpha,g,D,\omega_{1},c,A))\rightarrow
r=s\right].$

  3. (c)

$T+\mathsf{ZFC}_{\Delta_{0}}$ proves that

$\forall
r\subseteq\omega\,\exists\alpha,f,C\,\psi_{\mathrm{Moore}}(r,\alpha,f,C,\omega_{1},c,A)$

is forcible by a proper forcing.

By item c of the Theorem (and Thm. 4.1), if $S$ is
$T+\mathsf{ZFC}_{\Delta_{0}}+$_there are class many Woodin cardinals_

$\forall
r\subseteq\omega\exists\alpha\,\psi_{\mathrm{Moore}}(r,\alpha,\omega_{1},c,A)$

holds in a model of $S+R_{\forall}$ for any consistent
$\in_{\Delta_{0}}\cup\left\\{\omega_{1},c,A\right\\}$-theory $R$ extending
$S$.

######  Corollary 3.5.9.

Let $\theta_{\mathrm{Moore}}$ be the $\Pi_{2}$-sentence for $\in_{\Delta_{0}}$

$\forall
x,y,z\,\left[(\psi_{\aleph_{1}}(x)\wedge\psi_{\mathrm{ladder}}(y,x)\wedge\psi_{\mathrm{countpartstat}}(z,x))\rightarrow\forall
r\subseteq\omega\,\exists\alpha,f,C\,\psi_{\mathrm{Moore}}(r,\alpha,f,C,x,y,z)\right].$

Then the following holds:

  1. (1)

Assume $S\supseteq\mathsf{ZFC}$ and
$A\in\mathfrak{spec}_{\mathsf{AMC}}\left(S\right)$ are such that

$\exists x\,\psi_{\aleph_{1}}(x),\theta_{\mathrm{Moore}}$

are both in $\mathsf{AMC}(S,A)$. Let $\mathcal{M}$ be a model of
$\mathsf{AMC}(S,A)$ which satisfies the Replacement axiom schema; fix
$c,a,\omega_{1}^{\mathcal{M}}\in\mathcal{M}$ so that
$\psi_{\mathrm{countpartstat}}(a,\omega_{1}^{\mathcal{M}})$,
$\psi_{\mathrm{ladder}}(c,\omega_{1}^{\mathcal{M}})$ hold in $\mathcal{M}$.
Then $\exists f,C\,\psi_{Moore}(x,y,f,C,\omega_{1}^{\mathcal{M}},c,a)$ defines
in $\mathcal{M}$ the graph of a surjection of $\omega_{2}^{\mathcal{M}}$ onto
$\mathcal{P}\left(\omega\right)^{\mathcal{M}}$.

  2. (2)

There is at least one (recursive set) $B\subseteq\mathsf{Form}_{\in}\times 2$
with $\in_{B}\supseteq\in_{\Delta_{0}}$ such that for any $R$ extending

$\mathsf{ZFC}+\emph{there are class many Woodin cardinals and a supercompact
cardinal},$

$B\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ and

$\exists x\,\psi_{\aleph_{1}}(x),\theta_{\mathrm{Moore}}$

are both in $\mathsf{AMC}(R,B)$ with the latter implying the Replacement
schema.

  3. (3)

For any $R\supseteq\mathsf{ZFC}$ and
$A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ if

$\exists x\,\psi_{\aleph_{2}}(x),\theta_{\mathrm{Moore}}$

are both in $\mathsf{AMC}(R,A)$ and $\mathsf{AMC}(R,A)$ implies the
Replacement schema, then $\mathsf{AMC}(R,A)\models
2^{\aleph_{0}}\leq\aleph_{2}$.

  4. (4)

Assume $R\supseteq\mathsf{ZFC}$ and
$A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ are such that
$\mathsf{AMC}(R,A)$ implies the Replacement schema,
$(\psi_{\aleph_{2}}(x),1)\in A$, $\in_{A}\supseteq\in_{\Delta_{0}}$, and

$\mathsf{ZFC}+(R+T_{\in,A})_{\forall\vee\exists}+\theta_{\mathrm{Moore}}$

is consistent.

Then $2^{\aleph_{0}}>\aleph_{2}\not\in\mathsf{AMC}(R,A)$.

###### Proof.

__

1:

See the last item and adjust the proof from it.

2:

By Thm. 1.10.

3:

See the proof below and observe that the argument which below is given for
just one $\mathcal{M}$ which models $\mathsf{AMC}(R,A)$, in this case can be
repeated for all models of $\mathsf{AMC}(R,A)$.

4:

Since $\mathsf{AMC}(R,A)$ is the AMC of $R+T_{\in,A}$, we can find
$\mathcal{M}\prec_{1}\mathcal{N}$ with $\mathcal{N}$ an $\in_{A}$-model of

$T_{0}=\mathsf{ZFC}+(R+T_{\in,A})_{\forall\vee\exists}+\theta_{\mathrm{Moore}}$

and $\mathcal{M}$ an $\in_{A}$-model of $\mathsf{AMC}(R,A)$. Then

$\mathcal{M}\models\theta_{\mathrm{Moore}}+\mathsf{ZFC}^{-}_{A}.$

Fix $a,b,c,A\in\mathcal{M}$ such that $\mathcal{M}$ models:

  * •:

$\psi_{\aleph_{1}}(a)$,

  * •:

$\psi_{\aleph_{2}}(b)$,

  * •:

$\psi_{\mathrm{ladder}}(c,a)$,

  * •:

$\psi_{\mathrm{countpartstat}}(A,a)$.

Note that $\mathcal{N}$ models all of the above formulae with the same
parameters, since $\mathcal{M}\prec_{1}\mathcal{N}$.

Let $T$ be the $\in_{A}\cup\left\\{a,b\right\\}$\- theory

$T_{0}+\psi_{\aleph_{1}}(a)+\psi_{\aleph_{2}}(b).$

Note that $a,b$ are constant symbols interpreted according to the axioms
$\mathsf{Ax}^{1}_{\psi_{\aleph_{i}}}$ for $i=0,1$, that $\mathcal{M}$ and
$\mathcal{N}$ admit unique extensions to models
$\mathcal{M}^{*},\mathcal{N}^{*}$ of $T$, and that for such extensions it
still holds that $\mathcal{M}^{*}\prec_{1}\mathcal{N}^{*}$.

This gives that

(12) $\mathcal{M}^{*}\models\forall r\subseteq\omega\exists\alpha\in b\exists
f,C\,\psi_{\mathrm{Moore}}(r,\alpha,f,C,a,c,A).$

By Lemma 3.3.1 Replacement for $\in_{A}\cup\left\\{a,b\right\\}$-formulae
holds in $\mathcal{M}^{*}$. By Thm. 3.5.8b, $\mathcal{M}$ models

(13) $\forall\alpha\in b\exists!r\subseteq\omega\left[\exists
f,C\,\psi_{\mathrm{Moore}}(r,\alpha,f,C,a,c,A)\vee(\neg\exists
s\subseteq\omega,f,C\,\psi_{\mathrm{Moore}}(s,\alpha,f,C,a,c,A)\wedge
r=\emptyset)\right]$

Applying it to the existential $\in_{A}\cup\left\\{a,b\right\\}$-formula
$\exists f,C\,\psi_{\mathrm{Moore}}(x,y,f,C,a,v,w)$ with $v,w$ replaced by
parameters $c,A$, we get an $F\in\mathcal{M}$ which (by 12 and 13) is a
function with domain

$\left\\{\alpha\in\mathcal{M}:\,\mathcal{M}\models\alpha\in b\right\\}$

and range exactly given by

$\left\\{r\in\mathcal{M}:\,\mathcal{M}\models r\subseteq\omega\right\\}.$

Hence $\mathcal{P}\left(\omega\right)$ according to $\mathcal{M}$ is in
$\mathcal{M}$ the set $\operatorname{ran}(F)$, which is the surjective image
(via $F$) of the second uncountable cardinal according to $\mathcal{M}$.
Therefore $2^{\aleph_{0}}\leq\aleph_{2}$ holds in $\mathcal{M}$ as witnessed
by $\mathcal{F}$.

∎

######  Remark 3.5.10.

We expect to be able to prove for $\theta_{\mathrm{Moore}}$ and
$2^{\aleph_{0}}>\aleph_{2}$ a result of a similar vein of Cor. 3.5.9 for any
$A\subseteq\mathsf{Form}_{\in}\times 2$ such that
$\in_{A}\supseteq\in_{\Delta_{0}}$ and replacing $\mathsf{AMC}(S,A)$ (which
may not exist for many such $A$) with the strong consistency hull of
$S+T_{\in,A}$. However the proof becomes rather intricated since we must check
that the needed instances of replacement hold in the required models of
$\mathsf{SCH}(S+T_{\in,A})$. Most instances of replacement for
$\Sigma_{1}$-formulae hold in such structures. One then should argue that all
instances of replacement used in the above proof are among those which can
follow from $\mathsf{SCH}(S+T_{\in,A})$.

##  4\. Generic invariance results for signatures of second and third order
arithmetic

We collect here generic absoluteness results results needed to prove Thm.
1.10. We prove all these results working in “standard” models of
$\mathsf{ZFC}$, i.e. we assume the models are well-founded. This is a practice
we already adopted in Section 3.4. We leave to the reader to remove this
unnecessary assumption.

###  4.1. Universally Baire sets and generic absoluteness for second order
number theory

We recall here the properties of universally Baire sets and the generic
absoluteness results for second order number theory we need to prove Thm.
1.10.

####  4.1.1. Universally Baire sets

Assuming large cardinals, there is a very large sample of projectively closed
families of subsets of $\mathcal{P}\left(\omega\right)$ which are are
“simple”, hence it is natural to consider elements of these families as atomic
predicates.

The exact definition of what is meant by a “simple” subset of $2^{\omega}$ is
captured by the notion of universally Baire set.

Given a topological space $(X,\tau)$, $A\subseteq X$ is nowhere dense if its
closure has a dense complement, meager if it is the countable union of nowhere
dense sets, with the Baire property if it has meager symmetric difference with
an open set. Recall that $(X,\tau)$ is Polish if $\tau$ is a completely
metrizable, separable topology on $X$.

######  Definition 4.1.1.

(Feng, Magidor, Woodin) Given a compact Polish space $(X,\tau)$, $A\subseteq
X$ is _universally Baire_ if for every compact Hausdorff space $(Y,\sigma)$
and every continuous $f:Y\to X$ we have that $f^{-1}[A]$ has the Baire
property in $Y$.

$\mathsf{UB}$ denotes the family of universally Baire subsets of $X$ for some
compact Polish space $X$.

We adopt the convention that $\mathsf{UB}$ denotes the class of universally
Baire sets and of all elements of $\bigcup_{n\in\omega+1}(2^{\omega})^{n}$
(since the singleton of such elements are universally Baire sets).

The list below outlines three simple examples of projectively closed families
of universally Baire sets containing $2^{\omega}$.

######  Remark 4.1.2 (Woodin).

Let:

  * •

$T_{0}$ be the $\in_{\Delta_{0}}$-theory $\mathsf{ZFC}_{\Delta_{0}}+$_there
are infinitely many Woodin cardinals and a measurable above_ ;

  * •

$T_{1}$ be the $\in_{\Delta_{0}}$-theory $\mathsf{ZFC}_{\Delta_{0}}+$_there
are class many Woodin cardinals_ ;

  * •

$T_{2}$ be the $\in_{\Delta_{0}}$-theory $\mathsf{ZFC}_{\Delta_{0}}+$_there
are class many Woodin cardinals which are a limit of Woodin cardinals_.

The following holds:

  1. (A)

[16, Thm. 3.1.12, Thm. 3.1.19] Assume $(V,\in_{\Delta_{0}})$ models $T_{0}$.
Then every projective subset of $2^{\omega}$ is universally Baire.

  2. (B)

[16, Thm. 3.3.3, Thm. 3.3.5, Thm. 3.3.6, Thm. 3.3.8, Thm. 3.3.13, Thm. 3.3.14]
Assume $(V,\in_{\Delta_{0}})\models T_{1}$. Then $\mathsf{UB}$ is projectively
closed.

  3. (C)

(Woodin, Unpublished) Assume $(V,\in_{\Delta_{0}})\models T_{2}$. Then the
family of subsets of $2^{\omega}$ which are definable in
$L(\mathrm{Ord}^{\omega})$ consists of universally Baire sets.

We now list some standard facts about universally Baire sets we will need:

  1. (i)

[12, Thm. 32.22] $A\subseteq 2^{\omega}$ is universally Baire if and only if
for each forcing notion $P$ there are trees $T_{A},S_{A}$ on $2\times\delta$
for some $\delta>|P|$ such that $A=p[[T_{A}]]$ (where
$p:(2\times\delta)^{\omega}\to 2^{\omega}$ denotes the projection on the first
component and $[T]$ denotes the body of the tree $T$), and

$P\Vdash T_{A}\text{ and }S_{A}\text{ project to complements},$

by this meaning that for all $G$ $V$-generic for $P$

$V[G]\models(p[[T_{A}]]\cap p[[S_{A}]]=\emptyset)\wedge(p[[T_{A}]]\cup
p[[S_{A}]]=(2^{\omega})^{V[G]})$

  2. (ii)

Any two Polish spaces $X,Y$ of the same cardinality are Borel isomorphic [13,
Thm. 15.6].

  3. (iii)

The universally Baire sets are a pointclass closed under preimages by Borel
functions (by [13, Thm. 13.1]) and (assuming the existence of class many
Woodin cardinals) also under images by Borel functions (for example by [16,
Thm. 3.3.7]).

  4. (iv)

Given $\phi:\mathbb{N}\to\mathbb{N}$, $\prod_{n\in\omega}2^{\phi(n)}$ is
compact and Polish (it is actually homemomorphic to the union of $2^{\omega}$
with a countable Hausdorff space) [13, Thm. 6.4, Thm. 7.4].

Hence it is not restrictive (assuming large cardinals) to focus just on
universally Baire subsets of $2^{\omega}$ and of its countable products, which
is what we will do in the sequel.

######  Notation 4.1.3.

Given $G$ a $V$-generic filter for some forcing $P\in V$, $A\in\text{{\sf
UB}}^{V[G]}$ and $H$ $V[G]$-generic filter for some forcing $Q\in V[G]$,

$A^{V[G][H]}=\left\\{r\in(2^{\omega})^{V[G][H]}:V[G][H]\models r\in
p[[T_{A}]]\right\\},$

where $(T_{A},S_{A})\in V[G]$ is any pair of trees as given in item i above
such that $p[[T_{A}]]=A$ holds in $V[G]$, and $(T_{A},S_{A})$ project to
complements in $V[G][H]$.

###  4.2. Generic absoluteness for second order number theory

The generic absoluteness results we need in this section follow readily from
[16, Thm. 3.1.2] and the assumption that there exists class many Woodin limits
of Woodin. The Theorem below reduces these large cardinal assumptions to the
existence of class many Woodin cardinals.

######  Theorem 4.2.1.

Assume in $V$ there are class many Woodin cardinals. Let $\mathcal{A}\in V$ be
a family of universally Baire sets of $V$, and $G$ be $V$-generic for some
forcing notion $P\in V$.

Then

$(H_{\omega_{1}},\in,A:A\in\mathcal{A})\prec(H_{\omega_{1}}^{V[G]},\in,A^{V[G]}:A\in\mathcal{A}).$

The reader can find a proof in the author’s webpage, it is an improvement of
[25, Thm. 3.1].

It is now convenient to reformulate projective closure in a semantic way which
is handy when dealing with a fixed complete first order axiomatization of set
theory.

######  Definition 4.2.2.

Let
$\mathcal{A}\subseteq\bigcup_{n\in\omega}\mathcal{P}\left(\omega\right)^{n}$.
$\mathcal{A}$ is $H_{\omega_{1}}$-closed if any definable subset of
$\mathcal{P}\left(\omega\right)^{n}$ for some $n\in\omega$ in the structure

$(H_{\omega_{1}},\in,U:U\in\mathcal{A})$

is in $\mathcal{A}$.

Given a family
$\mathcal{X}\subseteq\bigcup_{n\in\omega}\mathcal{P}\left(\omega\right)^{n}$
its projective closure $\mathsf{PC}(\mathcal{X})$ is the smallest
$H_{\omega_{1}}$-closed family of subsets of
$\bigcup_{n\in\omega}\mathcal{P}\left(\omega\right)^{n}$ containing
$\mathcal{X}$.

It is immediate to check that if $T$ is the theory of $(V,\in)$ and
$\mathcal{A}$ is a family of universally Baire subsets of $V$, $\mathcal{A}$
is projectively closed for $T$ for the signature
$\in_{\Delta_{0}}\cup\mathcal{A}$ (according to Def. 3.4.5) if and only if it
is $H_{\omega_{1}}$-closed.

We get the following:

######  Corollary 4.2.3.

Assume $(V,\in)$ models $\mathsf{ZFC}+$_there are class many Woodin
cardinals_. Let $\mathcal{A}\subseteq\text{{\sf UB}}^{V}$ be
$H_{\omega_{1}}$-closed and
$\in_{\mathcal{A}}=\in_{\Delta_{0}}\cup\mathcal{A}$ be the signature in which
each element of $\mathcal{A}$ contained in
$\mathcal{P}\left(\omega\right)^{k}$ is a predicate symbol of arity $k$. Then
for any $G$ $V$-generic for some forcing $P\in V$ the
$\in_{\mathcal{A}}$-theory of $H_{\omega_{1}}^{V}$ is the AMC of the
$\in_{\mathcal{A}}$-theory of $V[G]$ and
$\left\\{A^{V[G]}:A\in\mathcal{A}\right\\}$ is $H_{\omega_{1}}^{V[G]}$-closed.

###### Proof.

The assumptions grant that

$(H_{\omega_{1}}^{V},\in_{\Delta_{0}}^{V},A:A\in\mathcal{A})\prec(H_{\omega_{1}}^{V[G]},\in_{\Delta_{0}}^{V[G]},A^{V[G]}:A\in\mathcal{A})\prec_{1}(V[G],\in_{\Delta_{0}}^{V[G]},A:A^{V[G]}\in\mathcal{A})$

(by Thm. 4.2.1 and by Lemma 3.1 applied in $V[G]$). Now the theory of
$H_{\omega_{1}}^{V}$ in signature $\in_{\mathcal{A}}$ is complete and model
complete, and is also the $\in_{\mathcal{A}}$-theory of
$H_{\omega_{1}}^{V[G]}$. We conclude that it is the AMC of the
$\in_{\mathcal{A}}$-theory of $V[G]$. It is also easy to check that
$\left\\{A^{V[G]}:A\in\mathcal{A}\right\\}$ is $H_{\omega_{1}}^{V[G]}$-closed.
∎

###  4.3. Generic invariance of the universal fragment of the theory of $V$
with predicates for the non-stationary ideal and for universally Baire sets

The results of this section are the key to establish Thm. 1.10. The proofs
require some familiarity with the basics of the
$\mathbb{P}_{\mathrm{max}}$-technology and with Woodin’s stationary tower
forcing.

######  Notation 4.3.1.

__

  * •

$\in_{\mathbf{NS}_{\omega_{1}}}$ is the signature
$\in_{\Delta_{0}}\cup\left\\{\omega_{1}\right\\}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$
with $\omega_{1}$ a constant symbol, $\mathbf{NS}_{\omega_{1}}$ a unary
predicate symbol.

  * •

$T_{\mathbf{NS}_{\omega_{1}}}$ is the $\in_{\mathbf{NS}_{\omega_{1}}}$-theory
given by $T_{\Delta_{0}}$ together with the axioms

$\omega_{1}\text{ is the first uncountable cardinal},$ $\forall
x\;[(x\subseteq\omega_{1}\text{ is non-
stationary})\leftrightarrow\mathbf{NS}_{\omega_{1}}(x)].$

  * •

$\mathsf{ZFC}^{-}_{\mathbf{NS}_{\omega_{1}}}$ is the
$\in_{\mathbf{NS}_{\omega_{1}}}$-theory

$\mathsf{ZFC}^{-}_{\Delta_{0}}+T_{\mathbf{NS}_{\omega_{1}}}.$

  * •

Accordingly we define $\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}}}$.

Clearly the above axioms are of the form $\text{Ax}^{0}_{\phi}$ for $\phi$ the
formula defining the non-stationary ideal on $\omega_{1}$, and
$\text{Ax}^{1}_{\phi}$ for $\phi$ the formula defining the first uncountable
cardinal. Furthermore the above axioms are $\Pi_{2}$-sentences of the relevant
signature.

######  Theorem 4.1.

Assume $(V,\in)$ models $\mathsf{ZFC}+$_there are class many Woodin
cardinals_. Then the $\Pi_{1}$-theory of $V$ for the language
$\in_{\Delta_{1}}\cup\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ is
invariant under set sized forcings383838Here we consider any
$A\subseteq(2^{\omega})^{k}$ in $\text{{\sf UB}}^{V}$ as a predicate symbol of
arity $k$, and we clearly assume that the symbols of $\in_{\Delta_{1}}$ are
interpreted according to
$\mathsf{ZFC}^{-}_{\Delta_{1}}+T_{\mathbf{NS}_{\omega_{1}}}$ in $V$ and in its
generic extensions..

Asperó and Veličkovic̀ provided the following basic counterexample to the
conclusion of the Theorem if large cardinal assumptions are dropped.

######  Remark 4.3.2.

Let $\phi(y)$ be the $\Sigma_{1}$-property in $\in_{\mathbf{NS}_{\omega_{1}}}$

$\exists y(y=\omega_{1}\wedge L_{y+1}\models y=\omega_{1}).$

Then $L$ models this property, while the property fails in any forcing
extension of $L$ which collapses $\omega_{1}^{L}$ to become countable.

######  Remark 4.3.3.

Note that for any $T$ extending $\mathsf{ZFC}+$_there are class many Woodin
cardinals_ and any signature $\tau$ extending $\in$ by predicates and function
symbols as prescribed by Thm. 4.1, we obtain that a $\Pi_{2}$-sentence $\psi$
for $\tau$ such that $\psi^{H_{\aleph_{2}}}$ is forcible over any model of $T$
is in the strong consistency hull of $T$, hence it is realized in any $T$-ec
model. In particular the strong consistency hull of
$\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}}}+$_there are class many Woodin
cardinals_ in signature $\in_{\mathbf{NS}_{\omega_{1}}}$ contains
$\neg\text{{\sf CH}}$ and the definable version of $2^{\aleph_{0}}=\aleph_{2}$
implied by $\theta_{\mathrm{Moore}}$. We instantiated in detail this argument
in Section 3.5 for the specific $\Pi_{2}$-sentences for
$\in_{\mathbf{NS}_{\omega_{1}}}$ which express $\neg\text{{\sf CH}}$ and for
$\theta_{\text{Moore}}$. Mutatis mutandis the same argument work for any
$\Pi_{2}$-sentence expressed in a signature to which Proposition 3.5.3
applies.

In order to prove Thm. 4.1 we need to recall some basic terminology and facts
about iterations of countable structures.

####  4.3.1. Generic iterations of countable structures

Recall that for a transitive model $(M,\in)$ of a suitable fragment of
$\mathsf{ZFC}$, $j:M\to N$ is the ultrapower embedding with respect to the
non-stationary ideal on $\omega_{1}$ if

  * •

$N$ is the transitive collapse of the structure

$(\left\\{[f]_{G}:\,f\in M,\,f:\omega_{1}^{M}\to M\text{ is a
function}\right\\},\in_{G})$

where:

    * –

$G$ is an $M$-generic filter for the forcing
$(\mathcal{P}\left(\omega_{1}\right)\setminus\mathbf{NS})^{M}$;

    * –

$f\equiv_{G}h$ if
$\left\\{\alpha\in\omega_{1}^{M}:h(\alpha)=f(\alpha)\right\\}\in G$, $[f]_{G}$
is the equivalence class of $f$ with respect to $\equiv_{G}$, and
$[f]_{G}\in[h]_{G}$ if $\left\\{\alpha\in\omega_{1}^{M}:h(\alpha)\in
f(\alpha)\right\\}\in G$;

  * •

$j(a)=[c_{a}]_{G}$ where $c_{a}:\omega_{1}^{M}\to M$ is constant with value
$a$ for any $a\in M$.

######  Definition 4.3.4.

[15, Def. 1.2] Let $M$ be a transitive countable model of $\mathsf{ZFC}$. Let
$\gamma$ be an ordinal less than or equal to $\omega_{1}$. An iteration
$\mathcal{J}$ of $M$ of length $\gamma$ consists of transitive sets $\langle
M_{\alpha}:\,\alpha\leq\gamma\rangle$, sets $\langle
G_{\alpha}:\,\alpha<\gamma\rangle$ and a commuting family of elementary
embeddings

$\langle j_{\alpha\beta}:M_{\alpha}\to
M_{\beta}:\,\alpha\leq\beta\leq\gamma\rangle$

such that:

  * •

$M_{0}=M$,

  * •

each $G_{\alpha}$ is an $M_{\alpha}$-generic filter for
$(\mathcal{P}\left(\omega_{1}\right)\setminus\mathbf{NS}_{\omega_{1}})^{M_{\alpha}}$,

  * •

each $j_{\alpha\alpha}$ is the identity mapping,

  * •

each $j_{\alpha\alpha+1}$ is the ultrapower embedding with respect to the non-
stationary ideal on $\omega_{1}$ of $(M_{\alpha},\in)$ induced by
$G_{\alpha}$,

  * •

for each limit ordinal $\beta\leq\gamma$, $M_{\beta}$ is the transitive
collapse of the direct limit of the system
$\left\\{M_{\alpha},j_{\alpha\delta}:\,\alpha\leq\delta<\beta\right\\}$, and
for each $\alpha<\beta$, $j_{\alpha\beta}$ is the induced embedding.

We adopt the convention to denote an iteration $\mathcal{J}$ just by $\langle
j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma\rangle$, we also stipulate that if
$X$ denotes the domain of $j_{0\alpha}$, $X_{\alpha}$ or $j_{0\alpha}(X)$ will
denote the domain of $j_{\alpha\beta}$ for any $\alpha\leq\beta\leq\gamma$.

######  Definition 4.3.5.

Let $A$ be a universally Baire sets of reals. $M$ is $A$-iterable if:

  1. (1)

$M$ is transitive and such that $H_{\omega_{1}}^{M}$ is countable.

  2. (2)

$M\models\mathsf{ZFC}+\mathbf{NS}_{\omega_{1}}$_is precipitous_.

  3. (3)

Any iteration

$\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma\right\\}$

of $M$ is well founded and such that $A\cap M_{\beta}=j_{\alpha\beta}(A\cap
M_{0})$ for all $\beta\leq\gamma$.

####  4.3.2. Proof of Theorem 4.1

###### Proof.

In view of Facts 3.1.7, 3.1.9 it suffices to prove the Theorem for the
signature $\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}.$ Let $\phi$
be a $\Pi_{1}$-sentence for this signature.

Assume $\phi$ holds in $V$ but for some forcing notion $P$, $\phi$ fails in
$V[h]$ with $h$ $V$-generic for $P$. By forcing over $V[h]$ with the
appropriate stationary set preserving (in $V[h]$) forcing notion (using a
Woodin cardinal $\gamma$ of $V[h]$), we may assume that $V[h]$ is extended to
a generic extension $V[g]$ such that $V[g]$ models $\mathbf{NS}_{\omega_{1}}$
is saturated393939A result of Shelah whose outline can be found in [19,
Chapter XVI], or [26], or in an handout of Schindler available on his
webpage.. Since $V[g]$ is an extension of $V[h]$ by a stationary set
preserving forcing and there are in $V[h]$ class many Woodin cardinals, we get
that $V[h]\sqsubseteq V[g]$ with respect to the signature
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$.

Since $\Sigma_{1}$-properties are upward absolute and $\neg\phi$ holds in
$V[h]$, $\phi$ fails in $V[g]$ as well.

Let $\delta$ be inaccessible in $V[g]$ and let $\gamma>\delta$ be a Woodin
cardinal.

Let $G$ be $V$-generic for $\mathcal{T}^{\omega_{1}}_{\gamma}$ (the countable
tower $\mathbb{Q}_{<\gamma}$ according to [16, Section 2.7]) and such that
$g\in V[G]$. Let $j_{G}:V\to\operatorname{Ult}(V,G)$ be the induced ultrapower
embedding.

Now remark that $V_{\delta}[g]\in\operatorname{Ult}(V,G)$ is
$B^{V[G]}$-iterable in $\operatorname{Ult}(V,G)$ for all $B\in\mathsf{UB}^{V}$
(since $V_{\eta}[g]\in\operatorname{Ult}(V,G)$ for all $\eta<\gamma$, and this
suffices to check that $V_{\delta}[g]$ is $B^{V[G]}$-iterable for all
$B\in\mathsf{UB}^{V}$, see [15, Thm. 4.10]).

By [15, Lemma 2.8] applied in $\operatorname{Ult}(V,G)$, there exists in
$\operatorname{Ult}(V,G)$ an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma=\omega_{1}^{\operatorname{Ult}(V,G)}\right\\}$
of $V_{\delta}[g]$ such that
$\mathbf{NS}_{\omega_{1}}^{X_{\gamma}}=\mathbf{NS}_{\omega_{1}}^{\operatorname{Ult}(V,G)}\cap
X_{\gamma}$, where $X_{\alpha}=j_{0\alpha}(V_{\delta}[g])$ for all
$\alpha\leq\gamma=\omega_{1}^{\operatorname{Ult}(V,G)}$.

This gives that $X_{\gamma}\sqsubseteq\operatorname{Ult}(V,G)$ for
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}.$ Since
$V_{\delta}[g]\models\neg\phi$, so does $X_{\gamma}$, by elementarity. But
$\neg\phi$ is a $\Sigma_{1}$-sentence, hence it is upward absolute for
superstructures, therefore $\operatorname{Ult}(V,G)\models\neg\phi$. This is a
contradiction, since $\operatorname{Ult}(V,G)$ is elementarily equivalent to
$V$ for $\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$, and
$V\models\phi$.

A similar argument shows that if $V$ models a $\Sigma_{1}$-sentence $\phi$ for
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ this will remain true
in all of its generic extensions:

Assume $V[h]\models\neg\phi$ for some $h$ $V$-generic for some forcing notion
$P\in V$. Let $\gamma>|P|$ be a Woodin cardinal, and let $g$ be $V$-generic
for404040$\mathcal{T}_{\gamma}$ is the full stationary tower of height
$\gamma$ whose conditions are stationary sets in $V_{\gamma}$, denoted as
$\mathbb{P}_{<\gamma}$ in [16], see in particular [16, Section 2.5].
$\mathcal{T}_{\gamma}$ with $h\in V[g]$ and
$\operatorname{crit}(j_{g})=\omega_{1}^{V}$ (hence there is in $g$ some
stationary set of $V_{\gamma}$ concentrating on countable sets). Then
$V[g]\models\phi$ since:

  * •

$V_{\gamma}\models\phi$, since $V_{\gamma}\prec_{1}V$ for
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ by Lemma 3.1;

  * •

$V_{\gamma}^{\operatorname{Ult}(V,g)}=V_{\gamma}^{V[g]}$, since $V[g]$ models
that $\operatorname{Ult}(V,g)^{<\gamma}\subseteq\operatorname{Ult}(V,g)$;

  * •

$V_{\gamma}^{\operatorname{Ult}(V,g)}\models\phi$, by elementarity of $j_{g}$,
since $j_{g}(V_{\gamma})=V_{\gamma}^{\operatorname{Ult}(V,g)}$;

  * •

$V_{\gamma}^{V[g]}\prec_{1}V[g]$ with respect to
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\text{{\sf UB}}^{V}$ again by Lemma 3.1
applied in $V[g]$.

Now repeat the same argument as before for the $\Pi_{1}$-property $\neg\phi$,
with $V[h]$ in the place of $V$ and $V[g]$ in the place of $V[h]$. ∎

##  5\. Model companionship results for the theory of $H_{\aleph_{2}}$ and
axiom $(*)$

Let UB denote the family of universally Baire sets; for
$\mathcal{A}\subseteq\text{{\sf UB}}$ $L(\mathcal{A})$ denotes the smallest
transitive model of $\mathsf{ZF}$ which contains $\mathcal{A}$ (see for
details Section 4.1.1).

We will be interested in (what we will call _generically tame_) families
$\mathcal{A}$ of universally Baire sets which are constructibly closed (i.e.
$\mathcal{P}\left(2^{\omega}\right)^{L(\mathcal{A})}=\mathcal{A}$), countably
closed (e.g. $\mathcal{A}^{\omega}\subseteq L(\mathcal{A})$), and generically
invariant (e.g. the theory of $L(\mathcal{A})$ cannot be changed by set sized
forcing). One example will be given by
$\mathcal{A}=\mathcal{P}\left(2^{\omega}\right)^{L(\mathrm{Ord}^{\omega})}$
assuming large cardinals414141This remarkable result of Woodin is to my
knowledge unpublished. There are some handwritten notes of a proof sketch by
Larson, and it is mentioned in [16, Remark 3.3.12]., another by the class UB
itself as computed in any generic extension of $V$ collapsing a supercompact
to countable.

We aim to prove two model companionship results relating the theory of $V$ to
that of $H_{\omega_{2}}$. First of all we must include a certain finite and
explicit set of properties and definable functions which are
$\Delta_{1}(\mathsf{ZFC}^{-})$ to $\in_{\Delta_{0}}$ (in order to be able to
express by means of quantifier free formulae certain absolute concepts of set
theory). We may call this extended signature $\in_{\Delta_{1}}$. This is
harmless in view of Fact 3.1.9, since we will only consider $\in$-structures
which are models of $\mathsf{ZFC}^{-}$. We will then consider signatures
extending
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}},\omega_{1}\right\\}$
with predicate symbols for certain elements of a generically tame
$\mathcal{A}$: one with predicate symbols for _all_ elements of $\mathcal{A}$,
the other just for the _lightface definable_ elements of $\mathcal{A}$.

  * •

The first result establishes the equivalence between the conditional version
of Woodin’s axiom $(*)$ to the model $L(\mathcal{A})$ and the assertion that
the theory of $V$ is the model companion of the theory of $H_{\omega_{2}}$ in
signature424242We consider an element $A\subseteq(2^{\omega})^{k}$ of
$\mathcal{A}$ as a $k$-ary predicate symbol, $\mathbf{NS}_{\omega_{1}}$ as a
unary predicate symbol, $\omega_{1}$ as a constant symbol.
$\in_{\Delta_{1}}\cup\mathcal{A}\cup\left\\{\mathbf{NS}_{\omega_{1}},\omega_{1}\right\\}$.

This result however makes sense to a platonist but not to a formalist since it
subsumes the existence of $V$ and it is stated for a signature which is highly
non constructive (it includes predicate symbols for the —at least continuum
many— elements of $\mathcal{A}$ in $V$).

  * •

The second result is a model companionship result for $\mathsf{ZFC}+$_large
cardinals_ with respect to the signature
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ extending
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}},\omega_{1}\right\\}$
only with predicate symbols for those elements of $\mathcal{A}$ which are
provably the extension of an $\in$-formula $\phi(x)$ in no parameters. This is
a recursive signature and we can give a recursive axiomatization of the model
companion theory. This model companion theory is the common chore of the
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-theory of $H_{\omega_{2}}^{V}$ as
$(V,\in)$ ranges over the models of $\text{{\sf MM}}^{++}+$_suitable large
cardinal axioms_ ; moreover the $\Pi_{1}$-fragment of this theory is forcing
invariant.

We will actually need a generic invariance property for $\mathcal{A}$ which is
delicate to formulate as $\mathcal{A}^{V}$ and $\mathcal{A}^{V[G]}$ share
their defining $\in$-formula but, in general, have a trival intersection
(since for most $V$-generic filters $G$ on the one hand there could be
completely new universally Baire existing in
$\mathcal{A}^{V[G]}\setminus\mathcal{A}^{V}$ — even if $G$ does not adds new
reals, on the other hand $A^{V[G]}\neq A^{V}$ for any uncountable universally
Baire set $A\in\mathcal{A}^{V}$ — in case $G$ adds a new real). These
difficulties lead us to formulate the above properties by means of the
somewhat convoluted syntactic definitions given below.

######  Notation 5.1.

Let $(V,\in)$ be a model of $\mathsf{ZFC}$.

A family $\mathcal{A}$ of subsets of434343For the remainder of the paper to
avoid heavy notation we feel free to identify when needed $2^{\omega}$ with
$\mathcal{P}\left(\omega\right)$, $\mathcal{P}\left(\omega^{<\omega}\right)$
or any variation of these sets which is clearly a canonical representation of
the Cantor space. $\mathcal{P}\left(2^{\omega}\right)$ is:

  * •

_constructibly closed_ if

$\mathcal{P}\left(2^{\omega}\right)^{L(\mathcal{A})}=\mathcal{A},$

and every binary relation $R$ on $2^{\omega}\times 2^{\omega}$ in
$\mathcal{A}$ can be uniformized by a function $f:2^{\omega}\to 2^{\omega}$ in
$\mathcal{A}$.

  * •

_countably closed_ if $\mathcal{A}^{\omega}\subseteq L(\mathcal{A})$,

  * •

_lightface definable_ if it is the extension in $V$ of some $\in$-formula
$\phi_{\mathcal{A}}(x)$ without parameters.

Woodin calls slight variants of ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ as
defined below “Sealing” for $\mathcal{A}$.

######  Definition 5.2.

${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ is the conjuction of the following
three first order $\in$-sentences construed from some $\in$-formula
$\phi_{\mathcal{A}}(x)$ in one free variable (together with the axioms of set
theory needed to make sense of them444444To make sense of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ it suffices that $(V,\in)$ satisfies
all axioms of $\mathsf{ZFC}$ with the exception of replacement and
comprehension, with the latters replaced by the $\Sigma_{n}$-replacement
schema for some large enough $n$. It is well known that this theory is
finitely axiomatizable for each $n\in\omega$.):

  1. (1)

The extension of $\phi_{\mathcal{A}}(x)$ in any generic extension $V[G]$ is a
constructibly closed family of universally Baire sets of $V[G]$.

  2. (2)

As above but replacing _constructibly closed_ with _countably closed_.

  3. (3)

Whenever $G$ is $V$-generic for some forcing notion $P\in V$ and $H$ is
$V[G]$-generic for some forcing notion $Q\in V[G]$ there are class of ordinals
$I_{V[G]},I_{V[G][H]}$ such that any order preserving map of $I_{V[G]}$ into
$I_{V[G][H]}$, combined with the map $A^{V[G]}\mapsto A^{V[G][H]}$ extends
uniquely to an elementary embedding of $(L(\mathcal{A}^{V[G]}),\in)$ into
$(L(\mathcal{A}^{V[G][H]}),\in)$ which is the identity on
$H_{\omega_{1}}^{V[G]}$.

  * •

An $\in$-formula $\phi_{\mathcal{A}}(x)$ in one free variable is _generically
tame for $T$_ if $T\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}$.

  * •

$\mathcal{A}\in V$ is _generically tame for $T$_ if $\mathcal{A}$ is the
extension of some $\in$-formula $\phi_{\mathcal{A}}(x)$ in $(V,\in)$ such
that:

    * –

$T\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}$,

    * –

$(V,\in)\models T$.

  * •

$\mathcal{A}\in V$ is _generically tame_ if it is generically tame for the
theory of $(V,\in)$.

Note that if $(V,\in)\models T$,
$T\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}$ and $G$ is $V$-generic for some
forcing $P\in V$, we may have that $V[G]\not\models T$ nonetheless
$V[G]\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}$ holds for sure. In
particular ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ is a consequence of $T$
which is preserved by any forcing over any model of $T$, while other axioms of
$T$ may not (for example $T\supseteq\mathsf{ZFC}$ might have either one of CH
or $\neg\text{{\sf CH}}$ among its axioms, and neither of them is preserved
through forcing extensions over models of $T$).

By [16, Remark 3.3.12, Thm. 3.3.14, Thm. 3.3.19, Thm 3.4.18] and [13, Thm.
36.9] we get the following:

  1. (i)

Assume $T_{0}$ extends $\mathsf{ZFC}+$_there are class many Woodin cardinals
which are a limit of Woodin cardinals_.

Then the $\in$-formula $\phi_{0}(x)$ defining454545The _Chang model_
$L(\mathrm{Ord}^{\omega})$ is the smallest transitive model of $\mathsf{ZF}$
containing all the countable sequences of ordinals.
$\mathcal{P}\left(2^{\omega}\right)^{L(\mathrm{Ord}^{\omega})}$ defines a
generically tame family for $T_{0}$; e.g.
$T_{0}\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}$ for
$\mathcal{A}=\mathcal{P}\left(2^{\omega}\right)^{L(\mathrm{Ord}^{\omega})}$.

  2. (ii)

Assume $T_{1}$ extends $T_{0}$ with the axiom464646This is first order
expressible by [17].

> _Any model $(V,\in)$ of $T_{1}$ is a generic extension of some inner model
> $(W,\in)$ where some countable ordinal $\delta$ of $V$ is a supercompact
> cardinal in $W$._

Then the universally Baire sets are a generically tame family for $T_{1}$;
e.g. $T_{1}\models{\mathbf{MAX}(\mathsf{UB})}$ for $\phi_{\text{{\sf UB}}}(x)$
an $\in$-formula defining the universally Baire sets.

Let us now be precise in our definition of the relevant signatures of this
Section.

######  Notation 5.3.

Given an $\in$-formula $\phi_{\mathcal{A}}(x)$:

  * •

$\in_{\Delta_{0},\mathcal{A}}$ is the extension of $\in_{\Delta_{0}}$ in which
we add an $n$-ary relation symbol $S_{\phi}$ for any $\in$-formula $\phi$ of
arity $n$;

  * •

$T_{\Delta_{0},\mathcal{A}}$ is the extension of $T_{\Delta_{0}}$ by the
axioms

$\forall\vec{x}\,\left[S_{\phi}(\vec{x})\leftrightarrow(\phi^{L(\mathcal{A})}(\vec{x})\wedge\bigwedge_{i=1}^{n}x_{i}\in\mathcal{P}\left(\omega\right))\right]$

as $\phi$ ranges over the $\in$-formulae of some arity $n$;

  * •

$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=\in_{\Delta_{0},\mathcal{A}}\cup\left\\{\mathbf{NS}_{\omega_{1}},\omega_{1}\right\\}$;

  * •

$T_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=T_{\Delta_{0},\mathcal{A}}+T_{\mathbf{NS}_{\omega_{1}}}$
(recall Notation 4.3.1);
$\in_{\Delta_{1},\mathcal{A}}=\in_{\Delta_{0},\mathcal{A}}\cup\in_{\Delta_{1}}$;

  * •

$T_{\Delta_{1},\mathcal{A}}=T_{\Delta_{0},\mathcal{A}}+T_{\Delta_{1}}$.

  * •

$\mathsf{ZFC}^{-}_{\Delta_{i},\mathcal{A}}$,
$\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$,…are all defined as
expected.

$T_{\Delta_{0},\mathcal{A}}$ makes the subsets of
$\mathcal{P}\left(\omega\right)$ definable in no parameters in
$(L(\mathcal{A}),\in)$ the extension of an atomic formula of
$\in_{\Delta_{0},\mathcal{A}}$.

######  Remark 5.4.

It is clear that
$\in_{\Delta_{0},\mathcal{A}},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ can
all be expressed as $\in_{E}$ for suitably chosen recursive sets
$E\subseteq\mathsf{Form}_{\in}\times 2$, while $\in_{\Delta_{1},\mathcal{A}}$
and $T_{\Delta_{1},\mathcal{A}}$ are semi-recursive sets. Nonetheless in view
of Facts 3.1.7, 3.1.9 it is actually irrelevant in our set up if we work with
an $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ model of
$\mathsf{ZFC}^{-}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ or with its unique
expansion to a
$\in_{\Delta_{1}}\cup\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-model of
$\mathsf{ZFC}^{-}_{\Delta_{1}}$. More precisely, we will feel free to be
somewhat sloppy in stating some results about these signatures, by stating
them with respect to $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ and proving
them freely using
$\in_{\Delta_{1}}\cup\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$. Facts 3.1.7,
3.1.9 grant that in all cases where we do it, this is feasible and correct.

Key to all results of this section is an analysis of the properties of generic
extensions by $\mathbb{P}_{\mathrm{max}}$ of $L(\mathcal{A})$ for
$\mathcal{A}$ generically tame. Generic tameness of $\mathcal{A}$ is used to
argue (among other things) that most of the results established in [15] on the
properties of $\mathbb{P}_{\mathrm{max}}$ for $L(\mathbb{R})$ can be also
asserted for $L(\mathcal{A})$. We refrain to define the
$\mathbb{P}_{\mathrm{max}}$-forcing rightaway and we will introduce it when
needed in our proofs (see Def. 5.5.6). Meanwhile we assume the reader is
familiar with $\mathbb{P}_{\mathrm{max}}$ or can accept as a blackbox its
existence as a certain forcing notion; our reference on this topic is [15].

We now give a precise definition of $(*)\text{-}\mathcal{A}$ (modulo the
definition of $\mathbb{P}_{\mathrm{max}}$).

######  Definition 5.5.

$(*)\text{-}\mathcal{A}$ holds in some $\mathsf{ZFC}$-model $(V,\in)$ if in
$V$:

  * •

there are class many Woodin cardinals;

  * •

${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ holds for some $\in$-formula
$\phi_{\mathcal{A}}(x)$;

  * •

$\mathbf{NS}_{\omega_{1}}$ is precipitous474747See [16, Section 1.6, pag. 41]
for a definition of precipitousness and a discussion of its properties.;

  * •

there exists a filter $G$ on $\mathbb{P}_{\mathrm{max}}$ meeting all the dense
subsets of $\mathbb{P}_{\mathrm{max}}$ definable in $L(\mathcal{A})$ (where
$\mathcal{A}\in V$ is the extension of the formula $\phi_{\mathcal{A}}(x)$).

$(*)\text{-}\mathsf{UB}$ holds if $(*)\text{-}\mathcal{A}$ holds for
$\mathcal{A}=\text{{\sf UB}}$.

Woodin’s axiom $(*)$ as defined in [15, Def. 7.5] is $(*)\text{-}\mathcal{A}$
for $\mathcal{A}=\mathcal{P}\left(2^{\omega}\right)^{L(\mathbb{R})}$ with
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ omitted.

This is the first main result of this section:

######  Theorem 5.6.

Assume $(V,\in)$ models

$\mathsf{ZFC}+\emph{there is a supercompact cardinal and class many Woodin
cardinals}$

and $\mathcal{A}\in V$ is generically tame.

TFAE:

  1. (1)

$(V,\in)$ models $(*)\text{-}\mathcal{A}$;

  2. (2)

$\mathbf{NS}_{\omega_{1}}$ is precipitous484848A key observation is that
$\mathbf{NS}_{\omega_{1}}$ being precipitous is independent of $\mathsf{CH}$
(see for example [16, Thm. 1.6.24]), while $(*)\text{-}\mathcal{A}$ entails
$2^{\aleph_{0}}=\aleph_{2}$ (for example by the results of [15, Section 6]).
Another key point is that we stick to the formulation of
$\mathbb{P}_{\mathrm{max}}$ as in [15] so to be able in its proof to quote
verbatim from [15] all the relevant results on
$\mathbb{P}_{\mathrm{max}}$-preconditions we will use. It is however possible
to develop $\mathbb{P}_{\mathrm{max}}$ focusing on Woodin’s countable tower
rather than on the precipitousness of $\mathbf{NS}_{\omega_{1}}$ to define the
notion of $\mathbb{P}_{\mathrm{max}}$-precondition. Following this approach in
all its scopes, one should be able to reformulate Thm. 5.6(2) omitting the
request that $\mathbf{NS}_{\omega_{1}}$ is precipitous. We do not explore this
venue any further. in $V$ and the
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$-theory494949E.g. we regard
$A\subseteq(2^{\omega})^{k}$ as a $k$-ary predicate symbol for any
$A\in\mathcal{A}$. of $V$ has as model companion the
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$-theory of
$H_{\omega_{2}}$.

Thm. 1.13 follows immediately from the above Theorem.

Note the difference between $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ which
is a countable recursive signature, and
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$ for $(V,\in)$ a model of
$\mathsf{ZFC}$. The latter has predicate symbols for the at least continuum
(in the sense of $V$) many elements of $\mathcal{A}$ existing in $V$.

An objection to Thm. 5.6 is that it subsumes the Platonist standpoint that
there exists a definite universe of sets. We can prove a version of Thm. 5.6
which makes perfect sense also to a formalist and from which we immediately
derive Thm. 1.10. This is one of the reasons we paid attention to give a
syntactically meaningful definition of generic tameness and of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$.

######  Theorem 5.7.

Assume $T$ is an $\in$-theory extending505050With
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ predicated for the $\in$-formula
$\phi_{\mathcal{A}}(x)$.

$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+\text{ there is a
supercompact cardinal and class many Woodin cardinals}.$

Let $B\subseteq\mathsf{Form}\times 2$ be such that $\in_{B}$ is exactly
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$. Then
$B\in\mathfrak{spec}_{\mathsf{AMC}}\left(T\right)$.

Moreover TFAE for any for any $\Pi_{2}$-sentence $\psi$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$:

  1. (A)

$\mathsf{AMC}(T,B)\vdash\psi$.

  2. (B)

$(V[G],\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V[G]})\models\psi^{H_{\omega_{2}}}$
whenever $(V,\in)\models T$, $V[G]$ is a forcing extension of $V$, and
$(V[G],\in)\models(*)\text{-}\mathcal{A}$.

  3. (C)

$T$ proves515151$\dot{H}_{\omega_{2}}$ denotes a canonical $P$-name for
$H_{\omega_{2}}$ as computed in generic extension by $P$.
$\Vdash_{P}\psi^{\dot{H}_{\omega_{2}}}$ stands for:
$(V[G],\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V[G]})\models\psi^{H_{\omega_{2}}}$
whenever $G$ is $V$-generic for $P$.

$\exists P\,(P\text{ is a \emph{stationary set preserving} partial order
}\wedge\Vdash_{P}\psi^{\dot{H}_{\omega_{2}}}).$

  4. (D)

$T$ proves

$\exists P\,(P\text{ is a partial order
}\wedge\Vdash_{P}\psi^{\dot{H}_{\omega_{2}}}).$

  5. (E)

$T$
proves525252$L(\mathcal{A})\models[\mathbb{P}_{\mathrm{max}}\Vdash\psi^{\dot{H}_{\omega_{2}}}]$
stands for:
$(L(\mathcal{A})[G],\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{L(\mathcal{A})[G]})\models\psi^{H_{\omega_{2}}}$
whenever $G$ is $L(\mathcal{A})$-generic for $P$.

$L(\mathcal{A})\models[\mathbb{P}_{\mathrm{max}}\Vdash\psi^{\dot{H}_{\omega_{2}}}].$

  6. (F)

If $(V,\in)\models T$ and $\psi$ is $\forall x\exists y\,\phi(x,y)$ with
$\phi$ quantifier free $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-formula,
then for all535353See [2, Def. 1.8] for the notion of honest consistency. It
can be equivalently stated as: _For some $\kappa$ and $G$ $V$-generic for
$\operatorname{Coll}(\omega,\kappa)$, there is a transitive set $M\in V[G]$
such that:_
$(H_{\omega_{2}}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V},\text{{\sf
UB}}^{V})\sqsubseteq(M,\in_{\Delta_{1}}^{M},\mathbf{NS}_{\omega_{1}}^{M},B^{V[G]}\cap
M:B\in\text{{\sf UB}}^{V}),$ $(M,\in_{\Delta_{1}}^{M},B^{V[G]}\cap
M:B\in\text{{\sf
UB}}^{V})\sqsubseteq(V[G],\in_{\Delta_{1}}^{V[G]},B^{V[G]}:B\in\text{{\sf
UB}}^{V}),$ _and_
$(M,\in_{\Delta_{1}}^{M},\mathbf{NS}_{\omega_{1}}^{M},B^{V[G]}\cap
M:B\in\text{{\sf UB}}^{V})\models\exists y\,\phi(A,y).$ $A\in
H_{\omega_{2}}^{V}$

$\exists y\,\phi(A,y)\text{ is \emph{honestly consistent} according to $V$}.$

  7. (G)

For any consistent $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-theory

$S\supseteq T+T_{\in,B},$

$S_{\forall\vee\exists}+\psi$ is consistent.

Note that even if $T\models\text{{\sf CH}}$, $\neg\text{{\sf CH}}$ is in
$\mathsf{AMC}(T,B)$ (for example by E above). In particular the model
companion $\mathsf{AMC}(T,B)$ of $T$ describes a theory of $H_{\aleph_{2}}$
which can be completely unrelated to that given by models of $T$. Moreover
recall again that CH is not expressible as a boolean combination of
$\Pi_{1}$-sentences in $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ for models
of $T$: it is not preserved by forcing, while $T_{\forall\vee\exists}$ is.

The rest of this section is devoted to proof of Theorems 5.6 and 5.7. Crucial
to their proof is the recent breakthrough of Asperó and Schindler [3]
establishing that$(*)\text{-}\mathcal{A}$ follows from $\text{{\sf MM}}^{++}$
for any generically tame $\mathcal{A}$. First of all it is convenient to
detail more on ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ and its use in our
proofs.

###  5.1. Some remarks on ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$

From now on we will need in several occasions that
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ holds in $V$ for a generically tame
$\mathcal{A}$ (recall Def. 5.2). We will always explicitly state where this
assumption is used, hence if a statement does not mention it in the
hypothesis, the assumption is not needed for its thesis.

We will use all three properties of ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$
crucially: (1) and (2) are used in the proof of Lemma 5.4.6; (3) in the proof
of Fact 5.4.7. Similarly they are essentially used in Remark 5.5.2.
Specifically we will need (1) and (2) of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ to prove that certain countable
families of subsets of $H_{\omega_{1}}$ simply definable using an existential
$\in$-formula quantifying over $L(\mathcal{A})$ with parameters in
$H_{\omega_{1}}\cup\mathcal{A}$ are coded by a universally Baire set in
$\mathcal{A}$, and (3) to prove that this coding is absolute between generic
extensions: i.e. for any family $\left\\{\phi_{n}:n\in\mathbb{N}\right\\}$ of
$\in$-formulae545454Note that the structures
$(H_{\omega_{1}}\cup\mathcal{A},\in)$,
$(H_{\omega_{1}}\cup\mathcal{A},\in_{\Delta_{0}}^{V})$,
$(H_{\omega_{1}}\cup\mathcal{A},\in_{\Delta_{1}}^{V})$ have the same algebra
of definable sets, hence we will use one or the other as we deem most
convenient, since any set definable by some formula in one of these structures
is also defined by a possibly different formula in the other. The formulation
of ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ is unaffacted if we choose any of
the two structures as the one for which we predicate it. with parameters in
$H_{\omega_{1}}^{V}\cup\mathcal{A}$, if

$A_{n}=\left\\{x\in
H_{\omega_{1}}^{V}:(H_{\omega_{1}}\cup\mathcal{A},\in^{V})\models\phi_{n}(x)\right\\}$

and $\left\\{A_{n}:n\in\mathbb{N}\right\\}$ is coded by $A\in\mathcal{A}$,
letting

$A_{n}^{V[G]}=\left\\{x\in
H_{\omega_{1}}^{V[G]}:(H_{\omega_{1}}^{V[G]}\cup\mathcal{A}^{V[G]},\in^{V[G]})\models\phi_{n}(x)\right\\}$

$\left\\{A_{n}^{V[G]}:n\in\mathbb{N}\right\\}$ is coded by
$A^{V[G]}\in\mathcal{A}^{V[G]}$.

It is useful to outline what is the different expressive power of the
structures
$(H_{\omega_{1}},\in_{\Delta_{0}}^{V},A:A\in\mathcal{A}^{V})$ and
$(H_{\omega_{1}}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$. The latter can be
seen as a second order extension of $H_{\omega_{1}}$, where we also allow
formulae to quantify over the family of universally Baire sets given by
$\mathcal{A}$; in the former quantifiers only range over elements of
$H_{\omega_{1}}$, but we can use the subsets of $H_{\omega_{1}}$ whose
univerally Baire code is in $\mathcal{A}$ as parameters. This is in exact
analogy between the comprehension scheme for the Morse-Kelley axiomatization
of set theory (where formulae with quantifiers ranging over classes are
allowed) and the comprehension scheme for Gödel-Bernays axiomatization of set
theory (where just formulae using classes as parameters and quantifiers
ranging only over sets are allowed). To appreciate the difference between the
two set-up, note that that the axiom of determinacy for universally Baire sets
in $\mathcal{A}$ is expressible in

$(H_{\omega_{1}}\cup\mathcal{A},\in_{\Delta_{0}}^{V})$

by the $\Pi_{2}$-sentence for $\in_{\Delta_{0}}$

> _For all $A\subseteq 2^{\omega}$ there is a winning strategy for one of the
> players in the game with payoff $A$_,

while in

$(H_{\omega_{1}},\in_{\Delta_{0}}^{V},A:A\in\mathcal{A}^{V})$

it is expressed by the axiom schema of $\Sigma_{2}$-sentences for
$\in_{\Delta_{0}}\cup\left\\{A\right\\}$

> _There is a winning strategy for some player in the game with payoff $A$_

as $A$ ranges over the universally Baire sets in $\mathcal{A}$.

We will crucially use the stronger expressive power of the structure
$(H_{\omega_{1}}\cup\mathcal{A},\in_{\Delta_{0}})$ to define certain
universally Baire sets as the extension in
$(H_{\omega_{1}}\cup\mathcal{A},\in_{\Delta_{0}}^{V})$ of lightface definable
properties (according to the Levy hierarchy); properties which require an
existential quantifier ranging over all universally Baire sets in
$\mathcal{A}$.

###  5.2. A streamline of the proofs

Let us give a general outline of the proofs before getting into details. From
now on we assume the reader is familiar with the basic theory of
$\mathbb{P}_{\mathrm{max}}$ as exposed in [15].

Much of our efforts will be now devoted to establish the model completeness of
the $\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}$-theory of
$(H_{\omega_{2}}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V},A:A\in\mathcal{A}^{V})$
(assuming $(*)\text{-}\mathcal{A}$ in $V$) and of the
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-thery given by the family of
models $(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V})$
as $V$ ranges over models of $(*)\text{-}\mathcal{A}$.

We will then leverage on Levy absoluteness to infer that in models of
$(*)\text{-}\mathcal{A}$ the theory of
$(H_{\omega_{2}}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V},A:A\in\mathcal{A}^{V})$
is the absolute model companion of the theory of
$(V,\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V},A:A\in\mathcal{A}^{V})$.
A similar strategy will work for the AMC-result for the signature
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$.

To prove these model completeness results we use Robinson’s test and we show
the following:

> Assuming ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ there is a _special_
> universally Baire set $\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ which
> belongs to $\mathcal{A}$ and is defined by an $\in$-formula _(in no
> parameters)_ relativized to $L(\mathcal{A})$ (hence represented by a
> relation symbol of $\in_{\Delta_{1},\mathcal{A}}$) coding a family of
> $\mathbb{P}_{\mathrm{max}}$-preconditions with the following fundamental
> property:
>
> _For any _existential_
> $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-formula
> $\psi(x_{1},\dots,x_{n})$ mentioning the universally Baire
> predicates555555Some of these $B_{j}$ may not appear in
> $\in_{\Delta_{1},\mathcal{A}}$ being not definable by a lightface property.
> $B_{1},\dots,B_{k}\in\mathcal{A}$, there is an algorithmic procedure which
> finds a _universal_
> $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-formula
> $\theta_{\psi}(x_{1},\dots,x_{n})$ mentioning just the universally Baire
> predicates
> $B_{1},\dots,B_{k},\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ such
> that_
>
>
> $(H_{\omega_{2}}^{L(\mathcal{A})[G]},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{L(\mathcal{A})[G]},B_{1},\dots,B_{k})\models\forall\vec{x}\,(\psi(x_{1},\dots,x_{n})\leftrightarrow\theta_{\psi}(x_{1},\dots,x_{n}))$
>
> _whenever $G$ is $L(\mathcal{A})$-generic for $\mathbb{P}_{\mathrm{max}}$._

Moreover the definition and properties of
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ and the computation of
$\theta_{\psi}(x_{1},\dots,x_{n})$ from $\psi(x_{1},\dots,x_{n})$ are just
based on the assumption that $(V,\in)$ is a model of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$, hence can be replicated mutatis-
mutandis in any model of $\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}$.
We will need that $(V,\in)$ is a model of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}+(*)\text{-}\mathcal{A}$ just to argue
that in $V$ there is an $L(\mathcal{A})$-generic filter $G$ for
$\mathbb{P}_{\mathrm{max}}$ such that565656It is this part of our argument
where the result of Asperò and Schindler establishing the consistency of
$(*)\text{-}\mathcal{A}$ relative to a supercompact is used in an essential
way. We will address again the role of Asperò and Schindler’s result in all
our proofs in some closing remarks.
$H_{\omega_{2}}^{L(\mathcal{A})[G]}=H_{\omega_{2}}^{V}$. Since in all our
arguments we will only use that $(V,\in)$ is a model of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ and (in some of them also of
$(*)\text{-}\mathcal{A}$), we will be in the position to conclude easily for
the truth of Theorems 5.6 and 5.7.

We condense the above information in the following:

######  Theorem 5.2.1.

Let $\phi_{\mathcal{A}}(x)$ be an $\in$-formula such that in some model
$(V,\in)$ of $\mathsf{ZFC}$

$(V,\in)\models{\mathbf{MAX}(\mathsf{\mathcal{A}})}+\emph{there are class many
Woodin cardinals}$

with $\mathcal{A}^{V}$ being the extension in $V$ of $\phi_{\mathcal{A}}(x)$.

Then there is an $\in$-formula
$\phi_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}(x)$ in one free
variable575757Whose canonical interpretation in models of
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ will be the magic set
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$. Note also that
$\phi_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}(x)$ is computable in terms of
$\phi_{\mathcal{A}}$. such that:

  1. (1)

Whenever
$(V,\in)\models\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+$_there are
class many Woodin cardinals_ ,

$\left\\{z\in
V:\,(V,\in)\models\phi_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}(z)\right\\}$

is the extension of some element of $\mathcal{A}^{V}$.

  2. (2)

Given predicate symbols $B_{1},\dots,B_{k}$ of arity $n_{1},\dots,n_{k}$, and
the theory $T_{B_{1},\dots,B_{k}}$ in signature
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
extending

$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+\emph{there are class many
Woodin cardinals}$

by the axioms585858Axioms stating that $B_{j}$ is an element of $\mathcal{A}$
for all the new predicate symbols $B_{1},\dots,B_{k}$.:

$\exists!y\subseteq(2^{\omega})^{n_{j}}\,\left[\forall
z_{1},\dots,z_{n_{j}}\,(\langle z_{1},\dots,z_{n_{j}}\rangle\in
y\leftrightarrow
B_{j}(z_{1},\dots,z_{n_{j}}))\wedge\phi_{\mathcal{A}}(y)\right],$

there is a recursive procedure assigning to any _existential_ formula
$\varphi(x_{1},\dots,x_{n})$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
a _universal_ formula $\theta_{\varphi}(x_{1},\dots,x_{n})$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
(mentioning just the predicate symbols occurring in $\varphi$ and
$S_{\phi_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}}$) such that
$T_{B_{1},\dots,B_{k}}$ proves that595959$\dot{G}\in L(\mathcal{A})$ is the
canonical $\mathbb{P}_{\mathrm{max}}$-name for the generic filter.

$\mathbb{P}_{\mathrm{max}}\Vdash[(H_{\omega_{2}}^{L(\mathcal{A})[\dot{G}]},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{L(\mathcal{A})[\dot{G}]},\check{B}_{1},\dots,\check{B}_{k})\models\forall\vec{x}\;(\varphi(x_{1},\dots,x_{n})\leftrightarrow\theta_{\varphi}(x_{1},\dots,x_{n}))]$

###  5.3. Proofs of Thm. 5.7, and of (1)$\to$(2) of Thm. 5.6

__

Theorem 5.7, and (1)$\to$(2) of Theorem 5.6 are immediate corollaries of the
above theorem combined with:

  * •

Asperò and Schindler’s proof that $\text{{\sf
MM}}^{++}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+$_there are class many Woodin
cardinals_ implies $(*)\text{-}\mathcal{A}$,

  * •

Theorem 4.1.

We start with the proof of (1)$\to$(2) of Thm. 5.6 assuming Thm. 5.2.1 and
Thm. 4.1:

###### Proof.

Assume $(V,\in)$ models $(*)\text{-}\mathcal{A}$. Then there is a
$\mathbb{P}_{\mathrm{max}}$-filter $G\in V$ such that
$H_{\omega_{2}}^{L(\mathcal{A})[G]}=H_{\omega_{2}}^{V}$. By Thm. 5.2.1 and
Robinson’s test, we get that the first order
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$-theory of
$H_{\omega_{2}}^{L(\mathcal{A})[G]}$ is model complete. By Levy’s absoluteness
(Lemma 3.1), $H_{\omega_{2}}^{L(\mathcal{A})[G]}$ is a $\Sigma_{1}$-elementary
substructure of $V$ also according to the signature
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}$. We conclude since the two
theories are $\Pi_{1}$-complete and share the same $\Pi_{1}$ and $\Sigma_{1}$
fragments. ∎

The proof of the converse implication requires more information on
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ then what is conveyed in Thm.
5.2.1. We defer it to a later stage.

We now prove Thm. 5.7:

###### Proof.

Let $R$ be the theory given by the $\Pi_{2}$-sentences $\psi$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ which hold in every model of the
form
$(H_{\omega_{2}}^{V[G]},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V[G]})$
obtained by forcing over some model $(V,\in)$ of $T$ with $V[G]$ a generic
extension of $(V,\in)$ such that $(V[G],\in)\models(*)\text{-}\mathcal{A}$.
Recall that $B\subseteq\mathsf{Form}_{\left\\{\in\right\\}}\times 2$ is such
that $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=\in_{B}$.

We show that $R=\mathsf{AMC}(T,B)$.

$R$ is consistent:

by Schindler and Asperò’s result [3]

$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+\text{{\sf
MM}}^{++}+\emph{there are class many Woodin cardinals}$

implies $(*)\text{-}\mathcal{A}$.

$\text{{\sf MM}}^{++}$ is forcible over a model of

$\mathsf{ZFC}+\emph{there is a supercompact}$

and

$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+\emph{there are class many
Woodin cardinals}$

is preserved in forcing extensions.

$R$ is model complete:

__

  * •:

for any existential $\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-formula
$\phi(\vec{x})$, the $\Pi_{2}$-sentence for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$

$\forall\vec{x}\,(\phi(\vec{x})\leftrightarrow\theta_{\phi}(\vec{x}))$

is in $R$: it holds in all the structures used to define $R$ (by Thm. 5.2.1).

  * •:

By Robinson’s test those axioms suffice to establish the model completeness of
$R$.

$(T+T_{\in,B})_{\forall\vee\exists}$ and $R_{\forall\vee\exists}$ are the
same:

By the very definition of $R$, we get that $R_{\forall\vee\exists}$ is equal
to

(14)
$\left\\{\psi\in(\in_{B})_{\forall\vee\exists}:\,\forall\,(V,\in),\mathsf{B}\in
V\,\left[\left[(V,\in)\models T+\mathsf{B}\text{ is a
cba}+\left\llbracket(*)\text{-}\mathcal{A}\right\rrbracket_{\mathsf{B}}=1_{\mathsf{B}}\right]\rightarrow\left\llbracket\psi^{H_{\omega_{2}}}\right\rrbracket_{\mathsf{B}}=1_{\mathsf{B}}\right]\right\\}.$

In view of Thm. 4.1 and Lemma 3.1, the formulae $\psi$ in the set displayed in
14 are exactly the same $\psi$ which belong to
$(T+T_{\in,B})_{\forall\vee\exists}$: by Thm. 4.1 the $\Pi_{1}$-theory of
$(V,\in_{B}^{V})$ for $(V,\in)$ a model of $T$ is exactly the same
$\Pi_{1}$-theory of $(V[G],\in_{B}^{V[G]})$ for $G$ $V$-generic for some cba
forcing $(*)\text{-}\mathcal{A}$ over $(V,\in)$; by Lemma 3.1 applied in
$V[G]$ this $\Pi_{1}$-theory is exactly equal to that of the structure of
$(H_{\omega_{2}}^{V[G]},\in_{B}^{V[G]})$.

This immediately gives A$\Longleftrightarrow$G for $(T+T_{\in,B})$ and $R$.

We are left with the proof of the remaining equivalences between A, B, C, D,
E, F, G.

A$\Longrightarrow$B:

By definition of $R$.

B$\Longrightarrow$C:

Given an $\in$-model $(V,\in)$ of $T$, by the results of [9], we can find a
stationary set preserving forcing extension $V[G]$ of $V$ which models
$\text{{\sf MM}}^{++}$. By the key result of Asperó and Schindler [3],
$V[G]\models(*)\text{-}\mathcal{A}$. By B
$(V[G],\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V[G]})$ models
$\psi^{H_{\omega_{2}}^{V[G]}}$, and we are done.

C$\Longrightarrow$D:

Trivial.

D$\Longrightarrow$E:

By606060${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ implies that the same
assumption used in the cited theorem for $L(\mathbb{R})$ holds for
$L(\mathcal{A})$. [15, Thm. 7.3], if some $P$ forces
$\psi^{\dot{H}_{\omega_{2}}}$, we get that
$L(\mathcal{A})\models\left[\mathbb{P}_{\mathrm{max}}\Vdash\psi^{\dot{H}_{\omega_{2}}}\right]$.

E$\Longleftrightarrow$F:

By [2, Thm. 2.7, Thm. 2.8].

E$\Longrightarrow$G:

Given some complete $S\supseteq T+T_{\in,B}$, and a model $\mathcal{M}$ of
$S$, find $\mathcal{N}$ forcing extension of $\mathcal{M}$ which models
$\psi^{H_{\omega_{2}}^{\mathcal{N}}}$. By Thm. 4.1 and Levy’s absoluteness
Lemma 3.1, $H_{\omega_{2}}^{\mathcal{N}}$ models
$\psi+S_{\forall\vee\exists}$, and we are done.

∎

###  5.4. Proof of Thm. 5.2.1

The rest of this section is devoted to the proof of Thm. 5.2.1.

What we will do first is to sketch the key intuition on how to define
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$.

####  5.4.1. More ideas on the proof of Thm. 5.2.1.

Recall the notion of generic tameness introduced in Def. 5.2. Let $M$ be a
countable transitive model of
$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+$_there are class many
Woodin cardinals_ for some generically tame $\mathcal{A}$. Then it will model
that $\mathcal{A}^{M}$ is generically tame for $M$.

Now assume that there is a countable family $\mathcal{A}_{M}$ of universally
Baire sets in $L(\mathcal{A})$ which is such that
$\mathcal{A}^{M}=\left\\{B\cap M:B\in\mathcal{A}_{M}\right\\}$. Furthermore
assume that the map $B\cap M\mapsto B$ extends the identity on
$H_{\omega_{1}}^{M}$ to an elementary embedding of

$(H_{\omega_{1}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{0}}^{M})$

into

$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V}).$

The setup described above is quite easy to realize (for example $M$ could the
transitive collapse of some countable $X\prec V_{\theta}$ for some large
enough $\theta$); in particular for any $a\in H_{\omega_{1}}$ and
$B_{1},\dots,B_{k}\in\mathcal{A}$, we can find $M$ countable transitive model
of a suitable fragment of $\mathsf{ZFC}$ with $a\in H_{\omega_{1}}^{M}$ and
$\mathcal{A}_{M}\supseteq\left\\{B_{1},\dots,B_{k}\right\\}$ countable and
$H_{\omega_{1}}$-closed family of sets in $\mathcal{A}$ such that
$\mathcal{A}^{M}=\left\\{B\cap M:B\in\mathcal{A}_{M}\right\\}$.

Letting $B_{M}=\prod\mathcal{A}_{M}$, $B_{M}\in\mathcal{A}$ since
$\mathcal{A}$ is countably closed; hence $(L(\mathcal{A}),\in_{\Delta_{0}})$
is able to compute correctly whether $B_{M}$ encodes a set $\mathcal{A}_{M}$
such that the pair $(\mathcal{A}_{M},M)$ satisfies the above list of
requirements, e.g.:

  * •

$M$ is a countable transitive model of
$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}+$_there are class many
Woodin cardinals_.

  * •

the map $B\cap M\mapsto B$ extends the identity on $H_{\omega_{1}}^{M}$ to an
elementary embedding of

$(H_{\omega_{1}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{0}}^{M})$

into

$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V}).$

In particular $(L(\mathcal{A}),\in_{\Delta_{0}})$ correctly computes616161Note
that $(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$ cannot
define $D_{\mathcal{A}}$ since the notion that
$(H_{\omega_{1}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{0}}^{M})$ is an
elementary substructure of
$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$ cannot be
defined in $(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V}).$ the
set $D_{\mathcal{A}}$ of $M\in H_{\omega_{1}}$ such that there exists a
universally Baire set $B_{M}=\prod\mathcal{A}_{M}$ with the property that the
pair $(M,\mathcal{A}_{M})$ realizes the above set of requirements. By
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$, $\bar{D}_{\mathcal{A}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathcal{A}}]$ is a universally Baire set in
$\mathcal{A}$.

Note moreover that $\bar{D}_{\mathcal{A}}$ is defined by a $\in$-formula
$\phi^{*}_{\mathcal{A}}(x)$ in no extra parameters; in particular for any
model $\mathcal{W}=(W,E)$ of
$\mathsf{ZFC}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}$, we can define
$\bar{D}_{\mathcal{A}}$ in $\mathcal{W}$ and all its properties outlined above
will hold relativized to $\mathcal{W}$.

We will consider the set $D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ of $M\in
D_{\mathcal{A}}$ such that:

  * •

$(M,\mathbf{NS}_{\omega_{1}}^{M})$ is a
$\mathbb{P}_{\mathrm{max}}$-precondition which is $B$-iterable for all
$B\in\mathcal{A}_{M}$ (according to [15, Def. 4.1]);

  * •

$j_{0\omega_{1}}$ is a $\Sigma_{1}$-elementary embedding of
$(H_{\omega_{2}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{1}},\mathbf{NS}_{\omega_{1}}^{M})$
into
$(H_{\omega_{2}}^{V}\cup\mathcal{A},\in_{\Delta_{1}},\mathbf{NS}_{\omega_{1}}^{V})$
whenever
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}\right\\}$
is an iteration of $M$ with
$j_{0\omega_{1}}(\mathbf{NS}_{\omega_{1}}^{M})=\mathbf{NS}_{\omega_{1}}^{V}\cap
j_{0\omega_{1}}(H_{\omega_{2}}^{M})$.

It will take a certain effort to prove that (assuming
$(*)\text{-}\mathcal{A}$):

  * •

for any $A\in H_{\omega_{2}}$ and $B\in\mathcal{A}$, we can find $M\in
D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ with $B\in\mathcal{A}_{M}$, some
$a\in H_{\omega_{2}}^{M}$, and an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ with
$j_{0\omega_{1}}(\mathbf{NS}_{\omega_{1}}^{M})=\mathbf{NS}_{\omega_{1}}^{V}\cap
j_{0\omega_{1}}(H_{\omega_{2}}^{M})$ such that $j_{0\omega_{1}}(a)=A$.

  * •

$D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ is correctly626262The key point is
that we will be able to define by a formula in parameter $D_{\mathcal{A}}$ an
element of $\mathcal{A}$ which is exactly
$D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ if $(*)\text{-}\mathcal{A}$ holds.
computable in $(L(\mathcal{A}),\in_{\Delta_{0}})$ using $D_{\mathcal{A}}$.

This effort will pay off, since we will then be able to prove the model
completeness of the
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$-theory636363Note that aiming for
the model completeness result for the theory of $H_{\omega_{2}}$ we are
loosing a certain flavour of second order logic:
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ is a signature considered in a
structure where universally Baire sets are considered only as parameters, but
not as objects over which we can quantify.

$(H_{\omega_{2}},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V})$

using Robinson’s test with
$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}]$ being the
extension of the formula $\phi_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}(x)$
mentioned in Thm. 5.2.1, and showing that for a quantifier free
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\mathcal{A}^{V}$-formula
$\phi(\vec{x},y)$, $A\in H_{\omega_{2}}$, $B_{1},\dots,B_{k}\in\mathcal{A}$ we
have that

$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V},B_{1}^{V},\dots,B_{k}^{V})\models\exists\vec{x}\,\phi(\vec{x},A)$

if and only if

> _For all $M\in D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ and for all
> $\mathcal{J}$ iteration of $M$ of length $\omega_{1}$ mapping correctly the
> nonstationary ideal, if there is $a\in M$ mapped via $\mathcal{J}$ to $A$,
> then_
>
>
> $(H_{\omega_{2}}^{M},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{M},,B_{1}^{V}\cap
> M,\dots,B_{k}^{V}\cap M)\models\exists\vec{x}\,\phi(\vec{x},a).$

A bit of work shows that the latter quoted statement is formalized by a
$\Pi_{1}$-formula for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
using the predicate symbol $\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}]$ together with
those appearing in $\phi$.

We now get into the details.

####  5.4.2. $\mathcal{A}$-correct models

######  Definition 5.4.1.

Given $M,N$ iterable structures, $M\geq N$ if $M\in(H_{\omega_{1}})^{N}$ and
there is an iteration

$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma=(\omega_{1})^{N}\right\\}$

of $M$ with $\mathcal{J}\in N$ such that

$\mathbf{NS}_{\gamma}^{M_{\gamma}}=\mathbf{NS}_{\gamma}^{N}\cap M_{\gamma}.$

######  Notation 5.4.2.

Given a countable family $\mathcal{X}=\left\\{B_{n}:n\in\omega\right\\}$ of
universally Baire sets with each $B_{n}\subseteq(2^{\omega})^{k_{n}}$, we say
that
$B_{\mathcal{X}}=\prod_{n\in\omega}B_{n}\subseteq\prod_{n}(2^{\omega})^{k_{n}}$
is a code for $\left\\{B_{n}:n\in\omega\right\\}$.

Clearly $B_{\mathcal{X}}$ is a universally Baire subset of the compact Polish
space $\prod_{n}(2^{\omega})^{k_{n}}$.

######  Definition 5.4.3.

Let $\mathcal{A}$ be a generically tame family of $V$.

A transitive model of $\mathsf{ZFC}$ $(M,\in)$ is $\mathcal{A}$-correct if
there is $\mathcal{A}_{M}$ countable family of universally Baire sets in $V$
such that:

  * •

The map

$\displaystyle\Theta_{M}:$ $\displaystyle\mathcal{A}_{M}\to M$ $\displaystyle
A\mapsto A\cap M$

is injective.

  * •

$(M,\in)$ models that $\left\\{A\cap M:A\in\mathcal{A}_{M}\right\\}$ is the
family of sets in $M$ satisfying $\phi_{\mathcal{A}}(x)$ in $(M,\in)$.

  * •

$(H_{\omega_{1}}^{M}\cup\left\\{A\cap
M:A\in\mathcal{A}_{M}\right\\},\in)\prec(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in)$
via the identity on $H_{\omega_{1}}$ and $A\cap M\mapsto A$ on $\left\\{A\cap
M:A\in\mathcal{A}_{M}\right\\}$.

  * •

If $M$ is countable, $M$ is $A$-iterable for all $A\in\mathcal{A}_{M}$.

$M$ is absolutely $\mathcal{A}$-correct if for all $N\geq M$:

> _ $N$ is $\mathcal{A}$-correct in $V$ if and only if_ $(M,\in)\models
> N\text{ is $\mathcal{A}^{M}$-correct}.$

######  Notation 5.4.4.

$D_{\mathcal{A}}$ denotes the set of countable absolutely
$\mathcal{A}$-correct $M$; $\bar{D}_{\mathcal{A}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathcal{A}}]$.

For each $M\in D_{\mathcal{A}}$, $\mathcal{A}_{M}$ is a witness that $M$ is
$\mathcal{A}$-correct and $B_{\mathcal{A}_{M}}=\prod\mathcal{A}_{M}$ is a
universally Baire set in $\mathcal{A}$ coding this witness.

For universally Baire sets $B_{1},\dots,B_{k}\in\mathcal{A}$,
$E_{\mathcal{A},B_{1},\dots,B_{k}}$ denotes the set of $M\in D_{\mathcal{A}}$
with $B_{1},\dots,B_{k}\in\mathcal{A}_{M}$ for some witness $\mathcal{A}_{M}$
that $M\in D_{\mathcal{A}}$;
$\bar{E}_{\mathcal{A},B_{1},\dots,B_{k}}=\text{{\rm
Cod}}_{\omega}^{-1}[E_{\mathcal{A},B_{1},\dots,B_{k}}]$.

######  Fact 5.4.5.

$(V,\in)$ models $M$_is countable and (absolutely) $\mathcal{A}$-correct as
witnessed by646464Note that the map $M\mapsto\mathcal{A}_{M}$ can be defined
in $V$ but possibly not in $L(\mathcal{A})$, however the binary relation
$\left\\{(M,B):B\in\mathcal{A}\text{ is a witness that $M$ is
$\mathcal{A}$-correct}\right\\}$ is in $L(\mathcal{A})$. This suffices for all
our proofs. $\mathcal{A}_{M}$_ if and only if so does $(L(\mathcal{A}),\in)$.

Consequently the set $D_{\mathcal{A}}$ of countable absolutely
$\mathcal{A}$-correct $M$ is properly computed in $(L(\mathcal{A}),\in)$.

Therefore assuming ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$

$\bar{D}_{\mathcal{A}}=\mathrm{Cod}^{-1}[D_{\mathcal{A}}]$

is universally Baire.

The same holds for $\bar{E}_{\mathcal{A},B_{1},\dots,B_{k}}$ for given
universally Baire sets $B_{1},\dots,B_{k}$.

###### Proof.

The first part follows almost immediately by the definitions, since the
assertion in parameters $B,M$:

> $B=\prod_{n\in\omega}B_{n}$ codes an $H_{\omega_{1}}$-closed family
> $\mathcal{A}_{M}=\left\\{B_{n}:n\in\omega\right\\}$ of sets such that
>
>   * •
>
> $M$ is $A$-iterable for all $A\in\mathcal{A}_{M}$,
>
>   * •
>
> $M$ models that $\left\\{A\cap M:A\in\mathcal{A}_{M}\right\\}$ is the family
> of sets realizing $\phi_{\mathcal{A}}(x)$ in $M$ and is
> $H_{\omega_{1}}$-closed,
>
>   * •
>
> $(H_{\omega_{1}}^{M}\cup\left\\{A\cap
> M:A\in\mathcal{A}_{M}\right\\},\in_{\Delta_{0}}^{M})$ embeds elementarily
> into
> $(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$ via the map
> extending the identity on $H_{\omega_{1}}^{M}$ by $A\cap M\mapsto A$,
>
>

gets the same truth value in $(V,\in)$ and in $(L(\mathcal{A}),\in)$. Note
that $B=\prod\mathcal{A}_{M}\in\mathcal{A}$ by countable closure of
$\mathcal{A}$.

We conclude that $D_{\mathcal{A}}$ has the same extension in $(V,\in)$ and in
$(L(\mathcal{A}),\in)$. By ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$
$\bar{D}_{\mathcal{A}}$ is universally Baire.

The same argument can be replicated for
$\bar{E}_{\mathcal{A},B_{1},\dots,B_{k}}$. ∎

Note that while the notion of being absolutely $\mathcal{A}$-correct is a
priori not computable in
$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$, if $M\leq N$
are both absolutely $\mathcal{A}$-correct in $V$, there is an elementary
embedding of $(H_{\omega_{1}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{0}}^{N})$
into $(H_{\omega_{1}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{0}}^{M})$ which
belongs to $M$ and is coded by a universally set of $V$ which belongs to
$\mathcal{A}_{M}$ whenever the latter is a witness that $M$ is
$\mathcal{A}$-correct.

######  Lemma 5.4.6.

Assume $\mathbf{NS}_{\omega_{1}}$ is precipitous and there are class many
Woodin cardinals in $V$. Let $\delta$ be an inaccessible cardinal in $V$ and
$G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. Then $V_{\delta}$
is absolutely $\mathcal{A}^{V[G]}$-correct in $V[G]$ as witnessed by
$\prod\left\\{B^{V[G]}:B\in\mathcal{A}^{V}\right\\}$. Furthermore $V_{\delta}$
is $B^{V[G]}$-iterable for all $B\in\mathcal{A}^{V}$.

###### Proof.

$\left\\{B^{V[G]}:B\in\mathcal{A}^{V}\right\\}$ is a countable family of
universally Baire sets in $\mathcal{A}^{V[G]}$ hence its product belongs to
$\mathcal{A}^{V[G]}$ by countable closure of $\mathcal{A}$ in $V[G]$.

By ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ there is an elementary
$j:L(\mathcal{A}^{V})\to L(\mathcal{A}^{V[G]})$ which is the identity on
$H_{\omega_{1}}$ and maps $B\in\mathcal{A}$ to $B^{V[G]}$ in
$\mathcal{A}^{V[G]}$.

By Fact 5.4.5 $V$ and $L(\mathcal{A}^{V})$ (respectively $V[G]$ and
$L(\mathcal{A}^{V[G]})$) agree on which elements of $H_{\omega_{1}}^{V}$ are
$\mathcal{A}$-correct.

The above grants that the unique missing condition to get the
$\mathcal{A}^{V[G]}$-correctness of $V_{\delta}$ is to check that it is
$B^{V[G]}$-iterable for all $B\in\mathcal{A}^{V}$. This is a standard argument
which can be reconstrued looking for example at the proof of [15, Thm. 4.10].
∎

######  Fact 5.4.7.

$({\mathbf{MAX}(\mathsf{\mathcal{A}})})$ Assume $\mathbf{NS}_{\omega_{1}}$ is
precipitous and ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ holds. Then for any
iterable $M$, $B_{1},\dots,B_{k}\in\mathcal{A}$, there is $N\leq M$ such that:

  * •

$N$ is absolutely $\mathcal{A}$-correct as witnessed by $\mathcal{A}_{N}$;

  * •

$B_{1},\dots,B_{k}\in\mathcal{A}_{N}$.

###### Proof.

The assumptions grant that whenever $G$ is
$\operatorname{Coll}(\omega,\delta)$-generic for $V$, $V_{\delta}$ is
absolutely $\mathcal{A}^{V[G]}$-correct in $V[G]$ and $B^{V[G]}$ iterable for
all $B\in\mathcal{A}^{V}$ (i.e. Lemma 5.4.6).

By [15, Lemma 2.8], for any iterable $M\in H_{\omega_{1}}^{V}$ there is in $V$
an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\omega_{1}^{V}\right\\}$
of $M$ such that $\mathbf{NS}_{\omega_{1}}^{V}\cap
M_{\omega_{1}}=\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}$.

By ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$

$(L(\mathcal{A}^{V}),\in_{\Delta_{0}})\prec(L(\mathcal{A}^{V[G]}),\in_{\Delta_{0}})$

via a map which is the identity on $H_{\omega_{1}}$ and maps $B$ to $B^{V[G]}$
for $B\in\mathcal{A}$. Therefore we have that in $V[G]$
$\bar{E}_{\mathcal{A},B_{1},\dots,B_{k}}^{V[G]}$ is exactly
$\bar{E}_{\mathcal{A},B_{1}^{V[G]},\dots,B_{k}^{V[G]}}$.

Also for any $B\in\mathcal{A}^{V}$, $V_{\delta}$ is $B^{V[G]}$-iterable, and
$B^{V[G]}\in\mathcal{A}_{V_{\delta}}=\left\\{A^{V[G]}:A\in\mathcal{A}^{V}\right\\}$;

Hence for each iterable $M\in H_{\omega_{1}}^{V}$ and
$B_{1},\dots,B_{k}\in\mathcal{A}^{V}$, $N=V_{\delta}\leq M$ belongs to
$E_{\mathcal{A},B_{1},\dots,B_{k}}^{V[G]}$ as witnessed by
$\mathcal{A}_{V_{\delta}}=\left\\{U^{V[G]}:U\in\mathcal{A}\right\\}$ and
$B_{1}^{V[G]},\dots,B_{k}^{V[G]}\in\mathcal{A}^{V[G]}$.

Note that the $\Sigma_{1}$-formulae for $\in_{\Delta_{0}}$ we established to
be true in $H_{\omega_{1}}^{V[G]}\cup\mathcal{A}^{V[G]}$ as witnessed by
$V_{\delta}$ are in parameters $M\in H_{\omega_{1}}^{V}$,
$B_{1}^{V[G]},\dots,B_{k}^{V[G]},D_{\mathcal{A}}^{V[G]}$ as
$B_{1},\dots,B_{k}$ vary in $\mathcal{A}^{V}$. Hence these formulae reflect to
$H_{\omega_{1}}^{V}\cup\mathcal{A}^{V}$.

The Lemma is proved.

∎

###  5.5. Three characterizations of $(*)$-$\mathcal{A}$

######  Definition 5.5.1.

An absolutely $\mathcal{A}$-correct $M$ is
_$(\mathbf{NS}_{\omega_{1}},\mathcal{A})$ -ec_ if $(M,\in)$ models that
$\mathbf{NS}_{\omega_{1}}$ is precipitous and there is a witness
$\mathcal{A}_{M}$ that $M$ is $\mathcal{A}$-correct with the following
property:

> Assume an iterable $N\leq M$ is absolutely $\mathcal{A}$-correct with a
> witness $\mathcal{A}_{N}$ such that
> $\prod\mathcal{A}_{M}\in\mathcal{A}_{N}$.
>
> Then for all iterations
>
>
> $\mathcal{J}=\left\\{j_{\alpha\beta}:\alpha\leq\beta\leq\gamma=\omega_{1}^{N}\right\\}$
>
> in $N$ witnessing $M\geq N$, we have that $j_{0\gamma}$ defines a
> $\Sigma_{1}$-elementary embedding of
>
>
> $(H_{\omega_{2}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{1}}^{M},\mathbf{NS}_{\omega_{1}}^{M})$
>
> into
>
>
> $(H_{\omega_{2}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{1}}^{N},\mathbf{NS}_{\omega_{1}}^{N}).$

######  Remark 5.5.2.

A crucial observation is that _“ $x$ is
$(\mathbf{NS}_{\omega_{1}},\mathcal{A})$-ec”_ is a property correctly
definable in $(H_{\omega_{1}}\cup\mathcal{A},\in)$ using as parameter the
set656565Note however that $D_{\mathcal{A}}$ may not be lightface definable in
$(H_{\omega_{1}}\cup\mathcal{A},\in)$. $D_{\mathcal{A}}$. Therefore (assuming
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$)

$D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=\left\\{M\in
H_{\omega_{1}}:\,M\text{ is
$(\mathbf{NS}_{\omega_{1}},\mathcal{A})$-ec}\right\\}$

is such that $\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}]$ is a
universally Baire set in $\mathcal{A}$. Moreover letting for $V[G]$ a generic
extension of $V$

$D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]}}=\left\\{M\in
H_{\omega_{1}}^{V[G]}:\,M\text{ is
$(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec}\right\\},$

we have that

$\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}^{V[G]}=\text{{\rm
Cod}}_{\omega}^{-1}[D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]}}].$

######  Theorem 5.5.3.

Assume $V$ models ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$. The following are
equivalent:

  1. (1)

Woodin’s axiom $(*)$-$\mathcal{A}$ holds (i.e. $\mathbf{NS}_{\omega_{1}}$ is
precipitous, and there is an $L(\mathcal{A})$-generic filter $G$ for
$\mathbb{P}_{\mathrm{max}}$ such that
$L(\mathcal{A})[G]\supseteq\mathcal{P}\left(\omega_{1}\right)^{V}$).

  2. (2)

Let $\delta$ be inaccessible. Whenever $G$ is $V$-generic for
$\operatorname{Coll}(\omega,\delta)$, $V_{\delta}$ is
$(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec in $V[G]$.

  3. (3)

$\mathbf{NS}_{\omega_{1}}$ is precipitous and for all $A\in H_{\omega_{2}}$,
$B\in\mathcal{A}$, there is an $(\mathbf{NS}_{\omega_{1}},\mathcal{A})$-ec $M$
with witness $\mathcal{A}_{M}$, and an iteration
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ such that:

     * •

$A\in M_{\omega_{1}}$,

     * •

$B\in\mathcal{A}_{M}$,

     * •

$\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}=\mathbf{NS}_{\omega_{1}}\cap
M_{\omega_{1}}$.

Theorem 5.5.3 is the key to the proofs of Theorem 5.2.1 and to the missing
implication in the proof of Theorem 5.6.

####  5.5.1. Proof of Theorem 5.2.1

The theorem is an immediate corollary of the following:

######  Lemma 5.5.4.

Let $B_{1},\dots,B_{k}$ be new predicate symbols and
$S_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k}}$ be the
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-theory
$\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}+(*)\text{-}\mathcal{A}$
enriched with the sentences asserting that $B_{1},\dots,B_{k}$ have as
extension elements of $\mathcal{A}$.

Let $E_{B_{1},\dots,B_{k}}$ consists of the set of $M\in
D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ such that:

  * •

$M$ is $B_{j}$-iterable for all $j=1,\dots,k$;

  * •

there is $\mathcal{A}_{M}$ witnessing $M\in
D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$ with $B_{j}\in\mathcal{A}_{M}$ for
all $j$.

Let also $\bar{E}_{B_{1},\dots,B_{k}}=\text{{\rm
Cod}}_{\omega}^{-1}[E_{B_{1},\dots,B_{k}}]$.

Then $S_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k}}$ proves that
$\bar{E}_{B_{1},\dots,B_{k}}$ is in $\mathcal{A}$.

Moreover:

  * •

Let
$S_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}}$
be the natural extension of
$S_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k}}$ adding a
predicate symbol for $\bar{E}_{B_{1},\dots,B_{k}}$ to
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
and the axioms stating that the interpretation of
$\bar{E}_{B_{1},\dots,B_{k}}$ is given by its definition.

  * •

Let
$T_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}}$
be the family of $\Pi_{2}$-sentences $\psi$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$
such that
$S_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}}$
proves $\psi^{H_{\omega_{2}}}$.

Then
$T_{\mathbf{NS}_{\omega_{1}},\mathcal{A},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}}$
proves that every existential formula for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
is equivalent to a universal formula for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$.

###### Proof.

$\bar{E}_{B_{1},\dots,B_{k}}$ is universally Baire and in $\mathcal{A}$ by
${\mathbf{MAX}(\mathsf{\mathcal{A}})}$, since $E_{B_{1},\dots,B_{k}}$ is
definable in $(H_{\omega_{1}}\cup\mathcal{A},\in)$ with parameters the
universally Baire sets
$B_{1},\dots,B_{k},\bar{D}_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}$.

Given any $\Sigma_{1}$-formula $\phi(\vec{x})$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$
mentioning the universally Baire predicates $B_{1},\dots,B_{k}$, we want to
find a universal formula $\psi(\vec{x})$ for
$\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$
such that

$T_{\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\},\mathbf{NS}_{\omega_{1}}}\models\forall\vec{x}(\phi(\vec{x})\leftrightarrow\psi(\vec{x})).$

Let $\theta_{\phi}(\vec{x})$ be the formula asserting:

> _For all $M\in E_{B_{1},\dots,B_{k}}$, for all iterations
> $\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
> of $M$ such that:_
>
>   * •
>
> $\vec{x}=j_{0\omega_{1}}(\vec{a})$_for some_ $\vec{a}\in M$,
>
>   * •
>
> $\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}\cap
> j_{0\omega_{1}}(M)$,
>
>

>
> we have that
>
> $(H_{\omega_{2}}^{M},\in_{\mathbf{NS}_{\omega_{1}}}^{M},B_{j}\cap\text{{\rm
> Cod}}(r):j=1,\dots,k)\models\phi(\vec{a}).$

More formally:

$\displaystyle\forall r\,\forall\mathcal{J}$ $\displaystyle\\{$
$\displaystyle[$ $\displaystyle(r\in\bar{E}_{B_{1},\dots,B_{k}})\wedge$
$\displaystyle\wedge\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}\text{
is an iteration of }\text{{\rm Cod}}(r)\wedge$
$\displaystyle\wedge\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(\text{{\rm
Cod}}(r))}=\mathbf{NS}_{\omega_{1}}\cap j_{0\omega_{1}}(\text{{\rm
Cod}}(r))\wedge$ $\displaystyle\wedge\exists\vec{a}\in\text{{\rm
Cod}}(r)\,(\vec{x}=j_{0\omega_{1}}(\vec{a}))$ $\displaystyle]$
$\displaystyle\rightarrow$ $\displaystyle(H_{\omega_{2}}^{\text{{\rm
Cod}}(r)},\in_{\mathbf{NS}_{\omega_{1}}}^{\text{{\rm
Cod}}(r)},B_{j}\cap\text{{\rm Cod}}(r):j=1,\dots,k)\models\phi(\vec{a})$
$\displaystyle\\}.$

The above is a $\Pi_{1}$-formula for
$\in_{\Delta_{1}}\cup\left\\{\omega_{1},\mathbf{NS}_{\omega_{1}}\right\\}\cup\left\\{B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}}\right\\}$.

(We leave to the reader to check that the property

>
> _$\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
> is an iteration of $M$ such that
> $\mathbf{NS}_{\omega_{1}}^{j_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}\cap
> j_{0\omega_{1}}(M)$_

is definable by a $\Delta_{1}$-property in parameters $M,\mathcal{J}$ in the
signature
$\in_{\Delta_{0}}\cup\left\\{\omega_{1},\mathbf{NS}_{\omega_{1}}\right\\}$).

Now it is not hard to check that:

######  Claim 2.

For all $\vec{A}\in H_{\omega_{2}}$

$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})\models\phi(\vec{A})$

if and only if

$(H_{\omega_{2}},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k},\bar{E}_{B_{1},\dots,B_{k}})\models\theta_{\phi}(\vec{A}).$

###### Proof.

__

$\theta_{\phi}(\vec{A})\rightarrow\phi(\vec{A})$:

Take any $M$ and $\mathcal{J}$ satisfying the premises of the
implication666666At least one such $M$ exists by $(*)\text{-}\mathcal{A}$. in
$\theta_{\phi}(\vec{A})$, Then
$(H_{\omega_{2}}^{M},\in_{\mathbf{NS}_{\omega_{1}},\mathcal{A}^{M}}^{M})\models\phi(\vec{a})$
for some $\vec{a}$ such that $j_{0,\omega_{1}}(\vec{a})=\vec{A}$ and
$B_{j}\cap M_{\omega_{1}}=j_{0\omega_{1}}(B_{j}\cap M)$ for all $j=1,\dots,k$.

Since $\Sigma_{1}$-properties are upward absolute and
$(M_{\omega_{1}},\in_{\mathbf{NS}_{\omega_{1}}}^{M_{\omega_{1}}},B_{j}\cap
M_{\omega_{1}}:j=1,\dots,k)$ is a
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\left\\{B_{1},\dots,B_{k}\right\\}$-substructure
of $(H_{\omega_{2}},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B_{j}:j=1,\dots,k)$
which models $\phi(\vec{A})$, we get that $\phi(\vec{A})$ holds for
$(H_{\omega_{2}},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})$.

$\phi(\vec{A})\rightarrow\theta_{\phi}(\vec{A})$:

Assume

$(H_{\omega_{2}},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B_{1},\dots,B_{k})\models\phi(\vec{A}).$

Take any $(\mathbf{NS}_{\omega_{1}},\mathcal{A})$-ec $M\in V$ and any
iteration
$\mathcal{J}=\left\\{j_{\alpha}\beta:\alpha\leq\beta\leq\omega_{1}\right\\}$
of $M$ witnessing the premises of the implication676767Note that such $M$ and
$\mathcal{J}$ exists by Thm. 5.5.3(3) applied to $B_{1}\times\dots\times
B_{k}$ and $\vec{A}$. in $\theta_{\phi}(\vec{A})$, in particular such that:

  * •:

$\vec{A}=j_{0\omega_{1}}(\vec{a})\in M_{\omega_{1}}$ for some $\vec{a}\in M$,

  * •:

$\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}}=\mathbf{NS}_{\omega_{1}}\cap
M_{\omega_{1}}$,

  * •:

$M$ is $B_{j}$-iterable for $j=1,\dots,k$.

Let $G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$ with $\delta$
inaccessible. Then in $V[G]$, $V_{\delta}$ is $\mathcal{A}^{V[G]}$-correct, by
Lemma 5.4.6.

Therefore (since $M$ is $(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec
also in $V[G]$ by ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$), $V[G]$ models that
$j_{0\omega_{1}^{V}}$ is a $\Sigma_{1}$-elementary embedding of686868Actually
is $\Sigma_{1}$-elementary between the structures
$(H_{\omega_{2}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{0}},\mathbf{NS}_{\omega_{1}}^{M})$
and
$(H_{\omega_{2}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}},\mathbf{NS}_{\omega_{1}}^{V})$,
but we only need the weaker form of $\Sigma_{1}$-elementarity described in the
proof.

$(H_{\omega_{2}}^{M},\in_{\mathbf{NS}_{\omega_{1}}}^{M},B\cap
M:B\in\mathcal{A}_{M})$

into

$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}}}^{V},B:B\in\mathcal{A}_{M}).$

This grants that

$(H_{\omega_{2}}^{M},\in_{\mathbf{NS}_{\omega_{1}}}^{M},B\cap
M:B\in\mathcal{A}_{M})\models\phi(\vec{a}),$

as was to be shown.

∎

The Lemma is proved.

∎

######  Remark 5.5.5.

Note that by essentially the same proof we can argue that the
$\in_{\mathbf{NS}_{\omega_{1}}}$-theory of $H_{\aleph_{2}}\cup\mathcal{A}$ in
models of
$\mathsf{ZFC}_{\mathbf{NS}_{\omega_{1}}}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}$
is model complete. However this result is not relevant for the AMC-spectrum
results we are aiming for, since
$(H_{\aleph_{2}}\cup\mathcal{A},\in_{\Delta_{1}},\mathbf{NS}_{\omega_{1}})$ is
not a $\Sigma_{1}$-substructure of
$(V,\in_{\Delta_{1}},\mathbf{NS}_{\omega_{1}})$.

####  5.5.2. Proof of (2)$\to$(1) of Theorem 5.6

###### Proof.

Assume $\delta$ is supercompact, $P$ is a standard forcing notion to force
$\text{{\sf MM}}^{++}$ of size $\delta$ (such as the one introduced in [9] to
prove the consistency of Martin’s maximum), and $G$ is $V$-generic for $P$;
then $(*)$-$\mathcal{A}$ holds in $V[G]$ by Asperó and Schindler’s recent
breakthrough [3]. By Thm. 4.1 $V$ and $V[G]$ agree on the $\Pi_{1}$-fragment
of the $\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$-theory $\bar{T}$ of
$V$, therefore so do $H_{\omega_{2}}^{V}$ and $H_{\omega_{2}}^{V[G]}$ (by
Lemma 3.1 applied in $V$ and $V[G]$ respectively).

Since $P\in\text{{\sf SSP}}$

$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}}}^{V},A:A\in\mathcal{A}^{V})\sqsubseteq(H_{\omega_{2}}^{V[G]},\in_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\mathcal{A}^{V}).$

Now the model completeness of the
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$-theory $\bar{S}$ of
$H_{\omega_{2}}^{V}$ grants that $H_{\omega_{2}}^{V}$ is
$\bar{T}_{\forall}$-ec. This gives that:

$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}}}^{V},\mathcal{A}^{V})\prec_{\Sigma_{1}}(H_{\omega_{2}}^{V[G]},\in_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\mathcal{A}^{V}).$

Therefore any $\Pi_{2}$-property for
$\in_{\mathbf{NS}_{\omega_{1}}}\cup\mathcal{A}^{V}$ with parameters in
$H_{\omega_{2}}^{V}$ which holds in

$(H_{\omega_{2}}^{V[G]},\in_{\mathbf{NS}_{\omega_{1}}}^{V[G]},A^{V[G]}:A\in\mathcal{A})$

also holds in
$(H_{\omega_{2}}^{V},\in_{\mathbf{NS}_{\omega_{1}}}^{V},\mathcal{A}^{V})$.

Hence in $H_{\omega_{2}}^{V}$ it holds characterization (3) of
$(*)$-$\mathcal{A}$ given by Thm. 5.5.3 and we are done. ∎

####  5.5.3. Proof of Theorem 5.5.3

######  Definition 5.5.6.

[15, Def. 2.1] $\mathbb{P}_{\mathrm{max}}$ is the subset of $H_{\omega_{1}}$
given by the pairs $(M,a)$ such that

  * •

$M$ is iterable, countable, and models Martin’s axiom.

  * •

$a\in\mathcal{P}\left(\omega_{1}\right)^{M}\setminus L(\mathbb{R})^{M}$, and
there exists $r\in\mathcal{P}\left(\omega\right)\cap M$ such that
$\omega_{1}^{M}=\omega_{1}^{L[a,r]}$.

$(M,a)\leq(N,b)$ if there exists $\mathcal{J}=\langle
j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}^{M}\rangle$ in $M$ iteration
of $N$ of length $\omega_{1}^{M}$ such that $j_{0\omega_{1}^{M}}(b)=a$ and
$(M,\in)$ models that $\mathcal{J}$ is correct.

Note that $\mathbb{P}_{\mathrm{max}}$ is a definable class in
$(H_{\omega_{1}},\in)$; in particular it belongs to any transitive model of
$\mathsf{ZFC}^{-}$ containing $\mathcal{P}\left(\omega\right)$.

Our definition of $\mathbb{P}_{\mathrm{max}}$ is slightly different than the
one given in [15], but it defines a dense subset of the poset defined in [15,
Def. 2.1] in view of the following696969Much weaker large cardinals
assumptions are needed, we don’t spell the optimal hypothesis.:

######  Fact 5.5.7.

Let $\mathbb{P}_{\mathrm{max}}^{0}$ be the forcing defined707070E.g. the
forcing $\mathbb{P}_{\mathrm{max}}$ according to the terminology of [15]. in
[15, Def. 2.1]. Assume there are class many Woodin cardinals.

Then for every condition $(M,I,a)$ in $\mathbb{P}_{\mathrm{max}}^{0}$ there is
a condition $(N,b)$ in $\mathbb{P}_{\mathrm{max}}$ and an iteration $\langle
j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}^{N}\rangle\in N$ of $(M,I)$
according to [15, Def. 1.2] such that
$j_{0\omega_{1}^{N}}(I)=\mathbf{NS}_{\omega_{1}}^{N}\cap
j_{0\omega_{1}^{N}}[M]$ and $j_{0}(a)=b$. Hence
$(N,\mathbf{NS}_{\omega_{1}}^{N},b)$ refines $(M,I,a)$ in
$\mathbb{P}_{\mathrm{max}}^{0}$.

###### Proof.

Let $\gamma>\delta$ be two Woodin cardinals. Let $X\prec V_{\gamma}$ be
countable with $\delta,(M,I,a)\in X$. Let $N_{0}$ be the transitive collapse
of $X$, and $N$ be a generic extension of $N_{0}$ by a forcing collapsing
$\delta$ to become $\omega_{2}$ and forcing $\mathbf{NS}_{\omega_{1}}$ is
precipitous and Martin’s axiom. Since $\gamma$ is Woodin, there are class many
measurables in $V_{\gamma}$, hence $N_{0}$ is iterable and so is $N$ (by [15,
Thm. 4.10]).

By [15, Lemma 2.8] there is in $N$ the required iteration $\langle
j_{\alpha\beta}:\,\alpha\leq\beta\leq\omega_{1}^{N}\rangle$ of $(M,I)$ and we
can set $b=j_{0\omega_{1}^{N}}(a)$. ∎

In particular (at the prize of assuming the right large cardinal assumptions)
the forcings $\mathbb{P}_{\mathrm{max}}$ and $\mathbb{P}_{\mathrm{max}}^{0}$
are equivalent as the former sits inside the latter as a dense subset.

This is a key property of $\mathbb{P}_{\mathrm{max}}$ we will need, and is
based on Asperò and Schindler result that $\text{{\sf
MM}}^{++}+{\mathbf{MAX}(\mathsf{\mathcal{A}})}$ implies
$(*)\text{-}\mathcal{A}$:

######  Lemma 5.5.8.

Assume ${\mathbf{MAX}(\mathsf{\mathcal{A}})}$ and there is a supercompact
cardinal. Let $\dot{A}\in L(\mathbb{R})$ and $\dot{\mathcal{N}}\in
L(\mathcal{A})$ be the $\mathbb{P}_{\mathrm{max}}$-canonical names
respectively for:

  * •

$\bigcup\left\\{a:(N,a)\in G\right\\}$,

  * •

$H_{\omega_{2}}^{L(\mathcal{A})[G]}\cup\mathcal{A}$,

whenever $G$ is a $\mathbb{P}_{\mathrm{max}}$-generic filter for
$L(\mathcal{A})$.

For any quantifier free formula $\phi(x,y,z)$ for
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$ and
$B\in\mathcal{A}^{V}$

$L(\mathcal{A}^{V})\models\left[(N,a)\Vdash\exists
y\in\dot{\mathcal{N}}\phi(\dot{A},y,\check{B})\right]$

if and only if the set $D_{\phi}$ of $(M,e)$ such that

  * •

$M$ is $B$-iterable,

  * •

$(H_{\omega_{2}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{1}}^{M},\mathbf{NS}_{\omega_{1}}^{M})\models\exists
y\,\phi(e,y,B\cap M)$,

is dense below $(N,a)$.

###### Proof.

Let $G$ be a $\mathbb{P}_{\mathrm{max}}$-generic filter for
$L(\mathcal{A}^{V})$ and $A=\dot{A}_{G}$. Then (by [15, Lemma 2.7])

$G=\left\\{(N,a):\,\exists\mathcal{J}_{N}\,\mathbf{NS}\text{-correct iteration
of $N$ mapping $a$ to $A$}\right\\}.$

If some $(M,a)\in G$ is such that:

  * •

$M$ is $B$-iterable,

  * •

$(H_{\omega_{2}}^{M}\cup\mathcal{A}^{M},\in_{\Delta_{1}}^{M},\mathbf{NS}_{\omega_{1}}^{M})\models\exists
y\phi(a,y,B\cap M).$

Let

$\mathcal{J}_{M}=\left\\{j_{\alpha,\beta}:M_{\alpha}\to
M_{\beta}:\,\alpha\leq\beta\leq\omega_{1}\right\\}.$

Then

$\mathcal{M}=(H_{\omega_{2}}^{M_{\omega_{1}}}\cup\mathcal{A}^{M_{\omega_{1}}},\in_{\Delta_{1}}^{M_{\omega_{1}}},\mathbf{NS}_{\omega_{1}}^{M_{\omega_{1}}})\models\exists
y\,\phi(A,y,B\cap M_{\omega_{1}}).$

This yields that

$\mathcal{N}=(H_{\omega_{2}}^{L(\mathcal{A})[G]}\cup\mathcal{A},\in_{\Delta_{1}}^{L(\mathcal{A})[G]},\mathbf{NS}_{\omega_{1}}^{L(\mathcal{A})[G]})\models\exists
y\phi(A,y,B),$

since it is not hard to check717171If
$\phi(x_{1},\dots,x_{n},y_{1},\dots,y_{k})$ is provably
$\Delta_{1}(\mathsf{ZFC}^{-})$, one can prove by an induction on its
syintactic complexity that for each $A_{1},\dots,A_{n}\in
H_{\omega_{2}}^{M_{\omega_{1}}}$ and each
$B_{1},\dots,B_{k}\in\mathcal{A}^{M_{\omega_{1}}}$
$\phi(A_{1},\dots,A_{n},B_{1},\dots,B_{k})$ holds in $\mathcal{N}$ if and only
if $\phi(A_{1},\dots,A_{n},B_{1}\cap M_{\omega_{1}},\dots,B_{k}\cap
M_{\omega_{1}})$ holds in $\mathcal{M}$. that
$\mathcal{M}\sqsubseteq\mathcal{N}$ via the map which is the identity on
$H_{\omega_{2}}^{L(\mathcal{A})[G]}$ and maps $B\cap M_{\omega_{1}}$ to $B$ on
$\mathcal{A}^{M_{\omega_{1}}}$.

Otherwise note that

$D_{\phi}\cup\left\\{(M,b):\forall(N,c)\leq(M,b)\,(N,c)\not\in
D_{\phi})\right\\}$

is dense in $\mathbb{P}_{\mathrm{max}}$ and belongs to $L(\mathcal{A})$. Hence
for some $(M,a)\in G$, $L(\mathcal{A})$ models that for all $(N,c)\leq(M,a)$
which are $B$-iterable

$(H_{\omega_{2}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{1}}^{N},\mathbf{NS}_{\omega_{1}}^{N})\not\models\exists
y\,\phi(c,y,B\cap N).$

If the Lemma fails we can find $(N,c)\leq(M,a)$ in the above set such that

$L(\mathcal{A})\models\left[(N,c)\Vdash_{\mathbb{P}_{\mathrm{max}}}\exists
y\in\dot{\mathcal{N}}\,\phi(\dot{A},y,\check{B})\right].$

In particular for any $(P,d)\leq(N,c)$ we have that

(15)
$(H_{\omega_{2}}^{P}\cup\mathcal{A}^{P},\in_{\Delta_{1}}^{P},\mathbf{NS}_{\omega_{1}}^{P})\not\models\exists
y\,\phi(d,y,B\cap P),$

and

(16)
$L(\mathcal{A})\models\left[(P,d)\Vdash_{\mathbb{P}_{\mathrm{max}}}\exists
y\in\dot{\mathcal{N}}\,\phi(\dot{A},y,\check{B})\right].$

Fix in $V$

$\mathcal{K}=\left\\{k_{\alpha,\beta}:N_{\alpha}\to
N_{\beta}:\,\alpha\leq\beta\leq\omega_{1}\right\\}.$

$\mathbf{NS}$-correct iteration of $N$. Let $A=k_{0\omega_{1}}(c)$.

Now let $\gamma$ be a supercompact cardinal, $\delta>\gamma$ be inaccessible,
and $H$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. Then in
$V[H]$ we can find $K$ $V$-generic for some stationary set preserving forcing
of $V$ collapsing $\gamma$ to become $\omega_{2}$, together with $\text{{\sf
MM}}^{++}$ (and therefore $(*)\text{-}\mathcal{A}$ by Asperò and Schindler’s
result). Then $(V_{\delta}[K],A)$ is a $\mathbb{P}_{\mathrm{max}}$-condition
in $V[H]$ refining $(N,c)$ (as witnessed by $\mathcal{K}$), hence such that

$(H_{\omega_{2}}^{V_{\delta}[K]}\cup\mathcal{A}^{V_{\delta}[K]},\in_{\Delta_{1}}^{V_{\delta}[K]},\mathbf{NS}_{\omega_{1}}^{V_{\delta}[K]})\not\models\exists
y\,\phi(A,y,B^{V_{\delta}[K]})$

by 15. On the other hand since $(V_{\delta}[K],A)$ models
$(*)\text{-}\mathcal{A}$ and $A\in\mathcal{P}\left(\omega_{1}\right)\setminus
L(\mathcal{A}^{V[K]})$,

$G_{A}=\left\\{(P,d):\,\exists\mathcal{J}\text{ $\mathbf{NS}$-correct
iteration of $P$ mapping $d$ to $A$}\right\\}$

is $L(\mathcal{A}^{V[K]})$-generic for $\mathbb{P}_{\mathrm{max}}$, with
$(N,c)$ belonging to $G_{A}$; therefore

$(H_{\omega_{2}}^{L(\mathcal{A}^{V[K]})[G_{A}]}\cup\mathcal{A}^{V_{\delta}[K]},\in_{\Delta_{1}}^{L(\mathcal{A}^{V[K]})[G_{A}]},\mathbf{NS}_{\omega_{1}}^{L(\mathcal{A}^{V[K]})[G_{A}]})\models\exists
y\phi(A,y,B^{V_{\delta}[K]})$

by 16. Since
$H_{\omega_{2}}^{L(\mathcal{A}^{V[K]})[G_{A}]}=H_{\omega_{2}}^{V[K]}$, we have
reached a contradiction. ∎

We can now prove Thm. 5.5.3.

###### Proof.

(1) implies (2):

Let $G$ be $V$-generic for $\operatorname{Coll}(\omega,\delta)$. By Lemma
5.4.6, $V_{\delta}$ is absolutely $\mathcal{A}^{V[G]}$-correct in $V[G]$ as
witnessed by
$\left\\{B^{V[G]}:B\in\mathcal{A}^{V}\right\\}=\mathcal{A}_{V}=\left\\{B_{n}^{V[G]}:n\in\omega\right\\}$
and $B^{V[G]}$-iterable for all $B\in\mathcal{A}^{V}$.

######  Claim 3.

$V_{\delta}$ is $(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec in $V[G]$
as witnessed by $\mathcal{A}_{V}$.

###### Proof.

Let in $V[G]$ $B_{\mathcal{A}_{V}}=\prod_{n\in\omega}B_{n}^{V[G]}$ be the
universally Baire set in $\mathcal{A}^{V[G]}$ coding $\mathcal{A}_{V}$.

Let (by Fact 5.4.7) $N\leq V_{\delta}$ in $V[G]$ be absolutely
$\mathcal{A}^{V[G]}$-correct, $B_{\mathcal{A}_{V}}$-iterable, with
$B_{\mathcal{A}_{V}}\in\mathcal{A}_{N}$ for some $\mathcal{A}_{N}$ countable
subset of $\mathcal{A}^{V[G]}$ witnessing that $N$ is
$\mathcal{A}^{V[G]}$-correct.

Then it is not hard to check that

(17)
$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{1}}^{V})\prec(H_{\omega_{1}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{1}}^{N})$

via the map extending the identity on $H_{\omega_{1}}^{V}$ by $B\mapsto
B^{V[G]}\cap N$ for $B\in\mathcal{A}^{V}$. This holds since $N\in
D_{\mathcal{A}}^{V[G]}$ and $\left\\{B^{V[G]}\cap
N:\,B\in\mathcal{A}^{V}\right\\}$ in $N$.

Let

$\mathcal{J}=\left\\{j_{\alpha,\beta}:\alpha\leq\beta\leq\gamma=(\omega_{1})^{N}\right\\}\in
N$

be an iteration witnessing $V_{\delta}\geq N$ in $V[G]$.

We must show that

$j_{0\gamma}:H_{\omega_{2}}^{V}\cup\mathcal{A}^{V}\to
H_{\omega_{2}}^{N}\cup\mathcal{A}^{N}$

is $\Sigma_{1}$-elementary for
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$.

By simple coding tricks (e.g. coding any finite tuple of elements of
$H_{\omega_{2}}$ by a subset of $\omega_{1}$ via the map $\text{{\rm
Cod}}_{\omega_{1}}$ and any finite tuple of elements of $\mathcal{A}$ by their
product), it suffices to check that for any $\Sigma_{1}$-formula $\phi(x,y)$
for $\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$
$A\in\mathcal{P}\left(\omega_{1}\right)^{V}$, $B\in\mathcal{A}^{V}$

$(H_{\omega_{2}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{1}}^{N},\mathbf{NS}_{\omega_{1}}^{N})\models\phi(j_{0\gamma}(A),B^{V[G]}\cap
N).$

if and only if

$(H_{\omega_{2}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V})\models\phi(A,B).$

Now by $(*)\text{-}\mathcal{A}$ in $V$,

$G_{A}=\left\\{(N,a):\exists\mathcal{J}\,\mathbf{NS}\text{-correct iteration
of $N$ mapping $a$ to $A$}\right\\}$

is $L(\mathcal{A})$-generic for $\mathbb{P}_{\mathrm{max}}$.

Recall the set $D_{\phi}$ defined in 5.5.8 and the dense set

$E_{\phi}=D_{\phi}\cup\left\\{(M,b):\forall(N,c)\leq(M,b),\,(N,c)\not\in
D_{\phi}\right\\}$

which are both elements of $L(\mathcal{A})^{V}$, since both sets are the image
under $\text{{\rm Cod}}_{\omega}$ of universally Baire sets in
$\mathcal{A}^{V}$. Hence (by 17) we can argue in $V[G]$ that
$(N,j_{0,\gamma}(A))$ witnesses that any $(M,b)\in G_{A}\cap E_{\phi}\subseteq
E_{\phi}^{V[G]}$ is not in $E_{\phi}^{V[G]}\setminus D_{\phi}^{V[G]}$ (since
$(M,b)\geq(V_{\delta},A)\geq(N,j_{0,\gamma}(A))$. Therefore any such $(M,b)\in
G_{A}\cap E_{\phi}$ is in $D_{\phi}^{V[G]}\cap V=D_{\phi}$.

We conclude that

$(H_{\omega_{2}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\omega_{1}}^{V})\models\phi(A,B),$

by Lemma 5.5.8. ∎

(2) implies (3):

Our assumptions grants that the set

$D_{\mathcal{A}}=\left\\{M\in H_{\omega_{1}}^{V}:M\text{ is absolutely
$\mathcal{A}^{V}$-correct}\right\\}$

is coded by a universally Baire set $\bar{D}_{\mathcal{A}}$ in $V$. Moreover
we also get that whenever $G$ is $V$-generic for
$\operatorname{Coll}(\omega,\delta)$, the lift $\bar{D}_{\mathcal{A}}^{V[G]}$
of $\bar{D}_{\mathcal{A}}$ to $V[G]$ codes

$D_{\mathcal{A}^{V[G]}}^{V[G]}=\left\\{M\in H_{\omega_{1}}^{V[G]}:M\text{ is
absolutely $\mathcal{A}^{V[G]}$-correct}\right\\}.$

By (2) we get that $V_{\delta}\in
D_{\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]}}^{V[G]}$.

By Fact 5.4.7

$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})$

models

> _For all iterable $M$ there exists an absolutely $\mathcal{A}$-correct
> structure $\bar{M}\leq M$ with
> $\prod\mathcal{A}_{M}\in\mathcal{A}_{\bar{M}}$_.

Again since

$(H_{\omega_{1}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{0}}^{V})\prec(H_{\omega_{1}}^{V[G]}\cup\mathcal{A}^{V[G]},\in_{\Delta_{0}}^{V[G]}),$

and the latter is first order expressible in the predicate
$\bar{D}_{\mathcal{A}}\in\mathcal{A}^{V}$, we get that

$(H_{\omega_{1}}^{V[G]}\cup\mathcal{A}^{V[G]},\in_{\Delta_{0}}^{V[G]})$

models that there is an absolutely $\mathcal{A}$-correct structure $N\leq
V_{\delta}$ with

$\prod\mathcal{A}_{V_{\delta}}=\prod\left\\{B^{V[G]}:B\in\mathcal{A}^{V}\right\\}\in\mathcal{A}_{N}.$

Let
$\mathcal{J}=\left\\{j_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma=\omega_{1}^{N}\right\\}\in
H_{\omega_{2}}^{N}$ be an iteration witnessing $N\leq V_{\delta}$.

Now for any $A\in\mathcal{P}\left(\omega_{1}\right)^{V}$ and
$B\in\mathcal{A}^{V}$

$(H_{\omega_{2}}^{N}\cup\mathcal{A}^{N},\in_{\Delta_{1}}^{N},\mathbf{NS}_{\gamma}^{N})$

models

> _There exists an $(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec
> structure $M$ with $B^{V[G]}\cap N\in\mathcal{A}_{M}$ and an
> $\mathbf{NS}$-correct iteration
> $\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha\beta}:\,\alpha\leq\beta\leq\gamma\right\\}$
> of $M$ such that $\bar{j}_{0\gamma}(A)=j_{0\gamma}(A)$_.

This statement is witnessed exactly by $V_{\delta}$ in the place of $M$ (since
$B=B^{V[G]}\cap V_{\delta}$ is in $\mathcal{A}^{V}$ and
$\mathcal{A}^{V[G]}_{V_{\delta}}=\left\\{B^{V[G]}:\,B\in\mathcal{A}^{V}\right\\}$),
and $\mathcal{J}$ in the place of $\bar{\mathcal{J}}$.

Since $V_{\delta}$ is $(\mathbf{NS}_{\omega_{1}},\mathcal{A}^{V[G]})$-ec in
$V[G]$ we get that $j_{0\gamma}\restriction
H_{\omega_{2}}^{V}\cup\mathcal{A}^{V}$ is $\Sigma_{1}$-elementary for
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$ between
$H_{\omega_{2}}^{V}\cup\mathcal{A}^{V}$ and
$H_{\omega_{2}}^{N}\cup\mathcal{A}^{N}$.

Hence

$(H_{\omega_{2}}^{V}\cup\mathcal{A}^{V},\in_{\Delta_{1}}^{V},\mathbf{NS}_{\gamma}^{V})$

models

> _There exists an $(\mathbf{NS}_{\omega_{1}}^{V},\mathcal{A}^{V})$-ec
> structure $M$ with $B\in\mathcal{A}_{M}$ and an iteration
> $\bar{\mathcal{J}}=\left\\{\bar{j}_{\alpha\beta}:\,\alpha\leq\beta\leq(\omega_{1})^{V}\right\\}$
> of $M$ such that $\bar{j}_{0\omega_{1}}(a)=A$ and
> $\mathbf{NS}_{\omega_{1}}^{\bar{j}_{0\omega_{1}}(M)}=\mathbf{NS}_{\omega_{1}}^{V}\cap\bar{j}_{0\omega_{1}}(M)$_.

(3) implies (1):

The key point is to prove that if $M$ is $(\mathbf{NS},\mathcal{A})$-ec,
$a\in\mathcal{P}\left(\omega_{1}\right)^{M}\setminus L(\mathbb{R})^{M}$, $D$
is a dense open set of $\mathbb{P}_{\mathrm{max}}$ such that $D=\text{{\rm
Cod}}_{\omega}[\bar{D}]$ for some $\bar{D}\in\mathcal{A}_{M}$, then there is
some $(M_{0},a_{0})\geq(M,a)$ with $(M_{0},a_{0})\in D\cap M$.

Once this is achieved (3) gives immediately the desired conclusion.

So pick $D,a$ as above. Find $(P,b)\leq(M,a)\in D$ and $(N,c)\leq(P,b)$ with
$N$ $\mathcal{A}$-ec and $\mathcal{A}_{M}\subseteq\mathcal{A}_{N}$.

Then $N$ models (as witnessed by $(P,b)$) the $\Sigma_{1}$-statement for
$\in_{\Delta_{1}}\cup\left\\{\mathbf{NS}_{\omega_{1}}\right\\}$ in parameters
$\bar{D},c$:

> There exists a $\mathbf{NS}$-correct iteration of some $(M_{0},a_{0})\in
> D^{N}$ which maps $a_{0}$ to $c$.

Since $M$ is $(\mathbf{NS},\mathcal{A})$-ec and there is a unique
$\mathbf{NS}$-correct iteration of $M$ which maps $a$ to $c$, we get that $M$
models

> There exists a $\mathbf{NS}$-correct iteration of some $(M_{0},a_{0})\in
> D^{M}=D\cap M$ which maps $a_{0}$ to $a$.

Now the rest of the argument is routine and is left to the reader.

∎

## Some comments and open questions

We believe there is still room to improve the model completeness results one
can predicate from Woodin’s axiom $(*)$. Specifically we conjecture the
following:

######  Conjecture 5.5.9.

Assume ${\mathbf{MAX}(\mathsf{UB})}$ and $(*)\text{-}\mathsf{UB}$. Let
$\Theta$ be the supremum of the ordinals $\alpha$ which are the surjective
image of some $\phi:2^{\omega}\to\mathrm{Ord}$ which exists in $L(\text{{\sf
UB}})$. Then the theory of $L_{\Theta}(\text{{\sf UB}}^{\omega_{1}})$ is model
complete for the signature
$\in_{\Delta_{1}}\cup\left\\{\omega_{1},\mathbf{NS}_{\omega_{1}},\text{{\sf
UB}}\right\\}$ where UB is a predicate symbol which detects which subsets of
$2^{\omega}$ are universally Baire.

Note that the above conjecture does not say that $\Theta$ (the supremum of
Wadge rank of universally Baire sets) is regular in $V$ assuming $V$ models
${\mathbf{MAX}(\mathsf{UB})}+(*)\text{-}\mathsf{UB}$ (it could certainly have
cofinality $\omega_{2}$ in $V$, and we conjecture it could not have cofinality
$\omega_{1}$). In fact an argument of Woodin combined with the results of [24]
should bring that $\Theta$ cannot be regular in models of $\text{{\sf
MM}}^{+++}$.

A subtle question is whether for $\mathsf{ZFC}$ the AMC-spectrum and the model
companionship spectrum can be distinct. We conjecture the following:

######  Conjecture 5.5.10.

Assume $S+T_{\in,A}$ has a model companion for some $\in$-theory
$S\supseteq\mathsf{ZFC}$ and some $A$ with $\in_{A}\supseteq\in_{\Delta_{0}}$.
Then $A\in\mathfrak{spec}_{\mathsf{AMC}}\left(S\right)$.

Another set of open questions is whether the (generically invariant for
suitable classes of forcings) theories of $H_{\aleph_{2}}$ under bounded
category forcing axioms (or under iterated resurrection axioms) isolated in
[4, 5] produce model complete theories for some signature extending
$\in_{\Delta_{0}}$ and for some theory extending $\mathsf{ZFC}+$_large
cardinals_ with some $\Sigma_{2}$-sentence not holding assuming
$(*)\text{-}\mathsf{UB}$ (for example the assertion that canonical functions
are not dominating modulo clubs, or some other $\Sigma_{2}$-sentence whose
negation can only be forced using a stationary set preserving forcing which
cannot be proper).

We believe it can also be interesting to investigate more the notions of model
companionship spectrum or AMC-spectrum. For example: given a countable theory,
analyze the descriptive set theoretic complexity of the partial order given by
its AMC-spectrum under inclusion; can this be a useful measure to compare the
complexity of countable theories? Are there other model-theoretic properties
of a mathematical theory sensitive to the signature for which the spectrum
makes sense? In which case what type of information can we extract from this
spectrum?

## References

  * [1] D. Asperó, P. Larson, and J. T. Moore.  Forcing axioms and the continuum hypothesis.  Acta Math., 210(1):1–29, 2013.
  * [2] D. Asperó and R. Schindler.  Bounded martin’s maximum with an asterisk.  Notre Dame Journal of Formal Logic, 55(3):333–348, 2014.
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