# Forcing $``\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense” From Large
Cardinals

Andreas Lietz111Institut für Mathematische Logik und Grundlagenforschung,
Universität Münster, Einsteinstrasse 62, 48149 Münster, FRG.  222Current
address: Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, AT
This paper is part of the authors PhD thesis.

(October 2023)

###### Abstract

We answer a question of Woodin by showing that assuming an inaccessible
cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals
exists, there is a stationary set preserving forcing $\mathbb{P}$ so that
$V^{\mathbb{P}}\models``\mathrm{NS}_{\omega_{1}}\text{ is
}\omega_{1}\text{-dense}"$. We also introduce a new forcing axiom
$\mathrm{QM}$, show it is consistent assuming a supercompact limit of
supercompact cardinals and prove that it implies
$\mathbb{Q}_{\mathrm{max}}\text{-}(\ast)$. Consequently, $\mathrm{QM}$ implies
“$\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense”.

##  1 Introduction

###  1.1 History of “$\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense”

In 1930, Stanislaw Ulam published an influential paper [Ula30] dealing with a
question of Stefan Banach generalizing the measure problem of Lebesgue. He
proved the following theorem:

######  Theorem 1 (Ulam).

Suppose $\kappa$ is an uncountable cardinal and there is a $\sigma$-additive
real-valued measure on $\kappa$ which

  1. $(i)$

measures all subsets of $\kappa$ and

  2. $(ii)$

vanishes on points.

Then there is a weakly inaccessible cardinal $\leq\kappa$.

Ulam noticed that he could strengthen his conclusion if he replaces real-
valued by 0-1-valued. In more modern terminology, his second result reads:

######  Theorem 2 (Ulam).

Suppose $\kappa$ is an uncountable cardinal and there is a nonprincipal
$\sigma$-complete ultrafilter on $\kappa$. Then there is a (strongly)
inaccessible cardinal $\leq\kappa$.

These theorems gave birth to what are now known as real-valued measurable
cardinals and measurable cardinals respectively. In the interest of having all
subsets of some cardinal $\kappa$ be measured in some sense, instead of
increasing the size of $\kappa$, it is also possible to increase the number of
allowed filters that measure. Henceforth Ulam considered the following
question:

######  Question 3.

Suppose $\kappa$ is an uncountable cardinal below the least inaccessible. What
is the smallest possible size of a family $\mathcal{F}$ of $\sigma$-closed
nonprincipal filters on $\kappa$ so that every subset of $\kappa$ is measured
by some filter in $\mathcal{F}$?

Let us call the cardinal in question the Ulam number of $\kappa$,
$\mathrm{Ulam}(\kappa)$. Ulam’s second theorem above can be rephrased as
“$\mathrm{Ulam}(\kappa)>1$”. Indeed, Ulam proved in unpublished work that
$\mathrm{Ulam}(\kappa)\geq\omega$. At some point, Ulam proposed this question
to Paul Erdős, who, together with Leonidas Alaoglu, improved Ulam’s result to
“$\mathrm{Ulam}(\kappa)\geq\omega_{1}$” [Erd50]. The problem, this time in the
special case $\kappa=\omega_{1}$, was apparently revitalized by appearing in
the 1971 collection of unsolved problems in set theory popularized by Erdős
and Hajnal [EH71]: Shortly after, Karel Prikry [Pri72] produced a model in
which $\mathrm{Ulam}(\omega_{1})=2^{\omega_{1}}=\omega_{2}$, and did the same
again with a different method in [Pri76].
A critical step towards a model in which
$\mathrm{Ulam}(\omega_{1})=\omega_{1}$ was taken by Alan D. Taylor: Building
on earlier work of Baumgartner-Hajnal-Maté [BHM75], Taylor provided [Tay79] an
impressive amount of statements equivalent to a natural strengthening of
$``\mathrm{Ulam}(\omega_{1})=\omega_{1}"$, here is a shortened list.

######  Theorem 4 (Taylor).

The following are equivalent:

  1. $(i)$

There is a family of normal filters witnessing
$\mathrm{Ulam}(\omega_{1})=\omega_{1}$.

  2. $(ii)$

There is a $\sigma$-closed uniform $\omega_{1}$-dense ideal on $\omega_{1}$.

  3. $(iii)$

There is a normal uniform $\omega_{1}$-dense ideal on $\omega_{1}$.

The formulation $(iii)$ is much better suited for set-theoretical arguments.
We also mention that Taylor proved that all the above statements fail under
$\mathrm{MA}_{\omega_{1}}$.
Thus what remains of Ulam’s original question was reduced to: Is the existence
of a normal uniform $\omega_{1}$-dense ideal on $\omega_{1}$ consistent with
$\mathrm{ZFC}$? This was answered positively by W. Hugh Woodin in three
different ways. The first was by forcing over a model of
$\mathrm{AD}_{\mathbb{R}}$+$``\Theta\text{ is regular}"$, already in the fall
of 1978. (unpublished). At that time, this theory was not yet known to be
consistent relative to large cardinals. Naturally, somewhat later he did so
from large cardinals:

######  Theorem 5 (Woodin, unpublished333A proof can be found in Foreman’s
handbook article [For10].).

Assume there is an almost-huge cardinal $\kappa$. Then there is a forcing
extension in which there is a normal uniform $\omega_{1}$-dense ideal on
$\omega_{1}=\kappa$.

This finally resolved the question relative to large cardinals. But can the
canonical normal uniform ideal, namely $\mathrm{NS}_{\omega_{1}}$, have this
property? It is known that $\mathrm{NS}_{\omega_{1}}$ behaves a little
different in this context.

######  Theorem 6 (Shelah, [She86]).

If $\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense then
$2^{\omega}=2^{\omega_{1}}$. In particular $\mathrm{CH}$ fails.

This is not true for other normal uniform ideals on $\omega_{1}$, for example
$\mathrm{CH}$ holds in the model Woodin constructs from an almost huge
cardinal. One can also ask about the exact consistency strength of the
existence of such a normal uniform $\omega_{1}$-dense ideal on $\omega_{1}$.
Both these questions were answered in subsequent work by Woodin, building on
his $\mathbb{P}_{\mathrm{max}}$-technique.

######  Theorem 7 (Woodin, [Woo10, Corollary 6.150]).

The following theories are equiconsistent:

  1. $(i)$

$\mathrm{ZFC}+``\text{There are infinitely many Woodin cardinals.}"$

  2. $(ii)$

$\mathrm{ZFC}+``\mathrm{NS}_{\omega_{1}}\text{ is }\omega_{1}\text{-dense.}"$

  3. $(iii)$

$\mathrm{ZFC}+``\text{There is a normal uniform }\omega_{1}\text{-dense ideal
on }\omega_{1}."$

The direction $(iii)\Rightarrow(i)$ makes use of Woodin’s core model induction
technique, the argument is unpublished. We refer the interested reader to
[RS14] where part of this is proven. Woodin’s method for $(i)\Rightarrow(ii)$
is by forcing over $L(\mathbb{R})$, assuming $\mathrm{AD}$ there, with the
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{Q}_{\mathrm{max}}$. This
approach has one downside: It is a forcing construction over a canonical
determinacy model. $L(\mathbb{R})$ can be replaced by larger determinacy
models, but $\mathbb{Q}_{\mathrm{max}}$ relies on a good understanding of the
model in question. In practice, this is akin to an anti large cardinal
assumption and leaves open questions along the lines of: Is
“$\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense” consistent together with
all natural large cardinals, e.g. supercompact cardinals? Is it consistent
with powerful combinatorial principles, for example $\mathrm{SRP}$?
Woodin’s original motivation for these results was in fact the question of
generic large cardinal properties of $\omega_{1}$: For example $\omega_{1}$ is
not measurable by Ulam’s theorem, but there can be a generic extension of $V$
with an elementary embedding $j\mathrel{\mathop{\mathchar
58\relax}}V\rightarrow M$ with transitive $M$ and critical point
$\omega_{1}^{V}$. This leads to precipitous ideals on $\omega_{1}$.

######  Definition 8.

A uniform ideal $I$ on $\omega_{1}$ is precipitous if, whenever $G$ is generic
for $(\mathcal{P}(\omega_{1})/I)^{+}$ then $\mathrm{Ult}(V,U_{G})$ is
wellfounded444$U_{G}$ denotes the $V$-ultrafilter induced by $G$..

The existence of an $\omega_{1}$-dense ideal is a much stronger assumption
than the existence of a precipitous ideal. There is a natural well-studied
intermediate principle.

######  Definition 9.

A uniform ideal $I$ on $\omega_{1}$ is saturated if
$(\mathcal{P}(\omega_{1})/I)^{+}$ is $\omega_{2}$-c.c..

Here is a short history of similar result for these principles:

  1. $(i)$

Mitchell forces a precipitous ideal on $\omega_{1}$ from a measurable in the
mid 70s, see [JMMP80].

  2. $(ii)$

Magidor forces “$\mathrm{NS}_{\omega_{1}}$ is precipitous” from a measurable,
published in [JMMP80].

  3. $(iii)$

Kunen [Kun78] forces a saturated ideal on $\omega_{1}$ from a huge cardinal,
which he invented for this purpose.

  4. $(iv)$

Steel-Van Wesep [SVW82] force “$\mathrm{NS}_{\omega_{1}}$ is saturated” over a
model of555Woodin [Woo83] subsequently reduced the assumption to just
$\mathrm{AD}$ . $\mathrm{AD}+\mathrm{AC}_{\mathbb{R}}$.

  5. $(v)$

Foreman-Magidor-Shelah [FMS88] force “$\mathrm{NS}_{\omega_{1}}$ is saturated”
from a supercompact with semiproper forcing. Later reduced to one Woodin
cardinal by Shelah666The main ideas for the argument are in [She98, XVI], a
write-up by Schindler can be found in [Sch11]..

Woodin’s results continue this line of research for $\omega_{1}$-dense ideals.
But the analog of the step from $(iv)$ to $(v)$ for $\omega_{1}$-dense ideals
was missing. Accordingly, Woodin posed the following question:

######  Question 10 (Woodin, [Woo99, Chapter 11 Question 18 b)]).

Assuming the existence of some large cardinal: Must there exist some
semiproper partial order $\mathbb{P}$ such that

$V^{\mathbb{P}}\models``\mathrm{NS}_{\omega_{1}}\text{ is
}\omega_{1}\text{-dense}"\ \text{?}$

We will answer this positively in this thesis.

######  Theorem 11.

Assume there is an inaccessible cardinal $\kappa$ which is the limit of
cardinals which are ${<}\kappa$-supercompact. Then there is a stationary set
preserving forcing $\mathbb{P}$ so that

$V^{\mathbb{P}}\models``\mathrm{NS}_{\omega_{1}}\text{ is
}\omega_{1}\text{-dense}".$

If there is an additional supercompact cardinal below $\kappa$, we can find
such $\mathbb{P}$ that is semiproper.

On a different note, there has been significant interest recently into the
possible $\boldsymbol{\Delta}_{1}$-definability of $\mathrm{NS}_{\omega_{1}}$
(with parameters), in particular in the presence of forcing axioms. Note that
$\mathrm{NS}_{\omega_{1}}$ is trivially $\Sigma_{1}(\omega_{1})$-definable,
but it is independent of $\mathrm{ZFC}$ whether $\mathrm{NS}_{\omega_{1}}$ is
$\boldsymbol{\Pi}_{1}$-definable. Hoffelner-Larson-Schindler-Wu [HLSW22] show:

  1. $(i)$

If $\mathrm{BMM}$ holds and there is a Woodin cardinal then
$\mathrm{NS}_{\omega_{1}}$ is not $\boldsymbol{\Delta}_{1}$-definable.

  2. $(ii)$

If $(\ast)$ holds then $\mathrm{NS}_{\omega_{1}}$ is not
$\boldsymbol{\Delta}_{1}$-definable.

  3. $(iii)$

Thus by Asperó-Schindler [AS21], if $\mathrm{MM}^{++}$ holds,
$\mathrm{NS}_{\omega_{1}}$ is not $\boldsymbol{\Delta}_{1}$-definable.

  4. $(iv)$

It is consistent relative to large cardinals that $\mathrm{BPFA}$ holds and
$\mathrm{NS}_{\omega_{1}}$ is $\boldsymbol{\Delta}_{1}$-definable.

There is also a forthcoming paper by Ralf Schindler and Xiuyuan Sun [SS22]
showing that in $(iii)$, $\mathrm{MM}^{++}$ can be relaxed to $\mathrm{MM}$.
If $\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense then
$\mathrm{NS}_{\omega_{1}}$ is automatically
$\boldsymbol{\Delta}_{1}$-definable: If $\mathcal{S}$ is a set of
$\omega_{1}$-many stationary sets witnessing the density, then
$T\subseteq\omega_{1}$ is stationary iff

$\exists C\subseteq\omega_{1}\text{ a club, }\exists S\in\mathcal{S}\ C\cap
S\subseteq T.$

This was first observed by Friedman-Wu-Zdomskyy [FWZ15]. In this context, two
interesting points arise from our results here: First, we isolate for the
first time a forcing axiom which implies “$\mathrm{NS}_{\omega_{1}}$ is
$\boldsymbol{\Delta}_{1}$-definable”. Second, it is well known that many of
the structural consequences of $\mathrm{MM}$ follow already from
$\mathrm{SRP}$, for example “$\mathrm{NS}_{\omega_{1}}$ is saturated”,
$2^{\omega}=\omega_{2}$, $\mathrm{SCH}$, etc. In contrast, in the result of
Schindler-Sun, $\mathrm{MM}$ cannot be replaced by $\mathrm{SRP}$: If
appropriate large cardinals are consistent, then so is $\mathrm{SRP}$ together
with “$\mathrm{NS}_{\omega_{1}}$ is $\boldsymbol{\Delta}_{1}$-definable”.

### Acknowledgements

I want to thank my supervisor Ralf Schindler for his continued support that
led to the results presented here. Most importantly, for relentlessly asking
me to force “$\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense” from large
cardinals from the beginning of my PhD on. It should be obvious to the reader
that this work would not have been possible without Ralf’s great supervision.
Moreover, I am grateful to David Asperó for a number of very helpful remarks
and suggestions.
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC 2044 390685587,
Mathematics Münster: Dynamics - Geometry - Structure.

##  2 Notation

First, we fix some notation. We will extensively deal with countable
elementary substructures $X\prec H_{\theta}$ for large regular $\theta$. We
will make frequent use of the following notation:

######  Definition 2.1.

Suppose $X$ is any extensional set.

  1. $(i)$

$M_{X}$ denotes the transitive isomorph of $X$.

  2. $(ii)$

$\pi_{X}\colon M_{X}\rightarrow X$ denotes the inverse collapse.

  3. $(iii)$

$\delta^{X}\coloneqq\omega_{1}\cap X$.

In almost all cases, we will apply this definition to a countable elementary
substructure $X\prec H_{\theta}$ for some uncountable cardinal $\theta$. In
some cases, the $X$ we care about lives in a generic extension of $V$, even
though it is a substructure of $H_{\theta}^{V}$. In that case, $\delta^{X}$
will always mean $X\cap\omega_{1}^{V}$.

We will also sometimes make use of the following convention in order to
“unclutter” arguments.

######  Convention 2.2.

If $X\prec H_{\theta}$ is an elementary substructure and some object $a$ has
been defined before and $a\in X$ then we denote $\pi_{X}^{-1}(a)$ by
$\bar{a}$.

We will make use of this notation only if it is unambiguous.

######  Definition 2.3.

If $X,Y$ are sets then $X\sqsubseteq Y$ holds just in case

  1. $(i)$

$X\subseteq Y$ and

  2. $(ii)$

$\delta^{X}=\delta^{Y}$.

We use the following notions of clubs and stationarity on
$[H_{\theta}]^{\omega}$:

######  Definition 2.4.

Suppose $A$ is an uncountable set.

  1. $(i)$

$[A]^{\omega}$ is the set of countable subsets of $A$.

  2. $(ii)$

$\mathcal{C}\subseteq[A]^{\omega}$ is a club in $[A]^{\omega}$ if

    1. $a)$

for any $X\in[A]^{\omega}$ there is a $Y\in\mathcal{C}$ with $X\subseteq Y$
and

    2. $b)$

if $\langle Y_{n}\mid n<\omega\rangle$ is a $\subseteq$-increasing sequence of
sets in $\mathcal{C}$ then $\bigcup_{n<\omega}Y_{n}\in\mathcal{C}$.

  3. $(iii)$

$\mathcal{S}\subseteq[A]^{\omega}$ is stationary in $[A]^{\omega}$ if
$\mathcal{S}\cap\mathcal{C}\neq\emptyset$ for any club $\mathcal{C}$ in
$[A]^{\omega}$.

Next, we explain our notation for forcing iterations.

######  Definition 2.5.

Suppose
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is an iteration and $\beta\leq\gamma$. We consider elements of $\mathbb{P}$ as
functions of domain (or length) $\gamma$.

  1. $(i)$

If $p\in\mathbb{P}_{\beta}$ then $\mathrm{lh}(p)=\beta$.

  2. $(ii)$

If $G$ is $\mathbb{P}$-generic then $G_{\beta}$ denotes the restriction of $G$
to $\mathbb{P}_{\beta}$, i.e.

$G_{\beta}=\\{p\upharpoonright\beta\mid p\in G\\}.$

Moreover, $\dot{G}_{\beta}$ is the canonical $\mathbb{P}$-name for
$G_{\beta}$.

  3. $(iii)$

If $G_{\beta}$ is $\mathbb{P}_{\beta}$-generic then
$\mathbb{P}_{\beta,\gamma}$ denotes (by slight abuse of notation) the
remainder of the iteration, that is

$\mathbb{P}_{\beta,\gamma}=\\{p\in\mathbb{P}_{\gamma}\mid
p\upharpoonright\beta\in G_{\beta}\\}.$

$\dot{\mathbb{P}}_{\beta,\gamma}$ denotes a name for
$\mathbb{P}_{\beta,\gamma}$ in $V$.

  4. $(iv)$

If $G$ is $\mathbb{P}$-generic and $\alpha<\beta$ then $G_{\alpha,\beta}$
denotes the projection of $G$ onto $\mathbb{P}_{\alpha,\beta}$.

There will be a number of instances were we need a structure to satisfy a
sufficiently large fragment of $\mathrm{ZFC}$. For completeness, we make this
precise.

######  Definition 2.6.

Sufficiently much of $\mathrm{ZFC}$ is the fragment
$\mathrm{ZFC}^{-}+``\omega_{1}\text{ exists}"$. Here, $\mathrm{ZFC}^{-}$ is
$\mathrm{ZFC}$ without the powerset axiom and with the collection scheme
instead of the replacement scheme.

##  3 $\diamondsuit(\omega_{1}^{{<}\omega})$ and
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$

We introduce the central combinatorial principle which is due to Woodin. The
relevancy is motivated by the following observation: If
$\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense, then there is a dense
embedding

$\eta\colon\mathrm{Col}(\omega,\omega_{1})\rightarrow\mathcal{(}P(\omega_{1})/\mathrm{NS}_{\omega_{1}})^{+}.$

We aim to force a forcing axiom that implies this. As usual, the forcing
achieving this is an iteration $\mathbb{P}$ of some large cardinal length
$\kappa$ which preserves $\omega_{1}$ and iterates forcings of size
${<}\kappa$ with countable support-style supports. $\mathbb{P}$ will thus be
$\kappa$-c.c. and this means that some “representation”

$\eta_{0}\colon\mathrm{Col}(\omega,\omega_{1})\rightarrow\mathrm{NS}_{\omega_{1}}^{+}$

of $\eta$ exists already in an intermediate extension. By “representation” we
mean that in $V^{\mathbb{P}}$,

$[\eta_{0}(p)]_{\mathrm{NS}_{\omega_{1}}}=\eta(p)$

for all $p\in\mathrm{Col}(\omega,\omega_{1})$777For $S\subseteq\omega_{1}$ and
$I$ an ideal on $\omega_{1}$, $[S]_{I}$ denotes the equivalence class of $S$
induced by the equivalence relation $T\sim T^{\prime}\Leftrightarrow
T\triangle T^{\prime}\in I$.. With this in mind, one should isolate the
relevant $\Pi_{1}$-properties which $\eta_{0}$ possesses in $V^{\mathbb{P}}$.
Consequently, $\eta_{0}$ satisfies these properties in the intermediate
extension. It is hopefully easier to first force an object with this
$\Pi_{1}$-fragment and we should subsequently only force with partial orders
that preserve this property. This is exactly what we will do. The relevant
combinatorial properties are $\diamondsuit(\omega_{1}^{{<}\omega})$ and
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ and were already isolated by Woodin
in his study of $\mathbb{Q}_{\mathrm{max}}$ [Woo10, Section 6.2]. We remark
that the definition we use here is slightly stronger than Woodin’s original
principle in a technical way that turns out to be convenient for our purposes.
Most results in this Section are essentially due to Woodin and proven in
[Woo10, Section 6.2].

######  Definition 3.1.

  1. $(i)$

We say that $f$ guesses $\mathrm{Col}(\omega,\omega_{1})$-filters if $f$ is a
function

$f\colon\omega_{1}\rightarrow H_{\omega_{1}}$

and for all $\alpha<\omega_{1}$, $f(\alpha)$ is a
$\mathrm{Col}(\omega,\omega_{1})\cap\alpha$-filter888We consider the empty set
to be a filter..

  2. $(ii)$

Suppose $\theta\geq\omega_{2}$ is regular and $X\prec H_{\theta}$ is an
elementary substructure. We say $X$ is $f\text{{-slim}}$999We use the
adjective “slim” for the following reason: An $f\text{-slim}$ $X\prec
H_{\theta}$ cannot be too fat compared to its height below $\omega_{1}$, i.e.
$\delta^{X}$. If $X\sqsubseteq Y\prec H_{\theta}$ and $Y$ is $f\text{-slim}$
then $X$ is $f\text{-slim}$ as well, but the converse can fail. if

    1. $(X.i)$

$X$ is countable,

    2. $(X.ii)$

$f,\mathrm{Col}(\omega,\omega_{1})\in X$ and

    3. $(X.iii)$

$f(\delta^{X})$ is $\mathrm{Col}(\omega,\omega_{1})\cap\delta^{X}$-generic
over $M_{X}$.

######  Definition 3.2.

$\diamondsuit(\omega_{1}^{{<}\omega})$ states that there is a function $f$ so
that

  1. $(i)$

$f$ guesses $\mathrm{Col}(\omega,\omega_{1})$-filters and

  2. $(ii)$

for any $b\in\mathrm{Col}(\omega,\omega_{1})$ and regular
$\theta\geq\omega_{2}$

$\displaystyle\\{X\prec H_{\theta}\mid X\text{ is }f\text{-slim}\wedge$
$\displaystyle b\in f(\delta^{X})\\}$

is stationary in $[H_{\theta}]^{\omega}$.

$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ is the strengthening of
$\diamondsuit(\omega_{1}^{{<}\omega})$ where $(ii)$ is replaced by:

  1. $(ii)^{+}$

For any regular $\theta\geq\omega_{2}$

$\\{X\prec H_{\theta}\mid X\text{ is }f\text{-slim}\\}$

contains a club of $[H_{\theta}]^{\omega}$. Moreover, for any
$b\in\mathrm{Col}(\omega,\omega_{1})$

$\\{\alpha<\omega_{1}\mid b\in f(\alpha)\\}$

is stationary.

We say that $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$,
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ respectively.

We introduce some convenient shorthand notation.

######  Definition 3.3.

If $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$ and
$b\in\mathrm{Col}(\omega,\omega_{1})$ then

$S^{f}_{b}\coloneqq\\{\alpha<\omega_{1}\mid b\in f(\alpha)\\}.$

If $f$ is clear from context we will sometimes omit the superscript $f$.

Note that if $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$, then
$S^{f}_{b}$ is stationary for all $b\in\mathrm{Col}(\omega,\omega_{1})$. This
is made explicit for $\diamondsuit^{+}(\omega_{1}^{{<}\omega})$. This is
exactly the technical strengthening over Woodin’s original definition of
$\diamondsuit(\omega_{1}^{{<}\omega}),\diamondsuit^{+}(\omega_{1}^{{<}\omega})$.

######  Definition 3.4.

If $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$ and $\mathbb{P}$ is a
forcing, we say that $\mathbb{P}$ preserves $f$ if whenever $G$ is
$\mathbb{P}$-generic then $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$
in $V[G]$.

We remark that if $f$ witnesses $\diamondsuit^{+}(\omega_{1}^{{<}\omega})$
then “$\mathbb{P}$ preserves $f$” still only means that $f$ witnesses
$\diamondsuit(\omega_{1}^{{<}\omega})$ in $V^{\mathbb{P}}$.

Next, we define a variant of stationary sets related to a witness of
$\diamondsuit(\omega_{1}^{{<}\omega})$. Suppose $\theta\geq\omega_{2}$ is
regular. Then $S\subseteq\omega_{1}$ is stationary iff for any club
$\mathcal{C}\subseteq[H_{\theta}]^{\omega}$, there is some $X\in\mathcal{C}$
with $\delta^{X}\in S$. $f$-stationarity results from restricting to $f$-slim
$X\prec H_{\theta}$ only.

######  Definition 3.5.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$.

  1. $(i)$

A subset $S\subseteq\omega_{1}$ is $f$-stationary iff whenever
$\theta\geq\omega_{2}$ is regular and
$\mathcal{C}\subseteq[H_{\theta}]^{\omega}$ is club then there is some
$f\text{-slim}$ $X\in\mathcal{C}$ with $\delta^{X}\in S$.

  2. $(ii)$

A forcing $\mathbb{P}$ preserves $f$-stationary sets iff any $f$-stationary
set is still $f$-stationary in $V^{\mathbb{P}}$.

Note that all $f$-stationary sets are stationary, but the converse might fail.
$f$-stationary sets are the correct replacement of stationary set in our
context.

We mention a few basic facts about $\diamondsuit(\omega_{1}^{{<}\omega})$ and
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ which are all essentially due to
Woodin [Woo10], although he did not use the notion of $f$-stationary sets
explicitly.

######  Proposition 3.6.

Suppose $f$ guesses $\mathrm{Col}(\omega,\omega_{1})$-filters. The following
are equivalent for any set $S\subseteq\omega_{1}$:

  1. $(i)$

$S$ is $f$-stationary.

  2. $(ii)$

Whenever $\langle D_{\alpha}\mid\alpha<\omega_{1}\rangle$ is a sequence of
dense subsets of $\mathrm{Col}(\omega,\omega_{1})$, the set

$\\{\alpha\in S\mid\forall\beta<\alpha\ f(\alpha)\cap
D_{\beta}\neq\emptyset\\}$

is stationary.

######  Proposition 3.7.

Suppose $f$ guesses $\mathrm{Col}(\omega,\omega_{1})$-filters. The following
are equivalent:

  1. $(i)$

$f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$.

  2. $(ii)$

$S^{f}_{b}$ is $f$-stationary for all $b\in\mathrm{Col}(\omega,\omega_{1})$.

  3. $(iii)$

For any $b\in\mathrm{Col}(\omega,\omega_{1})$ and sequence $\langle
D_{\alpha}\mid\alpha<\omega_{1}\rangle$ of dense subsets of
$\mathrm{Col}(\omega,\omega_{1})$,

$\\{\alpha\in S^{f}_{b}\mid\forall\beta<\alpha\ f(\alpha)\cap
D_{\beta}\neq\emptyset\\}$

is stationary.

###### Proof.

The equivalence of $(i)$ and $(ii)$ follows from the definitions. $(ii)$ and
$(iii)$ are equivalent by the equivalent formulation of $f$-stationarity
provided by Proposition 3.6. ∎

We mention a handy corollary.

######  Corollary 3.8.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. Any forcing
preserving $f$-stationary sets preserves $f$.

######  Proposition 3.9.

Suppose $f$ guesses $\mathrm{Col}(\omega,\omega_{1})$-filters. The following
are equivalent:

  1. $(i)$

$f$ witnesses $\diamondsuit^{+}(\omega_{1}^{{<}\omega})$.

  2. $(ii)$

For any $b\in\mathrm{Col}(\omega,\omega_{1})$, $S^{f}_{b}$ is stationary and
all stationary sets are $f$-stationary.

  3. $(iii)$

If $D$ is dense in $\mathrm{Col}(\omega,\omega_{1})$ then

$\\{\alpha<\omega_{1}\mid\ f(\alpha)\cap D\neq\emptyset\\}$

contains a club and for all $b\in\mathrm{Col}(\omega,\omega_{1})$, $S^{f}_{b}$
is stationary.

  4. $(iv)$

All countable $X\prec H_{\theta}$ with $f\in X$ and $\theta\geq\omega_{2}$
regular are $f\text{-slim}$ and moreover for all
$b\in\mathrm{Col}(\omega,\omega_{1})$, $S^{f}_{b}$ is stationary.

We will now give a natural equivalent formulation of
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$. Witnesses of
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ are simply codes for regular
embeddings101010Regular embeddings, also known as complete embeddings, are
embeddings between partial orders which preserve maximal antichains. of
$\mathrm{Col}(\omega,\omega_{1})$ into $\mathrm{NS}_{\omega_{1}}^{+}$.

######  Lemma 3.10.

The following are equivalent:

  1. $(i)$

$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$.

  2. $(ii)$

There is a regular embedding
$\eta\colon\mathrm{Col}(\omega,\omega_{1})\rightarrow(\mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}})^{+}$.

The argument above suggests the following definition.

######  Definition 3.11.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. We define

$\eta_{f}\colon\mathrm{Col}(\omega,\omega_{1})\rightarrow(\mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}})^{+}$

by $b\mapsto[S^{f}_{b}]_{\mathrm{NS}_{\omega_{1}}}$ and call $\eta_{f}$ the
embedding associated to $f$.

######  Definition 3.12.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$.
$\mathrm{NS}_{f}$ is the ideal of $f$-nonstationary sets, that is

$\mathrm{NS}_{f}=\\{N\subseteq\omega_{1}\mid N\text{ is not
}f\text{-stationary}\\}.$

######  Lemma 3.13.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$.
$\mathrm{NS}_{f}$ is a normal uniform ideal.

To each witness $f$ of $\diamondsuit(\omega_{1}^{{<}\omega})$, one can
associate a version of semiproperness.

######  Definition 3.14.

  1. $(i)$

Let $\theta$ be a sufficiently large regular cardinal and $X\prec H_{\theta}$
$f\text{-slim}$ with $\mathbb{P}\in X$. A condition $q\in\mathbb{P}$ is
$(X,\mathrm{Col}(\omega,\omega_{1}),f)$-semigeneric if $q$ is
$(X,\mathrm{Col}(\omega,\omega_{1})$-semigeneric and

$q\Vdash``\check{X}[\dot{G}]\text{ is }f\text{-slim}"$

  2. $(ii)$

$\mathbb{P}$ is $f$-semiproper if for any sufficiently large regular $\theta$
and any $f\text{-slim}$ $X\prec H_{\theta}$ with $\mathbb{P}\in X$ as well as
all $p\in\mathbb{P}\cap X$ there is $q\leq p$ that is
$(X,\mathbb{P},f)$-semigeneric.

An $f$-semiproper forcing $\mathbb{P}$ need not preserve stationary sets,
however it will preserve $f$-stationary sets as $f$-stationary sets and hence
$f$ will still witness $\diamondsuit(\omega_{1}^{{<}\omega})$ in
$V^{\mathbb{P}}$.

However, just as for semiproperness, $f$-semiproper forcings can be iterated.

######  Theorem 3.15 (Lietz, [Lie23]).

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. Any nice
iteration of $f$-semiproper forcings is $f$-semiproper.

We refer to [Miy02] for the definition of nice iterations. For all intents and
purposes, nice iterations can be replaced by RCS iterations here.

##  4 A Forcing Axiom That Implies “$\mathrm{NS}_{\omega_{1}}$ Is
$\omega_{1}$-Dense”

We formulate a forcing axiom that implies
$\mathbb{Q}_{\mathrm{max}}\text{-}(\ast)$. We go on and show that it can be
forced from a supercompact limit of supercompact cardinals.

###  4.1 $\mathrm{Q}$-Maximum

######  Definition 4.1.

$\mathrm{Q}$-Maximum, denoted $\mathrm{QM}$, holds if there is a witness $f$
of $\diamondsuit(\omega_{1}^{{<}\omega})$ and $\mathrm{FA}(\Gamma)$ holds
where

$\Gamma=\\{\mathbb{P}\mid\mathbb{P}\text{ preserves
}f\\}=\\{\mathbb{P}\mid\forall p\in\mathrm{Col}(\omega,\omega_{1})\
S^{f}_{p}\in(\mathrm{NS}_{f}^{+})^{V^{\mathbb{P}}}\\}.$

We remark that the consistency of $\mathrm{QM}$ is a subtle matter, for
example any “$++$”-version of $\mathrm{QM}$ would be inconsistent. It is
however relevant to our purposes.

######  Lemma 4.2.

If $f$ witnesses $\mathrm{QM}$ then $\eta_{f}$ is a dense embedding. In
particular, $\mathrm{NS}_{\omega_{1}}$ is $\omega_{1}$-dense.

###### Proof.

Suppose $S\subseteq\omega_{1}$ is so that

$S^{f}_{p}\nsubseteq S\mod\mathrm{NS}_{\omega_{1}}$

for all $p\in\mathrm{Col}(\omega,\omega_{1})$. Let $\mathbb{P}$ be the
canonical forcing that shoots a club through $T\coloneqq\omega_{1}-S$. That is
$p\in\mathbb{P}$ iff $p\subseteq T$ is closed and bounded and $p\leq q$ iff
$q$ is an initial segment of $p$.

######  Claim 4.3.

$\mathbb{P}$ preserves $f$.

###### Proof.

Let $b\in\mathrm{Col}(\omega,\omega_{1})$, we have to show that $S^{f}_{b}$ is
$f$-stationary in $V^{\mathbb{P}}$. Let $p\in\mathbb{P}$, $\dot{C}$ a
$\mathbb{P}$-name for a club and $\langle\dot{D}_{i}\mid i<\omega_{1}\rangle$
a sequence of $\mathbb{P}$-names for dense subsets of
$\mathrm{Col}(\omega,\omega_{1})$. We will find $q\leq p$ with

$q\Vdash\exists\xi\in\dot{C}\cap S^{\check{f}}_{\check{b}}\ \forall i<\xi\
\check{f}(\xi)\cap\dot{D}_{i}\neq\emptyset.$ ($q$)

Let $\theta$ be large and regular. Note that $\mathrm{MM}(f)$ holds and hence
$f$ witnesses $\diamondsuit^{+}(\omega_{1}^{{<}\omega})$. As $T\cap S^{f}_{b}$
is stationary, $T\cap S^{f}_{b}$ is $f$-stationary and we can find some
$X\prec H_{\theta}$ with

  1. $(X.i)$

$X$ is $f\text{-slim}$,

  2. $(X.ii)$

$\mathbb{P},p,\dot{C},\langle\dot{D}_{i}\mid i<\omega_{1}\rangle\in X$ and

  3. $(X.iii)$

$\delta^{X}\in T\cap S^{f}_{b}$.

Now find a decreasing sequence $\langle p_{n}\mid n<\omega\rangle$ with

  1. $(\vec{p}.i)$

$p_{0}=p$,

  2. $(\vec{p}.ii)$

$\forall n<\omega\ p_{n}\in\mathbb{P}\cap X$ and

  3. $(\vec{p}.iii)$

for all $D\in M_{X}[f(\delta^{X})]$ dense in $\pi_{X}^{{}_{1}}(\mathbb{P})$,
there is $n<\omega$ with $p_{n}\in\pi_{X}[D]$.

Set $q=\bigcup_{n<\omega}p_{n}\cup\\{\delta^{X}\\}$ and note that
$q\in\mathbb{P}$ as $\delta^{X}\in T$. It is clear that $q$ is
$(X,\mathbb{P},f)$-semigeneric so that if $G$ is $\mathbb{P}$-generic with
$q\in G$ then

$\forall i<\delta^{X}=\delta^{X[G]}\
f(\delta^{X})\cap\dot{D}_{i}^{G}\neq\emptyset$

as well as $\delta^{X}\in\dot{C}^{G}\cap S^{f}_{b}$. Thus $q$ indeed satisfies
($q$). ∎

Thus $\mathrm{FA}(\\{\mathbb{P}\\})$ holds. This implies that if $G$ is
$\mathbb{P}$-generic then

$(H_{\omega_{2}};\in)^{V}\prec_{\Sigma_{1}}(H_{\omega_{2}};\in)^{V[G]}$

and as $T$ contains a club in $V[G]$, this must already be true in $V$. This
means $S$ is nonstationary which is what we had to show. ∎

We will prove eventually that $\mathrm{QM}$ can be forced from large
cardinals.

######  Theorem 4.4.

Suppose there is a supercompact limit of supercompact cardinals. Then
$\mathrm{QM}$ holds in a forcing extension by stationary set preserving
forcing.

###  4.2 $\mathrm{Q}$-iterations

Our strategy to force $\mathrm{QM}$, or “$\mathrm{NS}_{\omega_{1}}$ is
$\omega_{1}$-dense” for that matter has to make use of an iteration theorem
that allows us to iterate essentially arbitrary $f$-preserving forcings for a
witness $f$ of $\diamondsuit(\omega_{1}^{{<}\omega})$ so that $f$ is
preserved. We have proven in [Lie23] a more general version of the following
theorem.

######  Theorem 4.5 (Lietz, [Lie23]).

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$ and
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a nice iteration of $f$-preserving forcings. Suppose that

  1. $(\mathbb{P}.i)$

$\Vdash_{\mathbb{P}_{\alpha+2}}\mathrm{SRP}$ for all $\alpha+2\leq\gamma$ and

  2. $(\mathbb{P}.ii)$

$\Vdash_{\mathbb{P}_{\alpha}}``\dot{\mathbb{Q}}_{\alpha}\text{ preserves
}f\text{-stationary sets from }\bigcup_{\beta<\alpha}V[\dot{G}_{\beta}]$.

Then $\mathbb{P}$ preserves $f$.

The immediate problem is that $(\mathbb{P}.ii)$ puts an undesired additional
requirement on the forcings we want to iterate. Luckily, there is a small
trick to still get away with this: Note that an $f$-preserving forcing must
preserve the $f$-stationarity of the set $S^{f}_{b}$ for
$b\in\mathrm{Col}(\omega,\omega_{1})$. Suppose that at all successor steps, we
arrange that any $f$-stationary set from the previous extension contains some
$S^{f}_{b}$ modulo a non-stationary set. Now at a limit step, suddenly every
$f$-preserving forcing will satisfy requirement $(\mathbb{P}.ii)$. As
$(\mathbb{P}.i)$ does not ask anything of us at limit steps either, we are
free to use any $f$-preserving forcing we desire at limit steps.

######  Definition 4.6.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. We say that a
forcing $\mathbb{P}$ freezes $\mathrm{NS}_{\omega_{1}}$ along $f$ if for any
$\mathbb{P}$-generic $G$ we have

  1. $(i)$

$f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$ in $V[G]$ and

  2. $(ii)$

for any $S\in\mathcal{P}(\omega_{1})\cap V$, we either have
$S\in\mathrm{NS}_{\omega_{1}}^{V[G]}$ or there is
$p\in\mathrm{Col}(\omega,\omega_{1})$ with $S^{f}_{p}\subseteq
S\mod\mathrm{NS}_{\omega_{1}}^{V[G]}$.

We hope to have motivated the following definition.

######  Definition 4.7.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. A $Q$-iteration
(w.r.t. $f$) is a nice iteration
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
which satisfies

  1. $(i)$

$\Vdash_{\mathbb{P}_{\alpha}}``\dot{\mathbb{Q}}_{\alpha}\text{ preserves }f"$,

  2. $(ii)$

$\Vdash_{\mathbb{P}_{\alpha+1}}(\ddagger)$ and

  3. $(iii)$

if $\alpha+1<\gamma$ then
$\Vdash_{\mathbb{P}_{\alpha+1}}``\dot{\mathbb{Q}}_{\alpha+1}\text{ freezes
}\mathrm{NS}_{\omega_{1}}\text{ along }f"$

for all $\alpha<\gamma$.

As an immediate consequence of Theorem 4.5, we get the following “iteration
theorem”.

######  Theorem 4.8.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$. All
$Q$-iterations (w.r.t. $f$) preserve $f$.

Provided we find enough forcings which allow us to continue a $Q$-iteration up
to a supercompact cardinal, we are able to force $\mathrm{QM}$. To be precise,
we will prove the following two lemmas.

######  Lemma 4.9.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$, there is a
Woodin cardinal and $V$ is closed under $X\mapsto M_{1}^{\sharp}(X)$. Then
there is a $f$-preserving forcing which freezes $\mathrm{NS}_{\omega_{1}}$
along $f$.

######  Lemma 4.10.

Suppose $f$ witnesses $\diamondsuit(\omega_{1}^{{<}\omega})$ and there is a
supercompact cardinal. Then there is an $f$-preserving forcing $\mathbb{P}$
with $V^{\mathbb{P}}\models\mathrm{SRP}$.

We can show this right away.

###### Proof.

The same construction which forces $\mathrm{SRP}$ via semiproper forcing from
a supercompact cardinal can be used. A small change in the proof gives that
any forcing for an instance of $\mathrm{SRP}$ is not only semiproper, but also
$f$-semiproper. Now use Theorem 3.15 instead of Shelah’s iteration theorem for
semiproper forcings and do a nice iteration instead of a RCS iteration. ∎

We will eventually prove Lemma 4.9 in the next section. The basic idea is to
use a version of the Asperó-Schindler $(\ast)$-forcing with
$\mathbb{P}_{\mathrm{max}}$ replaced by $\mathbb{Q}_{\mathrm{max}}$. However,
we will run into a number of problems we need to solve first.

##  5 Blueprints for Instances of “$\mathrm{MM}^{++}\Rightarrow(\ast)$”

We modify the $(\ast)$-forcing method of Asperó-Schindler in a way that allows
us to prove a variety of instances of $\mathrm{MM}^{++}\Rightarrow(\ast)$,
though our main interest lies in Lemma 4.9.

######  Definition 5.1.

Let $\mathbb{P}\in L(\mathbb{R})$ be a forcing. $\mathbb{P}\text{-}(\ast)$
asserts that $\mathrm{AD}$ holds in $L(\mathbb{R})$ and there is a filter
$g\subseteq\mathbb{P}$ with

  1. $(i)$

$g$ is $\mathbb{P}$-generic over $L(\mathbb{R})$ and

  2. $(ii)$

$\mathcal{P}(\omega_{1})\subseteq L(\mathbb{R})[g]$.

$(\ast)$ is $\mathbb{P}_{\mathrm{max}}\text{-}(\ast)$.
$\mathbb{P}_{\mathrm{max}}$ is the most prominent of a number of similar
forcing notions defined and analyzed by Woodin in [Woo10]. A central notion to
all of them is that of a generically iterable structure.

######  Definition 5.2.

Suppose the following holds:

  1. $(M.i)$

$(M;\in,I)$ is a countable transitive model of (sufficiently much of)
$\mathrm{ZFC}$ where $I$ is allowed as a class parameter in the schemes.

  2. $(M.ii)$

$(M;\in,I)\models\text{``}I$ is a normal uniform ideal on $\omega_{1}$”.

  3. $(M.iii)$

$a_{0},\dots,a_{n}\in M$.

In this case, we call $(M,I,a_{0},\dots,a_{n})$ a potentially iterable
structure. A generic iteration of $(M,I,a_{0},\dots,a_{n})$ is a sequence

$\langle(M_{\alpha},I_{\alpha},a_{0,\alpha},\dots,a_{n,\alpha}),\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$

with

  * •

$(M_{0},I_{0})=(M,I)$,

  * •

$a_{i,\alpha}=\mu_{0,\alpha}(a_{i})$ for $i\leq n$,

  * •

$\mu_{\alpha,\alpha+1}\colon(M_{\alpha};\in,I_{\alpha})\rightarrow(M_{\alpha+1};\in,I_{\alpha+1})$
is a generic ultrapower of $M_{\alpha}$ w.r.t $I_{\alpha}$ and

  * •

if $\alpha\in\mathrm{Lim}$ then

$\langle(M_{\alpha};\in,I_{\alpha}),\mu_{\beta,\alpha}\mid\beta<\alpha\rangle=\varinjlim\langle(M_{\beta};\in,I_{\beta}),M_{\beta,\xi}\mid\beta\leq\xi<\alpha\rangle$

for all $\alpha\leq\gamma$. $(M,I,a_{0},\dots,a_{n})$ is a generically
iterable structure if all (countable) generic iterations of
$(M,I,a_{0},\dots,a_{n})$ produce wellfounded models. Note that this only
depends on $(M,I)$ and that we do not require $I\in M$.

######  Remark 5.3.

A generic iteration
$\langle(M_{\alpha},I_{\alpha},a_{0,\alpha},\dots,a_{n,\alpha}),\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$
can be read off from the final map $\mu_{0,\gamma}\colon M_{0}\rightarrow
M_{\gamma}$, so we will frequently identify one with the other. We also
reserve the right to call generic iterations simply iterations.

######  Definition 5.4.

$\mathbb{P}_{\mathrm{max}}$-conditions are generically iterable structures
$(M,I,a)$ with $a\in\mathcal{P}(\omega_{1})^{M}$ and
$M\models\omega_{1}^{L[a]}=\omega_{1}$. $\mathbb{P}_{\mathrm{max}}$ is ordered
by $q=(N,J,b)<_{\mathbb{P}_{\mathrm{max}}}p$ iff there is a generic iteration

$\mu\colon p\rightarrow p^{\ast}=(M^{\ast},I^{\ast},a^{\ast})$

of length $\omega_{1}^{q}+1$ in $q$ so that

  1. $(<_{\mathbb{P}_{\mathrm{max}}}.i)$

$I^{\ast}=J\cap M^{\ast}$ and

  2. $(<_{\mathbb{P}_{\mathrm{max}}}.ii)$

$a^{\ast}=b$.

There are a number of ways this definition can be varied, leading to different
partial orders. We will work with such variants in a general context.

###  5.1 $\mathbb{P}_{\mathrm{max}}\text{-variations}$ and the
$\mathbb{V}_{\mathrm{max}}$-multiverse view

######  Definition 5.5.

A $\mathbb{P}_{\mathrm{max}}\text{-variation}$ is a nonempty projective
preorder $(\mathbb{V}_{\mathrm{max}},\leq_{\mathbb{V}_{\mathrm{max}}})$ with
the following properties:

  1. $(\mathbb{V}_{\mathrm{max}}.i)$

Conditions in $\mathbb{V}_{\mathrm{max}}$ are generically iterable structures
$(M,I,a_{0},\dots,a_{n})$ for some fixed
$n=n^{\mathbb{V}_{\mathrm{max}}}$111111Of course, not all structures of this
form are necessarily conditions..

  2. $(\mathbb{V}_{\mathrm{max}}.ii)$

There is a first order formula $\varphi^{\mathbb{V}_{\mathrm{max}}}$ in the
language121212When dealing with $\mathbb{P}_{\mathrm{max}}\text{-variation}$s,
we stick to the convention that capitalized symbols are unary predicates
symbols which are lower case are constants.
$\\{\in,\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$ so that $q=(N,J,b_{0},\dots
b_{n})<_{\mathbb{V}_{\mathrm{max}}}(M,I,a_{0},\dots a_{n})$ iff there is a
generic iteration

$j\colon p\rightarrow
p^{\ast}=(M^{\ast},I^{\ast},a_{0}^{\ast},\dots,a_{n}^{\ast})$

in $N$ of length $\omega_{1}^{N}+1$ with

$(N;\in,J,b_{0},\dots,b_{n})\models\varphi^{\mathbb{V}_{\mathrm{max}}}(p^{\ast}).$

  3. $(\mathbb{V}_{\mathrm{max}}.iii)$

If $\mu\colon p\rightarrow p^{\ast}$ witnesses
$q<_{\mathbb{V}_{\mathrm{max}}}p$ and $\sigma\colon q\rightarrow q^{\ast}$
witnesses $r<_{\mathbb{V}_{\mathrm{max}}}q$ then $\sigma(\mu)\colon
p\rightarrow\sigma(p^{\ast})$ witnesses $r<_{\mathbb{V}_{\mathrm{max}}}p$.

  4. $(\mathbb{V}_{\mathrm{max}}.iv)$

Suppose $(M,I)$ is generically iterable,
$j\colon(M,I)\rightarrow(M^{\ast},I^{\ast})$ is a generic iteration of $(M,I)$
of countable length and $a_{0},\dots a_{n}\in M$. Then

$(M,I,a_{0},\dots,a_{n})\in\mathbb{V}_{\mathrm{max}}\Leftrightarrow(M^{\ast},I^{\ast},j(a_{0}),\dots,j(a_{n}))\in\mathbb{V}_{\mathrm{max}}.$

  5. $(\mathbb{V}_{\mathrm{max}}.v)$

$\mathbb{V}_{\mathrm{max}}$ has no minimal conditions.

We always consider $\mathbb{P}_{\mathrm{max}}\text{-variations}$ as a class
defined by a projective formula, rather then the set itself. So if we mention
$\mathbb{V}_{\mathrm{max}}$ in, e.g. a forcing extension of $V$, then we mean
the evaluation of the projective formula in that model131313In practice this
extension will be projectively absolute so it does not matter which projective
formula we choose. Also all the variations we consider will have a
$\Pi^{1}_{2}$-definition..

######  Remark 5.6.

Typically, $\varphi^{\mathbb{V}_{\mathrm{max}}}$ dictates e.g. one or more of
the following:

  * •

$a_{0}^{\ast}=b_{0},\dots,a_{n}^{\ast}=b_{n}$.

  * •

$I^{\ast}=J\cap M^{\ast}$.

  * •

Some first order property is absolute between $M^{\ast}$ and $N$.

We want to relate forcing axioms to star axioms of the form
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ for
$\mathbb{P}_{\mathrm{max}}\text{-variation}$s $\mathbb{V}_{\mathrm{max}}$. To
explain this relationship heuristically we present the
$\mathbb{V}_{\mathrm{max}}$-Multiverse View:
Suppose $\mathbb{V}_{\mathrm{max}}$ is a
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ (with
$n^{\mathbb{V}_{\mathrm{max}}}=0$ for convenience) and

  * •

$V=(V_{\kappa})^{\mathcal{V}}$ for some large cardinal $\kappa$ in some larger
model $\mathcal{V}$ and

  * •

there are a proper class of Woodin cardinals both in $V$ and $\mathcal{V}$.

We will take the point of view of $\mathcal{V}^{\mathrm{Col}(\omega,\kappa)}$.
Note that our assumptions imply generic projective absoluteness (and more) in
$\mathcal{V}$, in particular $\mathbb{V}_{\mathrm{max}}$ is a
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ also in
$\mathcal{V}^{\mathrm{Col}(\omega,\kappa)}$ and
$\mathbb{V}_{\mathrm{max}}^{W}=\mathbb{V}_{\mathrm{max}}\cap W$ for any
generic extension of $V$. Pick some
$\vec{A}=(A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})\in
H_{\omega_{2}}^{V}$. Let $\mathcal{M}(V)$ denote the closure of $V$ under
generic extensions and grounds containing $\vec{A}$. Points
$W\in\mathcal{M}(V)$ may be considered as
$\mathbb{V}_{\mathrm{max}}$-conditions if

$(W,\mathrm{NS}_{\omega_{1}}^{W},A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})\in\mathbb{V}_{\mathrm{max}}.$

In this case we identify $W$ with this condition. In practice, this can only
reasonably hold if $\omega_{1}^{W}=\omega_{1}^{V}$ so we make this an explicit
condition. The $\mathbb{V}_{\mathrm{max}}$-multiverse of $V$ (w.r.t.
$\vec{A}$) is

$\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)=\\{W\in\mathcal{M}(V)\mid
W\in\mathbb{V}_{\mathrm{max}}\wedge\omega_{1}^{W}=\omega_{1}^{V}\\}.$

If we $\vec{A}$ picked with sufficient care then
$\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$ should be nonempty. If $W[G]$ is
a generic extension of $W$, both in
$\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$, then it is a good extension if

$W[G]\leq_{\mathbb{V}_{\mathrm{max}}}W.$

Here, $p\leq_{\mathbb{V}_{\mathrm{max}}}q$ means
$p\Vdash_{\mathbb{V}_{\mathrm{max}}}\check{q}\in\dot{G}$. The existence of a
proper class of Woodin cardinals in $V$ should guarantee that
$\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$ reversely ordered by good
extensions is “as rich as” $\mathbb{V}_{\mathrm{max}}$.
In this sense, iterated forcing along good extensions corresponds to building
descending sequences in $\mathbb{V}_{\mathrm{max}}$. In practice,
$\mathbb{P}_{\mathrm{max}}\text{-variation}$s are $\sigma$-closed. From this
point of view, $\sigma$-closure of $\mathbb{V}_{\mathrm{max}}$ becomes roughly
equivalent to a forcing iteration theorem: If

$\langle W[G_{\alpha}]\mid\alpha<\gamma\rangle$

is a chain of good extensions $W[G_{\alpha}]\subseteq W[G_{\beta}]$ of points

$W[G_{\alpha}],W[G_{\beta}]\in\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V),\
\alpha\leq\beta<\gamma\in V$

then this constitutes a countable decreasing chain141414Note that the size of
$\gamma$ in $V$ does not matter here. in $\mathbb{V}_{\mathrm{max}}$ in
$\mathcal{V}^{\mathrm{Col}(\omega,\kappa)}$. $\sigma$-closure of
$\mathbb{V}_{\mathrm{max}}$ suggests that there should be a further point

$W[G_{\gamma}]\in\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$

below all $W[G_{\alpha}]$, $\alpha<\gamma$. Thus the “forcing iteration along
$\langle W[G_{\alpha}]\mid\alpha<\gamma\rangle$” preserves $\omega_{1}$ and
enough structure to be able to be extended to a
$\mathbb{V}_{\mathrm{max}}$-condition below all $W[G_{\alpha}]$ without
collapsing $\omega_{1}$.
We should be able to find points satisfying
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ by constructing “closure points”
$W\in\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$ of sufficiently generic
$\leq_{\mathbb{V}_{\mathrm{max}}}$-decreasing sequences

$\langle W_{\alpha}\mid\alpha<\gamma\rangle$

in $\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$. To make that precise, we
want:

$\text{If }D\in L(\mathbb{R})^{W}\text{ is dense open in
}\mathbb{V}_{\mathrm{max}}^{W}\text{ then }W_{\alpha}\in D^{\ast}\text{ for
some }\alpha<\gamma.$ ($\bigstar$)

Here, $D^{\ast}$ is the reinterpretation of the universally Baire $D$ in
$\mathcal{V}^{\mathrm{Col}(\omega,\kappa)}$. The degree of closure of
$W\in\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$ under this procedure is
measured by

$g^{W}=\\{p\in\mathbb{V}_{\mathrm{max}}\mid
W<_{\mathbb{V}_{\mathrm{max}}}p\\}$

which should be a filter if $W$ is “sufficiently closed”. $g^{W}$ can be
defined in $W$ via

$g^{W}=\\{p\in\mathbb{V}_{\mathrm{max}}\mid\exists\mu\colon p\rightarrow
p^{\ast}\text{ of length }\omega_{1}+1\text{ with
}\varphi^{\mathbb{V}_{\mathrm{max}}}(p^{\ast})\\}^{W}$

if $\mathbb{V}_{\mathrm{max}}$ has unique iterations.

######  Definition 5.7.

$\mathbb{V}_{\mathrm{max}}$ has unique iterations if whenever
$q<_{\mathbb{V}_{\mathrm{max}}}p$ then there is a unique generic iteration of
$p$ witnessing this.

Under reasonable assumptions, ($\bigstar$ ‣ 5.1) implies that $g^{W}$ is
generic over $L(\mathbb{R})^{W}$. Finally, an additional property151515Often,
simply $(\neg\mathrm{CH})^{W}$ is enough. Woodin [Woo] (see also [Sch]) has
shown that if $\mathrm{AD}^{L(\mathbb{R})}$ holds, there is a filter
$g\subseteq\mathbb{P}_{\mathrm{max}}$ generic over $L(\mathbb{R})$ and
$\mathrm{CH}$ fails then $g$ witnesses $(\ast)$. like
$W\models``\mathrm{NS}_{\omega_{1}}\text{ is saturated}"$ should imply
$\mathcal{P}(\omega_{1})^{W}\subseteq L(\mathbb{R})^{W}[g^{W}]$.
Taking a step back, forcing a forcing axiom related to good extensions via
iterated forcing looks like it should produce such sequences $\langle
W_{\alpha}\mid\alpha<\gamma\rangle$ with ($\bigstar$ ‣ 5.1) and
$\mathrm{NS}_{\omega_{1}}$ saturated in $W$, so
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ should follow from such a forcing
axiom.
On the other hand, $W$ looks like an endpoint of an iteration liberally
incorporating forcings leading to good extensions: For $\alpha<\gamma$, if
$D\in L(\mathbb{R})^{W_{\alpha}}$ is dense open in
$\mathbb{V}_{\mathrm{max}}^{W_{\alpha}}$ then $D^{\ast}$ is dense open in the
full $\mathbb{V}_{\mathrm{max}}$. $D^{\ast}$ can also be considered as a dense
subset of $\mathcal{M}_{\mathbb{V}_{\mathrm{max}}}(V)$. As
$D^{\ast}\cap\mathbb{V}_{\mathrm{max}}^{W}\in L(\mathbb{R})^{W}$, by
($\bigstar$ ‣ 5.1), there will be some later $\alpha\leq\beta<\gamma$ with
$W_{\beta}\in D^{\ast}$. Thus one might expect a forcing axiom to hold at $W$.
This suggest that $\mathbb{V}_{\mathrm{max}}$ should in fact be equivalent to
a forcing axiom related to good extensions. The consistency of this forcing
axiom should follow from the iteration theorem suggested by the
$\sigma$-closure of $\mathbb{V}_{\mathrm{max}}$.
If we look at the case $\mathbb{V}_{\mathrm{max}}=\mathbb{P}_{\mathrm{max}}$
and let $A$ be some subset of $\omega_{1}$ so that
$\omega_{1}^{L[A]}=\omega_{1}^{V}$ then stationary set preserving extensions
are exactly the generic extensions intermediate to a good extension. The
$\mathbb{P}_{\mathrm{max}}$-Multiverse View is roughly correct in the sense
that:

  * •

(Woodin) $\mathbb{P}_{\mathrm{max}}$ is $\sigma$-closed assuming
$\mathrm{AD}^{L(\mathbb{R})}$.

  * •

(Shelah) Semiproper forcings can be iterated and the class of stationary set
preserving forcings and semiproper forcings coincide under $\mathrm{MM}$.

  * •

(Asperó-Schindler) If there is a proper class of Woodin cardinals then

$(\ast)\Leftrightarrow(\mathcal{P}(\mathbb{R})\cap
L(\mathbb{R}))\text{-}\mathrm{BMM}^{++}.$

The rest of this section distills this heuristic into rigorous mathematics
that relates more $\mathbb{P}_{\mathrm{max}}\text{-variation}$s to forcing
axioms. We will assume (two-step) generic absoluteness in this section, though
this is not fully necessary. Note that in this case, if
$\mathbb{V}_{\mathrm{max}}$ is a $\mathbb{P}_{\mathrm{max}}\text{-variation}$
then we have

$V^{\mathbb{P}}\models``\mathbb{V}_{\mathrm{max}}\text{ is a
}\mathbb{P}_{\mathrm{max}}\text{-variation}"$

in any generic extension $V^{\mathbb{P}}$, where $\mathbb{V}_{\mathrm{max}}$
is to be understood as defined by a projective formula. Usually,
$\mathbb{P}_{\mathrm{max}}\text{-variation}$s are $\Pi^{1}_{2}$.
We will from now on work with some fixed
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$ and
assume $n^{\mathbb{V}}_{\mathrm{max}}=0$ to ease notation.

######  Definition 5.8.

We say that a structure $\mathcal{H}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition if

$V^{\mathrm{Col}(\omega,\mathcal{H})}\models\widecheck{\mathcal{H}}\in\mathbb{V}_{\mathrm{max}}.$

For $A\in H_{\omega_{2}}$, $\mathcal{H}_{A}$ denotes the structure:

$\mathcal{H}_{A}\coloneqq(H_{\omega_{2}},\mathrm{NS}_{\omega_{1}},A)$

Suppose that for some fixed $A\in H_{\omega_{2}}$ we have that
$\mathcal{H}\coloneqq\mathcal{H}_{A}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition. We may define

$g_{A}=\\{p\in\mathbb{V}_{\mathrm{max}}\mid
V^{\mathrm{Col}(\omega,2^{\omega_{1}})}\models\mathcal{H}<_{\mathbb{V}_{\mathrm{max}}}p\\}.$

Our goal is to show that $g_{A}$ witnesses
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ under favorable circumstances. At
the very least, it should be a filter.

######  Proposition 5.9.

Suppose $g_{A}$ meets all projective dense
$D\subseteq\mathbb{V}_{\mathrm{max}}$. Then $g_{A}$ is a filter.

###### Proof.

It is easy to see that if $q<_{\mathbb{V}_{\mathrm{max}}}p$ and $q\in g_{A}$
then $p\in g_{A}$. So assume $p,q\in g_{A}$ and we have to find some $r\in
g_{A}$ with $r\leq_{\mathbb{V}_{\mathrm{max}}}p,q$. Consider

$D=\\{r\in\mathbb{V}_{\mathrm{max}}\mid
r\leq_{\mathbb{V}_{\mathrm{max}}}p,q\vee r\perp p\vee r\perp q\\}$

and note that $D$ is a projective dense subset of $\mathbb{V}_{\mathrm{max}}$,
so by assumption we can find some $r\in D\cap g_{A}$. Now in
$V^{\mathrm{Col}(\omega,2^{\omega_{1}})}$ we have
$r,p,q\leq_{\mathbb{V}_{\mathrm{max}}}\mathcal{H}$ and thus $r$ is compatible
with both $p$ and $q$. By generic absoluteness, this is true in $V$ as well so
that $r\leq_{\mathbb{V}_{\mathrm{max}}}p,q$ as $r\in D$. ∎

Even assuming that $g_{A}$ is a fully generic over $L(\mathbb{R})$, we still
have to arrange $\mathcal{P}(\omega_{1})\subseteq L(\mathbb{R})[g_{A}]$.

######  Definition 5.10.

Suppose that

  1. $(i)$

$g\subseteq\mathbb{V}_{\mathrm{max}}$ is a filter,

  2. $(ii)$

$p\in g$ and

  3. $(iii)$

$\langle p_{\alpha},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$ is
a generic iteration of $p_{0}=p$.

Then we say that $\langle
p_{\alpha},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$ is guided
by $g$ if $p_{\alpha}\in g$ for all countable $\alpha\leq\gamma$.

######  Lemma 5.11.

Suppose $\mathbb{V}_{\mathrm{max}}$ has unique iterations and
$g\subseteq\mathbb{V}_{\mathrm{max}}$ is a filter meeting all projective dense
$D\subseteq\mathbb{V}_{\mathrm{max}}$. For any $p\in g$ and any
$\gamma\leq\omega_{1}$, there is a unique iteration

$\langle p_{\alpha},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$

of $p_{0}=p$ of length $\gamma+1$ guided by $g$.

###### Proof.

First, we prove existence for all $\gamma<\omega_{1}$.

######  Claim 5.12.

There is $q\in g$ with $\omega_{1}^{q}>\gamma$.

###### Proof.

Let $D=\\{q\in\mathbb{V}_{\mathrm{max}}\mid\omega_{1}^{q}>\gamma\\}$. Clearly,
$D$ is projective and we will show that $D$ is dense. Let
$q\in\mathbb{V}_{\mathrm{max}}$ and using $(\mathbb{V}_{\mathrm{max}}.v)$,
find $r<_{\mathbb{V}_{\mathrm{max}}}q$ as witnessed by

$\sigma\colon q\rightarrow q^{\ast}.$

Now let

$\nu\colon r\rightarrow r^{\ast}$

be any generic iteration of $r$ of length $\gamma+2$, consequently
$\omega_{1}^{r^{\ast}}>\gamma$. We have $r^{\ast}\in\mathbb{V}_{\mathrm{max}}$
by $(\mathbb{V}_{\mathrm{max}}.iv)$. Note that the iteration $\nu\circ\sigma$
witnesses $r^{\ast}<_{\mathbb{V}_{\mathrm{max}}}q$. Again applying
$(\mathbb{V}_{\mathrm{max}}.v)$, there is
$s<_{\mathbb{V}_{\mathrm{max}}}r^{\ast}$ and thus
$s<_{\mathbb{V}_{\mathrm{max}}}q$ and $s\in D$.
Thus $g\cap D\neq\emptyset$. ∎

As $g$ is a filter, we can find $q<_{\mathbb{V}_{\mathrm{max}}}p$ with
$\omega_{1}^{q}>\gamma$. Thus if $\mu\colon p\rightarrow p^{\ast}$ witnesses
this then $\mu$ is an iteration

$\langle
p_{\alpha,\beta},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\omega_{1}^{q}\rangle$

of length $\omega_{1}^{q}+1>\gamma+1$ by $(\mathbb{V}_{\mathrm{max}}.ii)$.

######  Claim 5.13.

$\langle
p_{\alpha,\beta},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\gamma\rangle$ is
guided by $g$.

###### Proof.

Let $\alpha\leq\gamma$. Then $\mu_{\alpha,\omega_{1}^{q}}$ is an iteration of
length $\omega_{1}^{q}+1$ in $q$ and
$q\models\varphi^{\mathbb{V}_{\mathrm{max}}}(p_{\omega_{1}^{q}})$, thus
$q<_{\mathbb{V}_{\mathrm{max}}}p_{\alpha}$ and $p_{\alpha}\in g$. ∎

Next we prove uniqueness. By proceeding by induction on
$\gamma\leq\omega_{1}$, it is in fact enough to verify the case $\gamma=1$.
Suppose that $\mu_{i}\colon p\rightarrow p^{\ast}_{i}$ is a generic ultrapower
of $p$ with $p^{\ast}_{i}\in g$ for $i<2$. As $g$ is a filter and by
$(\mathbb{V}_{\mathrm{max}}.v)$, there is $q\in g$ with
$q<_{\mathbb{V}_{\mathrm{max}}}p^{\ast}_{i}$ as witnessed by some

$\mu^{\ast}_{i}\colon p^{\ast}_{i}\rightarrow p^{\ast\ast}_{i}$

for $i<2$ as well as $q<_{\mathbb{V}_{\mathrm{max}}}p$ as witnessed by

$\mu\colon p\rightarrow p^{\ast\ast}.$

Let $i<2$. We have that $p,p^{\ast}_{i}$ are countable in $q$. As

$``p^{\ast}_{i}\text{ is a generic ultrapower of }p"$

is a true $\Sigma^{1}_{1}(p,p^{\ast}_{i})$-statement, it is true in $q$ as
well. Thus there is a generic ultrapower

$\mu^{\prime}_{i}\colon p\rightarrow p^{\ast}_{i}$

in $q$. Both $\mu,\mu^{\ast}_{i}\circ\mu^{\prime}_{i}$ witness
$q<_{\mathbb{V}_{\mathrm{max}}}p$ and as $\mathbb{V}_{\mathrm{max}}$ has
unique iterations, $\mu=\mu^{\ast}_{i}\circ\mu^{\prime}_{i}$. It follows that
$p^{\ast}_{0}=p^{\ast}_{1}$.

######  Claim 5.14.

$\mu_{0}^{\ast}=\mu_{1}^{\ast}$.

###### Proof.

Assume this fails, then

$``\text{There are distinct generic ultrapower maps }p\rightarrow
p_{0}^{\ast}"$

is another true $\Sigma^{1}_{1}(p,p_{0}^{\ast})$-statement which accordingly
must hold in $q$. Thus there is a generic ultrapower map
$\mu_{0}^{\prime\prime}\colon p\rightarrow p^{\ast}_{0}$ in $q$ different from
$\mu_{0}^{\prime}$. But then both $\mu_{0}^{\ast}\circ\mu_{0}^{\prime}$ and
$\mu_{0}^{\ast}\circ\mu_{0}^{\prime\prime}$ witness
$q<_{\mathbb{V}_{\mathrm{max}}}p$, which contradicts that
$\mathbb{V}_{\mathrm{max}}$ has unique iterations. ∎

Finally, existence of a generic iteration of $p$ of length $\omega_{1}+1$
guided by $g$ follows from existence and uniqueness of generic iterations of
$p$ guided by $g$ of any countable length. ∎

This suggests the following definition:

######  Definition 5.15.

Suppose $\mathbb{V}_{\mathrm{max}}$ is a
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ with unique iterations and
$g\subseteq\mathbb{V}_{\mathrm{max}}$ is a filter. For $p\in g$, the
$g$-iteration of $p$ is the unique generic iteration of $p$ of length
$\omega_{1}+1$ that is guided by $g$ (if it exists).

######  Corollary 5.16.

Suppose that

  1. $(i)$

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

  2. $(ii)$

$\mathbb{V}_{\mathrm{max}}$ has unique iterations,

  3. $(iii)$

$\mathcal{H}_{A}$ is almost a $\mathbb{V}_{\mathrm{max}}$-condition,

  4. $(iv)$

$g_{A}\cap D\neq\emptyset$ for all dense
$D\subseteq\mathbb{V}_{\mathrm{max}}$, $D\in L(\mathbb{R})$ and

  5. $(v)$

$\mathcal{P}(\omega_{1})=\bigcup\\{\mathcal{P}(\omega_{1})\cap p^{\ast}\mid
p\in g_{A}\wedge\mu\colon p\rightarrow p^{\ast}\text{ is guided by }g_{A}\\}$.

Then $\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ holds and $g_{A}$ witnesses
this.

###### Proof.

$g_{A}$ is a filter by Proposition 5.9 and thus $L(\mathbb{R})$-generic by
assumption. To see that $\mathcal{P}(\omega_{1})\subseteq
L(\mathbb{R})[g_{A}]$, notice that for any $p\in g_{A}$, $L(\mathbb{R})$ knows
of all countable generic iterations of $p$. Hence, $L(\mathbb{R})[g_{A}]$ can
piece together the $g_{A}$-iteration of $p$ from the countable iterations of
$p$ that are guided by $g_{A}$. $\mathcal{P}(\omega_{1})\subseteq
L(\mathbb{R})[g_{A}]$ now follows immediately from $(v)$. ∎

The biggest obstacle by far is to get into a situation where $g_{A}\cap
D\neq\emptyset$ for all dense $D\subseteq\mathbb{V}_{\mathrm{max}}$, $D\in
L(\mathbb{R})$. The main idea is:

######  Lemma 5.17.

Suppose that all of the following hold:

  1. $(i)$

$D\subseteq\mathbb{V}_{\mathrm{max}}$ is dense.

  2. $(ii)$

$\mathcal{H}_{A}$ is almost a $\mathbb{V}_{\mathrm{max}}$-condition.

  3. $(iii)$

$\mathbb{P}$ is a forcing and $D$ is $|\mathbb{P}|$-universally Baire.

  4. $(iv)$

In $V^{\mathbb{P}}$ there is $q\in D^{\ast}$ and an iteration $\sigma\colon
q\rightarrow q^{\ast}$ with

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A)^{V^{\mathbb{P}}}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(q^{\ast}).$

  5. $(v)$

$\Gamma$ is a set of formulas in the language
$\\{\in,\dot{I},\dot{a},\dot{D}\\}$ so that

    1. $(\Gamma.i)$

$\varphi^{\mathbb{V}_{\mathrm{max}}}\in\Gamma$,

    2. $(\Gamma.ii)$

$\Sigma_{0}\subseteq\Gamma$, where $\Sigma_{0}$ is computed in the language
$\\{\in,\dot{D}\\}$ and

    3. $(\Gamma.iii)$

$\Gamma$ is closed under $\exists$ and $\wedge$.

  6. $(vi)$

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A,D)^{V}\prec_{\Gamma}(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A,D^{\ast})^{V^{\mathbb{P}}}$.

Then $g_{A}\cap D\neq\emptyset$.
If additionally

  1. $(vii)$

$H_{\omega_{2}}^{V}\subseteq q^{\ast}$

then $\mathcal{P}(\omega_{1})=\bigcup\\{\mathcal{P}(\omega_{1})\cap
p^{\ast}\mid p\in g_{A}\wedge\mu\colon p\rightarrow p^{\ast}\text{ is guided
by }g_{A}\\}$.

###### Proof.

Observe that
$(H_{\omega_{2}};\in)\prec_{\Sigma_{1}}(H_{\omega_{2}};\in)^{V^{\mathbb{P}}}$
implies that $\mathbb{P}$ preserves $\omega_{1}$. The statement

$\exists q\in\dot{D}\ \exists\sigma\colon q\rightarrow q^{\ast}\text{ an
iteration of length }\omega_{1}+1\text{ and
}\varphi^{\mathbb{V}_{\mathrm{max}}}(q^{\ast})$

is in $\Gamma$ and thus is true in

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A,D)^{V}$

as witnessed by some $p\in D$ and iteration $\mu\colon p\rightarrow p^{\ast}$.
It follows that $\mu$ witnesses
$\mathcal{H}_{A}<_{\mathbb{V}_{\mathrm{max}}}q$ in
$V^{\mathrm{Col}(\omega,2^{\omega_{1}})}$ so that $p\in D\cap g_{A}$.
Now assume $(vii)$, it is our duty to show

$\mathcal{P}(\omega_{1})=\bigcup\\{\mathcal{P}(\omega_{1})\cap p^{\ast}\mid
p\in g_{A}\wedge\mu\colon p\rightarrow p^{\ast}\text{ is guided by }g_{A}\\}.$

Let $X\subseteq\omega_{1}$. As above,

$\exists q\in\mathbb{V}_{\mathrm{max}}\ \exists\sigma\colon q\rightarrow
q^{\ast}\text{ an iteration of length }\omega_{1}+1\text{ and
}\varphi^{\mathbb{V}_{\mathrm{max}}}(q^{\ast})\wedge X\in q^{\ast}$

reflects down to $V$. The iteration witnessing this in $V$ is guided by
$g_{A}$ by the same argument that showed $p\in g_{A}$ above. ∎

Condition $(vi)$ is a typical consequence of a (bounded) forcing axiom. It is
left to construct forcings $\mathbb{P}$ with property $(iv)$ to which
hopefully a broad range of forcing axioms may apply.

###  5.2 Asperó-Schindler $(\ast)$-forcing

We describe the results of Asperó-Schindler[AS21]. Their results carry over to
any $\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$
though they were originally proven in the case of
$\mathbb{V}_{\mathrm{max}}=\mathbb{P}_{\mathrm{max}}$. Suppose that

  1. $(i)$

$\mathrm{NS}_{\omega_{1}}$ is saturated,

  2. $(ii)$

$A\in H_{\omega_{2}}$ is so that
$\mathcal{H}=(H_{\omega_{2}},\mathrm{NS}_{\omega_{1}},A)$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition and

  3. $(iii)$

$D\subseteq\mathbb{V}_{\mathrm{max}}$ is a $2^{\omega_{1}}$-universally Baire
dense subset of $\mathbb{V}_{\mathrm{max}}$ whose reinterpretation is still
dense in extensions by forcings of size $\leq 2^{\omega_{1}}$, as witnessed by
trees $T,S$ with $D=p[T]$.

Asperó-Schindler construct a partial order
$\mathbb{P}=\mathbb{P}(\mathbb{V}_{\mathrm{max}},A,D)$ so that in
$V^{\mathbb{P}}$ the following picture

$p[T]$$q_{0}=(N,I,b)$$q_{\omega_{1}}=(N^{*},I^{*},b)$$p_{0}$$p_{\omega_{1}^{N}}$$p_{\omega_{1}}$$((H_{\omega_{2}})^{V},\mathrm{NS}_{\omega_{1}}^{V},A)=\mathcal{H}$$\mathbb{V}_{\mathrm{max}}$$\in$$\sigma_{0,\omega_{1}}$$\in$$\in$$\mu_{0,\omega_{1}^{N}}$$\mu_{\omega_{1}^{N},\omega_{1}}$$=$$\in$

exists so that

  1. $(\mathbb{P}.i)$

$\mu_{0,\omega_{1}},\sigma_{0,\omega_{1}}$ are generic iterations of $p_{0}$,
$q_{0}$ respectively,

  2. $(\mathbb{P}.ii)$

$\mu_{0,\omega_{1}^{N}}$ witnesses $q_{0}<_{\mathbb{V}_{\mathrm{max}}}p_{0}$,

  3. $(\mathbb{P}.iii)$

$\mu_{0,\omega_{1}}=\sigma_{0,\omega_{1}}(\mu_{0,\omega_{1}^{N}})$ and

  4. $(\mathbb{P}.iv)$

the generic iteration $\sigma_{0,\omega_{1}}\colon q_{0}\rightarrow
q_{\omega_{1}}$ is correct, i.e.
$I^{\ast}=\mathrm{NS}_{\omega_{1}}^{V^{\mathbb{P}}}\cap N^{\ast}$.

If $\varphi^{\mathbb{V}_{\mathrm{max}}}((M,J,a))$ implies $J=\dot{I}\cap M$
then $\mathrm{NS}_{\omega_{1}}^{p_{\omega_{1}^{N}}}=I\cap p_{\omega_{1}^{N}}$.
This gets transported upwards along $\sigma_{0,\omega_{1}}$ and shows
$\mathrm{NS}_{\omega_{1}}^{V}=I^{\ast}\cap H_{\omega_{2}}^{V}$. Together with
$(\mathbb{P}.iv)$, this yields
$\mathrm{NS}_{\omega_{1}}^{V}=\mathrm{NS}_{\omega_{1}}^{V^{\mathbb{P}}}\cap
V$, i.e. $\mathbb{P}$ preserves stationary sets. If $\mathrm{MM}^{++}$ holds
in $V$ then

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A,D)^{V}\prec_{\Sigma_{1}}(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A,D^{\ast})^{V^{\mathbb{P}}}$

and it follows from Lemma 5.17 that $g_{A}\cap D\neq\emptyset$ (note that
$\varphi^{\mathbb{P}_{\mathrm{max}}}((M,I,a))``=I=\dot{I}\cap M\wedge
a=\dot{a}"$). This is how Asperó-Schindler prove
$\mathrm{MM}^{++}\Rightarrow(\ast)$.
An important observation is the following: To invoke a forcing axiom in the
case of $\mathbb{P}$ or variants thereof, typically $\mathbb{P}$ needs to
preserve certain structure, like stationary sets in the example above. This
preservation is proven in two steps:

  1. $(i)$

Preservation between $q_{\omega_{1}}$ and $V^{\mathbb{P}}$. This is governed
by the iteration $\sigma_{0,\omega_{1}}$ having certain properties in
$V^{\mathbb{P}}$, e.g. correctness.

  2. $(ii)$

Preservation between $p_{\omega_{1}}$ and $q_{\omega_{1}}$. This is governed
by the nature of $\mathbb{V}_{\mathrm{max}}$, specifically the formula
$\varphi^{\mathbb{V}_{\mathrm{max}}}$.

We will modify the construction of $\mathbb{P}$ and get a forcing
$\mathbb{P}^{\diamondsuit}$ which strengthens $(\mathbb{P}.iv)$ so that
$\mathbb{P}^{\diamondsuit}$ can have a variety of preservation properties
depending on the $\mathbb{P}_{\mathrm{max}}\text{-variation}$
$\mathbb{V}_{\mathrm{max}}$ in question, for example

  * •

preserving stationary sets as well as all Suslin trees or

  * •

preserving a witness $f$ of $\diamondsuit(\omega_{1}^{{<}\omega})$
($\rightsquigarrow\mathrm{QM}\Rightarrow\mathbb{Q}_{\mathrm{max}}\text{-}(\ast)$).

###  5.3 $\diamondsuit$-iterations

We introduce the concept that is roughly the equivalent of
$\diamondsuit$-forcing in the world of generic iterations.

######  Definition 5.18.

Suppose $(N,I)$ is generically iterable. A generic iteration

$\langle(N_{i},I_{i}),\sigma_{i,j}\mid i\leq j\leq\omega_{1}\rangle$

of $(N,I)=(N_{0},I_{0})$ is a $\diamondsuit$-iteration if for any

  1. $(i)$

sequence $\langle D_{i}\mid i<\omega_{1}\rangle$ of dense subsets of
$((\mathcal{P}(\omega_{1})/I_{\omega_{1}})^{+})^{N_{\omega_{1}}}$ and

  2. $(ii)$

$S\in\mathcal{P}(\omega_{1})^{N_{\omega_{1}}}-I_{\omega_{1}}$

the set

$\\{\xi\in S\mid\forall i<\xi\
g_{\xi}\cap\sigma_{\xi,\omega_{1}}^{-1}[D_{i}]\neq\emptyset\\}$

is stationary. Here, $g_{\xi}$ is the generic ultrafilter applied to $N_{\xi}$
for $\xi<\omega_{1}$.

If $(N,I)$ is generically iterable and $\diamondsuit$ holds then there is a
$\diamondsuit$-iteration of $(N,I)$. But this is not generally the case. Paul
Larson noted that if $(M,I)$ is generically iterable and

$\langle
M_{\alpha},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\omega_{1}\rangle$

is a generic generic iteration of $(M,I)=(M_{0},I_{0})$ of length $\omega_{1}$
then this is a $\diamondsuit$-iteration. By this we mean that this iteration
has been constructed generically by forcing with countable approximations
ordered by endextension.

######  Lemma 5.19.

Suppose

$\langle(N_{i},I_{i}),\sigma_{i,j},g_{i}\mid i\leq j\leq\omega_{1}\rangle$

is a $\diamondsuit$-iteration. If

$N_{\omega_{1}}\models``f\text{ witnesses
}\diamondsuit_{I_{\omega_{1}}}^{+}(\mathbb{B})"$

then $I_{\omega_{1}}=\mathrm{NS}_{f}\cap N_{\omega_{1}}$. In particular, $f$
witnesses $\diamondsuit(\mathbb{B})$.

###### Proof.

Let $S\in\mathcal{P}(\omega_{1})^{N_{\omega_{1}}}-I_{\omega_{1}}$, we have to
show that $S$ is $f$-stationary. Let $\langle D_{i}^{\prime}\mid
i<\omega_{1}\rangle$ be a sequence of dense subsets of $\mathbb{B}$. As $f$
witnesses $\diamondsuit_{I_{\omega_{1}}}^{+}(\mathbb{B})$ in $N_{\omega_{1}}$,
we have

$N_{\omega_{1}}\models``\eta_{f}\colon\mathbb{B}\rightarrow(\mathcal{P}(\omega_{1})/I_{\omega_{1}})^{+}\text{
is a complete embedding}"$

and notice that $\eta_{f}$ is a complete embedding in $V$ as well. Thus
$D_{i}=\eta_{f}[D_{i}^{\prime}]$ is dense for $i<\omega_{1}$. As
$\sigma_{0,\omega_{1}}\colon N_{0}\rightarrow N_{\omega_{1}}$ is a
$\diamondsuit$-iteration,

$T\coloneqq\\{\xi\in S\mid\forall i<\xi\
g_{\xi}\cap\sigma_{\xi,\omega_{1}}^{-1}[D_{i}]\neq\emptyset\\}$

is stationary. Thus if $C\subseteq\omega_{1}$ is club, we can find $\xi\in
C\cap T$ with $\omega_{1}^{N_{\xi}}=\xi$ and
$f\in\mathrm{ran}(\sigma_{\xi,\omega_{1}})$. It follows that

$f(\xi)=\eta_{\sigma_{\xi,\omega_{1}}^{-1}(f)}^{-1}[g_{\xi}]$

so that $f(\xi)\cap D_{i}^{\prime}\neq\emptyset$ for all $i<\xi$. ∎

###  5.4 $\diamondsuit\text{-}(\ast)$-forcing

######  Theorem 5.20.

Suppose that

  1. $(i)$

generic projective absoluteness holds for generic extensions by forcings of
size $2^{\omega_{1}}$,

  2. $(ii)$

$\mathbb{V}_{\mathrm{max}}$ is a $\mathbb{P}_{\mathrm{max}}\text{-variation}$,

  3. $(iii)$

$\mathrm{NS}_{\omega_{1}}$ is saturated and $\mathcal{P}(\omega_{1})^{\sharp}$
exists,

  4. $(iv)$

$(H_{\omega_{2}},\mathrm{NS}_{\omega_{1}},A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})$
is almost a $\mathbb{V}_{\mathrm{max}}$-condition and

  5. $(v)$

$D\subseteq\mathbb{V}_{\mathrm{max}}$ is $2^{\omega_{1}}$-universally Baire
and dense in $\mathbb{V}_{\mathrm{max}}$ in any generic extension by a forcing
of size $2^{\omega_{1}}$, as witnessed by trees $T,S$ with $p[T]=D$.

Then there is a forcing $\mathbb{P}^{\diamondsuit}$ so that in
$V^{\mathbb{P}^{\diamondsuit}}$ the following picture

$p[T]$$q_{0}$$q_{\omega_{1}}$$p_{0}$$p_{\omega_{1}^{N}}$$p_{\omega_{1}}$$((H_{\omega_{2}})^{V},\mathrm{NS}_{\omega_{1}}^{V},A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})=\mathcal{H}$$\mathbb{V}_{\mathrm{max}}$$\in$$\sigma_{0,\omega_{1}}$$\in$$\in$$\mu_{0,\omega_{1}^{N}}$$\mu_{\omega_{1}^{N},\omega_{1}}$$=$$\in$

exists so that

  1. $(\mathbb{P}^{\diamondsuit}.i)$

$\mu_{0,\omega_{1}},\sigma_{0,\omega_{1}}$ are generic iterations of $p_{0}$,
$q_{0}$ respectively,

  2. $(\mathbb{P}^{\diamondsuit}.ii)$

$\mu_{0,\omega_{1}^{N}}$ witnesses $q_{0}<_{\mathbb{V}_{\mathrm{max}}}p_{0}$,

  3. $(\mathbb{P}^{\diamondsuit}.iii)$

$\mu_{0,\omega_{1}}=\sigma_{0,\omega_{1}}(\mu_{0,\omega_{1}^{N}})$ and

  4. $(\mathbb{P}^{\diamondsuit}.iv)$

the generic iteration $\sigma_{0,\omega_{1}}\colon q_{0}\rightarrow
q_{\omega_{1}}$ is a $\diamondsuit$-iteration.

For the remainder of this section, $\omega_{1}$ will always denote
$\omega_{1}^{V}$.
So suppose $(i)$-$(v)$ holds. We will assume $n^{\mathbb{V}_{\mathrm{max}}}=0$
for notational purposes. For the most part, we will follow the construction of
$\mathbb{P}$ in [AS21] but will put additional constraints on the
certificates. The idea that guides us here is:

> In order for $\sigma_{0,\omega_{1}}\colon q\rightarrow q^{\ast}$ to be a
> $\diamondsuit$-iteration, the forcing $\mathbb{P}^{\diamondsuit}$ will have
> to anticipate dense subsets of the forcing $(I^{+})^{N_{\omega_{1}}}$ so
> that they have been “hit before”. This should be captured by the map
> $K\rightarrow C$. Formulating this correctly produces a strengthened version
> of the “genericity condition” put onto semantic certificates.

A reader who can compile the above paragraph without syntax error can probably
safely skip most the definition of $\mathbb{P}$ and go straight to 4.

We try to keep our notation here consistent with the notation in the paper
[AS21]. For this reason, we will identify a condition
$p=(M,I,a)\in\mathbb{V}_{\mathrm{max}}$ with its first coordinate $M$.
Additionally, by even more abuse of notation:

######  Convention 5.21.

If $(N,J,b)$ is (almost) a condition in $\mathbb{V}_{\mathrm{max}}$, then

  * •

$I^{N}$ denotes $J$,

  * •

$(I^{+})^{N}$ denotes $\mathcal{P}(\omega_{1})^{M}-J$ and

  * •

$a^{N}$ denotes $b$.

We will additionally assume both $2^{\omega_{1}}=\omega_{2}$ and
$\diamondsuit_{\omega_{3}}$ to hold. Otherwise, first force with
$\mathrm{Add}(\omega_{2},1)\ast\mathrm{Add}(((2^{\omega_{1}})^{+})^{V},1)$ and
note that $(i)$ and $(v)$ still hold for forcing with
$\mathrm{Col}(\omega,\omega_{2})$, which is all we need. Moreover, observe
that this preserves “$\mathrm{NS}_{\omega_{1}}$ is saturated”.
We will denote $\omega_{3}$ by $\kappa$ and pick a
$\diamondsuit_{\kappa}$-sequence
$\langle\bar{A}_{\lambda}\mid\lambda<\kappa\rangle$.

We may find $T_{0}\subseteq T$ of size $\omega_{2}$ so that

$V^{\mathrm{Col}(\omega,\omega_{2})}\models\exists q\in p[T_{0}]\
q<_{\mathbb{V}_{\mathrm{max}}}\mathcal{H}.$

Here we use that $\mathcal{H}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition as well as
$(\mathbb{V}_{\mathrm{max}}.v)$. Note that $p[T_{0}]\subseteq p[T]$ in any
outer model. Without loss of generality, we may assume that $T_{0}$ is a tree
on $\omega\times\omega_{2}$.
Fix a bijection

$c\colon\kappa\rightarrow H_{\kappa}.$

For $\lambda<\kappa$ let

$Q_{\lambda}\coloneqq c[\lambda]\text{ and }A_{\lambda}\coloneqq
c\left[\bar{A}_{\lambda}\right].$

There is then a club $C\subseteq\kappa$ with

  1. $(i)$

$T_{0},p\in Q_{\lambda}$ and $\omega_{2}+1\subseteq Q_{\lambda}$,

  2. $(ii)$

$Q_{\lambda}\cap\mathrm{Ord}=\lambda$ and

  3. $(iii)$

$(Q_{\lambda};\in)\prec(H_{\kappa};\in)$

for all $\lambda\in C$. We now have

$\displaystyle\text{For all }P,B\subseteq H_{\kappa}\text{ the set }$
$\displaystyle(\diamondsuit)\hskip 10.0pt$ $\displaystyle\hskip
10.0pt\\{\lambda\in C\mid(Q_{\lambda};\in,P\cap
Q_{\lambda},A_{\lambda})\prec(H_{\kappa};\in,P,B)\\}$ is stationary.

We will also define $Q_{\kappa}$ as $H_{\kappa}$. The forcing $\mathbb{P}$
will add some

$(N_{0},I_{0},a_{0})\in D^{\ast}$

together with a generic iteration

$\langle N_{i},\sigma_{i,j}\mid i\leq j\leq\omega_{1}\rangle$

by Henkin-style finite approximations. By abuse of notation, we let
$N_{i}=(N_{i};I_{i},a_{i})$. For readability we will also write

$N_{\omega_{1}}=(N_{\omega_{1}},I^{\ast},a^{\ast}).$

$\mathbb{P}^{\diamondsuit}$ will be the last element of an increasing sequence
$\langle\mathbb{P}^{\diamondsuit}_{\lambda}\mid\lambda\in
C\cup\\{\kappa\\}\rangle$ of forcings which we define inductively. We will
have:

  1. $(i)$

$\mathbb{P}^{\diamondsuit}_{\lambda}\subseteq Q_{\lambda}$,

  2. $(ii)$

conditions in $\mathbb{P}^{\diamondsuit}_{\lambda}$ will be finite sets of
formulae in a first order language $\mathcal{L}_{\lambda}$ and

  3. $(iii)$

the order on $\mathbb{P}^{\diamondsuit}_{\lambda}$ is reverse inclusion.

Suppose now that $\lambda\in C\cup\\{\kappa\\}$ and
$\mathbb{P}^{\diamondsuit}_{\nu}$ is defined for all $\nu\in C\cap\lambda$.

We will make use of the same convention as Asperó-Schindler.

######  Convention 5.22.

$x\subseteq\omega$ is a real code for $N_{0}=(N,I_{0},a_{0})$ if there is a
surjection $f\colon\omega\rightarrow N$ so that $x$ is the monotone
enumeration of Gödel numbers of all expressions of the form

$\ulcorner\dot{N}\models\varphi(\dot{n}_{1},\dots,\dot{n}_{l},\dot{I},\dot{a})\urcorner$

where $\varphi$ is a first order formula of the language associated to
$(N_{0},I_{0},a_{0})$(see below) and

$N\models\varphi(f(n_{1}),\dots,f(n_{l}),I_{0},a_{0})$

holds.

We will have conditions in $\mathbb{P}^{\diamondsuit}_{\lambda}$ be certified
in a concrete sense by objects $\mathfrak{C}$ which exist in generic
extensions of $V$ that satisfies projective absoluteness w.r.t. $V$. They are
of the form

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

where

  1. $(\mathfrak{C}.1)$

$M_{0}$, $N_{0}\in\mathbb{V}_{\mathrm{max}}$,

  2. $(\mathfrak{C}.2)$

$x=\langle k_{n}\mid n<\omega\rangle$ is a real code for
$N_{0}=(N_{0};\in,I,a_{0})$ and $\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle$ is a branch through $T_{0}$,

  3. $(\mathfrak{C}.3)$

$\langle M_{i},\mu_{i,j}\mid i\leq j\leq\omega_{1}^{N_{0}}\rangle\in N_{0}$ is
a generic iteration of $M_{0}$ witnessing
$N_{0}<_{\mathbb{V}_{\mathrm{max}}}M_{0}$,

  4. $(\mathfrak{C}.4)$

$\langle N_{i},\sigma_{i,j}\mid i\leq j\leq\omega_{1}\rangle$ is a generic
iteration of $N_{0}$,

  5. $(\mathfrak{C}.5)$

$\langle M_{i},\mu_{i,j}\mid i\leq
j\leq\omega_{1}\rangle=\sigma_{0,\omega_{1}}(\langle M_{i},\mu_{i,j}\mid i\leq
j\leq\omega_{1}^{N_{0}}\rangle)$ and

$M_{\omega_{1}}=((H_{\omega_{2}})^{V};\in,(\mathrm{NS}_{\omega_{1}})^{V},A),$

  6. $(\mathfrak{C}.6)$

$K\subseteq\omega_{1}$ and for all $\xi\in K$

    1. $(\mathfrak{C}.6.a)$

$\lambda_{\xi}\in\lambda\cap C$, and if $\gamma<\xi$ is in $K$ then
$\lambda_{\gamma}<\lambda_{\xi}$ and
$X_{\gamma}\cup\\{\lambda_{\gamma}\\}\subseteq X_{\xi}$,

    2. $(\mathfrak{C}.6.b)$

$X_{\xi}\prec(Q_{\lambda_{\xi}};\in,\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}},A_{\lambda_{\xi}})$
and $\delta^{X_{\xi}}=\xi$.

If $\mathfrak{C}$ has these properties, we call $\mathfrak{C}$ a potential
certificate.

Next up, we will define a certain first order language $\mathcal{L}$.
$\mathcal{L}$ will have the following distinguished constants

  * •

$x$ for any $x\in H_{\kappa}$,

  * •

$\dot{n}$ for any $n<\omega$,

  * •

$\dot{M}_{i}$ for $i<\omega_{1}$,

  * •

$\dot{\mu}_{i,j}$ for $i\leq j\leq\omega_{1}$,

  * •

$\dot{\vec{M}}$,

  * •

$\dot{N}_{i}$ for $i<\omega_{1}$,

  * •

$\dot{\sigma}_{i,j}$ for $i\leq j<\omega_{1}$,

  * •

$\dot{I}$, $\dot{a}$ and

  * •

$\dot{X}_{\xi}$ for $\xi<\omega_{1}$.

The constants $\dot{n}$ will eventually produce “Henkin-style” term models for
the $N_{i}$. Formulas in the language $\mathcal{L}$ are of the form

$\ulcorner\dot{N}_{i}\models\varphi(\underline{\gamma_{1}},\dots,\underline{\gamma_{k}},\dot{n}_{1},\dots,\dot{n}_{l},\dot{I},\dot{a},\dot{M}_{j_{1}},\dots,\dot{M}_{j_{m}},\dot{\mu}_{q_{1},r_{1}},\dots,\dot{\mu}_{q_{s},r_{s}},\dot{\vec{M}})\urcorner$

where

  * •

$i<\omega_{1}$,

  * •

$\gamma_{1},\dots\gamma_{k}<\omega_{1}$,

  * •

$n_{1},\dots,n_{l}<\omega$,

  * •

$j_{1},\dots,j_{m}<\omega_{1}$,

  * •

$q_{t}\leq r_{t}<\omega_{1}$ for $t\in\\{1,\dots,s\\}$

and $\varphi$ is a first order $\in$-formula. Moreover we allow as formulas

  * •

$\ulcorner\dot{\mu}_{i,\omega_{1}}(\dot{n})=\underline{x}\urcorner$ for
$i<\omega_{1},n<\omega$ and $x\in H_{\omega_{2}}$,

  * •

$\ulcorner\dot{\mu}_{\omega_{1},\omega_{1}}(\underline{x})=\underline{x}\urcorner$
for $x\in H_{\omega_{2}}$,

  * •

$\ulcorner\dot{\sigma}_{i,j}(\dot{n})=\dot{m}\urcorner$ for $i\leq
j<\omega_{1}$ and $n,m<\omega$,

  * •

$\ulcorner(\vec{\underline{k}},\vec{\underline{\alpha}})\in\underline{T}\urcorner$
for $\vec{k}\in\omega^{{<}\omega}$ and
$\vec{\alpha}\in\omega_{2}^{{<}\omega}$,

  * •

$\ulcorner\underline{\xi}\mapsto\underline{\nu}\urcorner$ for $\xi<\omega_{1}$
and $\nu<\kappa$ and

  * •

$\ulcorner\underline{x}\in\dot{X}_{\xi}\urcorner$ for $\xi<\omega_{1}$ and
$x\in H_{\kappa}$.

$\mathcal{L}^{\lambda}$ is the set of $\mathcal{L}$-formulae $\varphi$ so that
if $\underline{x}$ appears in $\varphi$ for some $x\in H_{\kappa}$ then $x\in
Q_{\lambda}$. We assume formulae in $\mathcal{L}^{\lambda}$ to be coded in a
reasonably way (ultimately uniform in $\lambda$) so that
$\mathcal{L}^{\lambda}=\mathcal{L}\cap Q_{\lambda}$. We will not make this
precise.

A potential certificate

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

is ($\lambda$-)precertified by $\Sigma\subseteq\mathcal{L}^{\lambda}$ if there
are surjections $e_{i}\colon\omega\rightarrow N_{i}$ for $i<\omega_{1}$ so
that

  1. ($\Sigma$.1)

$\ulcorner\dot{N}_{i}\models\varphi(\underline{\gamma_{1}},\dots,\underline{\gamma_{k}},\dot{n}_{1},\dots,\dot{n}_{l},\dot{I},\dot{a},\dot{M}_{j_{1}},\dots,\dot{M}_{j_{m}},\dot{\mu}_{q_{1},r_{1}},\dots,\dot{\mu}_{q_{s},r_{s}},\dot{\vec{M}})\urcorner\in\Sigma$
iff

    1. (a)

$i<\omega_{1}$,

    2. (b)

$\gamma_{1},\dots,\gamma_{k}\leq\omega_{1}^{N_{i}}$,

    3. (c)

$n_{1},\dots,n_{l}<\omega$,

    4. (d)

$j_{1},\dots,j_{m}\leq\omega_{1}^{N_{i}}$,

    5. (e)

$q_{t}\leq r_{t}\leq\omega_{1}^{N_{i}}$ for $t\in\\{1,\dots,s\\}$

and

$\displaystyle N_{i}\models\varphi($
$\displaystyle\gamma_{1},\dots,\gamma_{k},e_{i}(n_{1}),\dots,e_{i}(n_{l}),I^{N_{i}},a^{N_{i}},$
$\displaystyle
M_{j_{1}},\dots,M_{j_{m}},\mu_{q_{1},r_{1}},\dots,\mu_{q_{s},r_{s}},\vec{M})$

where $\vec{M}=\langle M_{j},\mu_{j,j^{\prime}}\mid j\leq
j^{\prime}\leq\omega_{1}^{N_{i}}\rangle$,

  2. ($\Sigma$.2)

$\ulcorner\dot{\mu}_{i,\omega_{1}}(\dot{n})=\underline{x}\urcorner\in\Sigma$
iff $i<\omega_{1}$, $n<\omega$ and $\mu_{i,\omega_{1}}(e_{i}(n))=x$,

  3. ($\Sigma$.3)

$\ulcorner\dot{\mu}_{\omega_{1},\omega_{1}}(\underline{x})=\underline{x}\urcorner\in\Sigma$
for all $x\in H_{\omega_{2}}$,

  4. ($\Sigma$.4)

$\ulcorner\dot{\sigma}_{i,j}(\dot{n})=\dot{m}\urcorner\in\Sigma$ iff $i\leq
j<\omega_{1}$ and $\sigma_{i,j}(e_{i}(n))=e_{j}(m)$,

  5. ($\Sigma$.5)

$\ulcorner(\vec{\underline{l}},\vec{\underline{\beta}})\in\underline{T}\urcorner\in\Sigma$
iff for some $n<\omega$, $\mathrm{lh}(\vec{l})=n=\mathrm{lh}(\vec{\beta})$ and
for all $m<n$ $l_{m}=k_{m}$, $\beta_{m}=\alpha_{m}$,

  6. ($\Sigma$.6)

$\ulcorner\underline{\xi}\mapsto\underline{\nu}\urcorner\in\Sigma$ iff $\xi\in
K$ and $\nu=\lambda_{\xi}$ and

  7. ($\Sigma$.7)

$\ulcorner\underline{x}\in\dot{X}_{\xi}\urcorner\in\Sigma$ iff $\xi\in K$ and
$x\in X_{\xi}$.

Note that $\mathfrak{C}$ can be “read off” from $\Sigma$ in a unique way via a
Henkin-style construction. For $i<\omega_{1}$ and $n,m<\omega$, let

$n\sim_{i}m\Leftrightarrow\ulcorner
N_{i}\models\dot{n}=\dot{m}\urcorner\in\Sigma$

and denote the equivalence class of $n$ modulo $\sim_{i}$ by
$[n]_{i}^{\Sigma}$. We will usually drop the superscript $\Sigma$ if it is
clear from context. Also let

$n\tilde{\in}_{i}m\Leftrightarrow\ulcorner
N_{i}\models\dot{n}\in\dot{m}\urcorner\in\Sigma.$

Then $(N_{i},\in)\cong(\omega,\tilde{\in}_{i})/\sim_{i}$. We call the latter
model the term model producing $N_{i}$. See Lemma 3.7 in [AS21] for more
details. For $x\in N_{i}$ we say $x$ is represented by $n$ if $x$ gets mapped
to $[n]_{i}$ by the unique isomorphism of $N_{i}$ to the term model. The term
model for $N_{\omega_{1}}$ is then the direct limit along the term models
producing the $N_{i}$, $i<\omega_{1}$ and elements can then be represented by
pairs $(i,n)$, $i<\omega_{1},n<\omega$ in the natural way.

To define certificates, we make use of the following concept:

######  Definition 5.23.

For $\bar{\lambda}\in C\cap\lambda$,

$Z\subseteq\mathbb{P}^{\diamondsuit}_{\bar{\lambda}}\times\omega_{1}\times\omega$

is a $\bar{\lambda}$-code for a dense subset of
$(I^{+})^{\dot{N}_{\omega_{1}}}$ given that

  1. $(i)$

if $(p,i,n)\in Z$ then

$\ulcorner\dot{N}_{i}\models``\dot{n}\in\dot{I}_{i}^{+}"\urcorner\in p,$

  2. $(ii)$

for any $(q,j,m)\in\mathbb{P}_{\bar{\lambda}}\times\omega_{1}\times\omega$
with

$\ulcorner\dot{N}_{j}\models``\dot{m}\in\dot{I}_{j}^{+}"\urcorner\in q$

there is $(p,i,n)\in Z$ with

    1. $(a)$

$p\leq q$, $j\leq i$ and

    2. $(b)$

$\ulcorner\dot{N}_{i}\models``\dot{n}\subseteq\dot{k}\mod\dot{I}_{i}"\urcorner,\ulcorner\dot{\sigma}_{j,i}(\dot{m})=\dot{k}\urcorner\in
p$ for some $k<\omega$,

  3. $(iii)$

and if $(p,i,n)\in Z$ as well as $q\leq p$ then $(q,i,n)\in Z$.

Suppose that

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

is $(\lambda-)$precertified by $\Sigma\subseteq\mathcal{L}^{\lambda}$ as
witnessed by $(e_{i})_{i<\omega_{1}}$. For $Z_{0}\subseteq Z$ we define the
evaluation of $Z_{0}$ by $\Sigma$ as

$Z_{0}^{\Sigma}\coloneqq\\{S\in N_{\omega_{1}}\mid\exists
p\in[\Sigma]^{{<}\omega}\exists i<\omega_{1}\exists n<\omega\ ((p,i,n)\in
Z_{0}\wedge S=\sigma_{i,\omega_{1}}(e_{i}(n)))\\}.$

A potential certificate $\mathfrak{C}$ is ($\lambda$-)certified by a
collection $\Sigma\subseteq\mathcal{L}^{\lambda}$ if $\mathfrak{C}$ is
$(\lambda$-)precertified by $\Sigma$ and additionally

  1. 4.

whenever $\xi\in K$ and $Z$ is a $\lambda_{\xi}$-code for a dense subset of
$(I^{+})^{\dot{N}_{\omega_{1}}}$ definable over

$(Q_{\lambda_{\xi}};\in,\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}},A_{\lambda_{\xi}})$

from parameters in $X_{\xi}$, then there is $S\in(Z\cap X_{\xi})^{\Sigma}$
with $\xi\in S$.

######  Definition 5.24.

In the case that 4 is satisfied, we call $\mathfrak{C}$ a semantic
certificate, and $\Sigma$ a syntactic certificate, relative to

$\mathbb{V}_{\mathrm{max}},A,H_{\omega_{2}},T_{0},\langle A_{\nu}\mid\nu\in
C\cap\lambda\rangle\text{ and
}\langle\mathbb{P}^{\diamondsuit}_{\nu}\mid\nu\in C\cap\lambda\rangle.$

######  Remark 5.25.

The genericity condition in [AS21] that is replaced here with 4 (adapted to
our context) is:

  1. $(\Sigma.8)^{\mathrm{AS}}$

If $\xi\in K$ and $E\subseteq\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}}$ is
dense and definable over

$(Q_{\lambda_{\xi}};\in,\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}},A_{\lambda_{\xi}})$

from parameters in $X_{\xi}$ then

$[\Sigma]^{<\omega}\cap E\cap X_{\xi}\neq\emptyset.$

Condition 4 is stronger than $(\Sigma.8)^{\mathrm{AS}}$: From any such $E$,

$Z=\\{(p,i,n)\in\mathbb{P}^{\diamondsuit}_{\bar{\lambda}}\times\omega_{1}\times\omega\mid\exists
q\in E\ p\leq
q\wedge\ulcorner\dot{N}_{i}\models``\dot{n}\in\dot{I}_{i}^{+}"\urcorner\in
p\\}$

is a $\lambda_{\xi}$-code for a dense subset of
$(I^{+})^{\dot{N}_{\omega_{1}}}$ definable over the same structure from the
same parameters. If $(Z\cap X_{\xi})^{\Sigma}\neq\emptyset$, it follows that

$[\Sigma]^{<\omega}\cap E\cap X_{\xi}\neq\emptyset.$

Suppose $\Sigma$ is a certificate that certifies

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle,$

$\xi\in K$ and $Z$ is a $\lambda_{\xi}$-code for a dense subset of
$(I^{+})^{\dot{N}_{\omega_{1}}}$ definable over

$(Q_{\lambda_{\xi}};\in,\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}},A_{\lambda_{\xi}}).$

$Z$ is supposed to represent a dense subset of $(I^{+})^{N_{\omega_{1}}}$
(w.r.t. inclusion $\mod I^{N_{\omega_{1}}}$) in
$V^{\mathbb{P}^{\diamondsuit}_{\lambda}}$. $\Sigma$ may not be “generic over
$V$”, so it may not be the case that $Z^{\Sigma}$ is dense in
$(I^{+})^{N_{\omega_{1}}}$. Nonetheless, already $(\Sigma.8)^{\mathrm{AS}}$
implies that

$D=\sigma_{\xi,\omega_{1}}^{-1}[(Z\cap
X_{\xi})^{\Sigma}]\subseteq(I^{+})^{N_{\xi}}$

is dense. $D$ may not be in $N_{\xi}$, so it is not guaranteed that $D$ is hit
by the ultrapower $\sigma_{\xi,\xi+1}\colon N_{\xi}\rightarrow N_{\xi+1}$ just
from genericity over $N_{\xi}$ alone, however 4 makes sure that this happens
(observe that $\omega_{1}^{N_{\xi}}=\xi$). So in essence, the idea of 4 is
that any dense subset of $(I^{+})^{N_{\omega_{1}}}$ that exists in the final
$V^{\mathbb{P}^{\diamondsuit}_{\kappa}}$ has been “hit” before at some point
along the iteration of $N_{0}$ to $N_{\omega_{1}}$.

######  Remark 5.26.

Note that for any syntactic certificate, there is a unique semantic
certificate it corresponds to. Given a semantic certificate, its corresponding
syntactic certificate is unique modulo the choice of the maps
$(e_{i})_{i<\omega}$.

A finite set $p$ of $\mathcal{L}^{\lambda}$-formulas is certified by $\Sigma$
iff $\Sigma$ is a syntactic certificate and $p\subseteq\Sigma$. If
$\mathfrak{C}$ is a semantic certificate then we also say $p$ is certified by
$\mathfrak{C}$ in case there is a syntactic certificate $\lambda$ certifying
both $\mathfrak{C}$ and $p$.

######  Definition 5.27.

Conditions $p\in\mathbb{P}^{\diamondsuit}_{\lambda}$ are finite sets of
$\mathcal{L}^{\lambda}$ formulae so that

$V^{\mathrm{Col}(\omega,\omega_{2})}\models``\exists\Sigma\subseteq\mathcal{L}^{\lambda}\
\Sigma\text{ certifies }p".$

This completes the construction of $\mathbb{P}^{\diamondsuit}_{\lambda}$.

######  Proposition 5.28.

Let $p\in[\mathcal{L}^{\lambda}]^{<\omega}$. If $p$ is certified in some outer
model, then $p$ is certified in $V^{\mathrm{Col}(\omega,\omega_{2})}$.

###### Proof.

Let $g$ be $\mathrm{Col}(\omega,\omega_{2})$-generic. If there is some outer
model in which $p$ is certified, then by Shoenfield absoluteness we can find
in $V[g]$ a set of $\mathcal{L}^{\lambda}$-formulas $\Sigma$ with
$p\in[\Sigma]^{<\omega}$ such that if

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

is the corresponding semantic interpretation then

  1. $(i)$

$\Sigma$ satisfies ($\Sigma$.1)-4,

  2. $(ii)$

$\mathfrak{C}$ satisfies $(\mathfrak{C}.2)$ as well as
$(\mathfrak{C}.4)$-$(\mathfrak{C}.6)$ and

  3. $(iii)$

$\mathfrak{C}$ satisfies $(\mathfrak{C}.3)$ in the sense that
$\mu_{0,\omega_{1}^{N_{0}}}\in N_{0}$ and
$N_{0}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(M_{\omega_{1}^{N_{0}}})$,

as this can be expressed by a $\mathbf{\Sigma}^{1}_{2}$-formula. It remains to
show that $(\mathfrak{C}.1)$ holds true as well, i.e.
$M_{0},N_{0}\in\mathbb{V}_{\mathrm{max}}$. For $N_{0}$ this follows as
$N_{0}\in p[T_{0}]$ and by assumption $(v)$,
$p[T_{0}]\subseteq\mathbb{V}_{\mathrm{max}}$ in $V[g]$. To see that
$M_{0}\in\mathbb{V}_{\mathrm{max}}$, note that
$\mathcal{H}\in\mathbb{V}_{\mathrm{max}}$ as $\mathcal{H}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition in $V$. By
$(\mathbb{V}_{\mathrm{max}}.iv)$, it is enough to see that $M_{0}$ is
generically iterable. This follows from (the proof of) Theorem 3.16 in
[Woo10], here we use $\mathcal{P}(\omega_{1})^{\sharp}$ exists in $V$. ∎

We let $\mathbb{P}^{\diamondsuit}=\mathbb{P}^{\diamondsuit}_{\kappa}$. As in
Asperó-Schindler, we conclude that there is a club $D\subseteq C$ so that for
all $\lambda\in D$

$\mathbb{P}^{\diamondsuit}_{\lambda}=\mathbb{P}^{\diamondsuit}\cap
Q_{\lambda}$

and hence we get

$\displaystyle\text{for all }B\subseteq H_{\kappa}\text{ the set }$
$\displaystyle(\diamondsuit(\mathbb{P}^{\diamondsuit}))\hskip 10.0pt$
$\displaystyle\hskip 10.0pt\\{\lambda\in
C\mid(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda})\prec(H_{\kappa};\in,\mathbb{P},B)\\}$
is stationary.

######  Lemma 5.29.

$\emptyset\in\mathbb{P}^{\diamondsuit}_{\min(C)}$.

The argument is essentially the same as the proof of Lemma 3.6 in [AS21]
modulo some details that arise from replacing $\mathbb{P}_{\mathrm{max}}$ by a
general $\mathbb{P}_{\mathrm{max}}\text{-variation}$.

###### Proof.

Let $g$ be generic for $\mathrm{Col}(\omega,\omega_{2})$. Note that
$\mathcal{H}\in\mathbb{V}_{\mathrm{max}}$ as $\mathcal{H}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition in $V$. By choice of $T_{0}$, we can
find $N_{0}=(N_{0},I_{0},a_{0})\in D^{\ast}$ with
$N_{0}<_{\mathbb{V}_{\mathrm{max}}}\mathcal{H}$. Let
$\langle(k_{n},\alpha_{n})\mid n<\omega\rangle$ witness $N_{0}\in p[T]$. Let
us denote $M_{0}=\mathcal{H}$ and let

$\mu_{0,\omega_{1}^{N_{0}}}\colon M_{0}\rightarrow M_{\omega_{1}^{N_{0}}}$

witness $N_{0}<_{\mathbb{V}_{\mathrm{max}}}M_{0}$. Now let

$\sigma_{0,\kappa}\colon N_{0}\rightarrow N_{\kappa}$

be a generic iteration of $N_{0}$ of length $\kappa+1=\omega_{1}^{V[g]}+1$ as
well as

$\mu_{0,\kappa}\coloneqq\sigma_{0,\kappa}(\mu_{0,\omega_{1}^{N_{0}}})\colon
M_{0}\rightarrow M_{\kappa}$

the stretch of $\mu_{0,\omega_{1}^{N_{0}}}$ by $\sigma_{0,\kappa}$. Note that
this is a generic iteration of $M_{0}$ of length $\kappa+1$.

######  Claim 5.30.

The generic iteration

$\langle M_{\alpha},\mu_{\alpha,\beta}\mid\alpha\leq\beta\leq\kappa\rangle$

can be extended to a generic iteration of
$M_{0}^{+}\coloneqq(V,\mathrm{NS}_{\omega_{1}}^{V})$ of length $\kappa+1$.
That is, there is a generic iteration

$\langle
M_{\alpha}^{+},\mu^{+}_{\alpha,\beta}\mid\alpha\leq\beta\leq\kappa\rangle$

of $M_{0}^{+}$ so that for all $\alpha\leq\beta\leq\kappa$

  1. $(+.i)$

$M_{\alpha}=\left(H_{\omega_{2}}\right)^{M_{\alpha}^{+}}$ and

  2. $(+.ii)$

$\mu_{\alpha,\beta}=\mu_{\alpha,\beta}^{+}\upharpoonright M_{\alpha}$.

###### Proof.

The iteration $\langle
M_{\alpha}^{+},\mu^{+}_{\alpha,\beta}\mid\alpha\leq\beta\leq\kappa\rangle$
arises by applying the same generic ultrafilter $g_{\alpha}$ which generates
$\mu_{\alpha,\alpha+1}\colon M_{\alpha}\rightarrow M_{\alpha+1}$ to
$M_{\alpha}^{+}$. By induction on $\alpha$, as
$M_{\alpha}=\left(H_{\omega_{2}}\right)^{M_{\alpha}^{+}}$, $g_{\alpha}$
measures all subsets of $\omega_{1}^{M_{\alpha}^{+}}$ in $M_{\alpha}^{+}$. It
is a generic ultrafilter as

$M_{\alpha}^{+}\models``\mathrm{NS}_{\omega_{1}}\text{ is saturated}"$

by elementarity of $\mu_{0,\alpha}^{+}$, and hence all maximal antichains in
$(\mathrm{NS}_{\omega_{1}}^{+})^{M_{\alpha}^{+}}$ are already in $M_{\alpha}$,
hence are met by $g_{\alpha}$. Now let

$\mu_{\alpha,\alpha+1}^{+}\colon M_{\alpha}^{+}\rightarrow
M_{\alpha+1}^{+}\coloneqq\mathrm{Ult}(M_{\alpha}^{+},g_{\alpha})$

be the ultrapower. Any $x\in(H_{\omega_{2}})^{M_{\alpha+1}^{+}}$ is
represented by some function
$f\colon\omega_{1}^{M_{\alpha}^{+}}\rightarrow\left(H_{\omega_{2}}\right)^{M_{\alpha}^{+}}$
which is an element of
$\left(H_{\omega_{2}}\right)^{M_{\alpha}^{+}}=M_{\alpha}$. It follows that
$\mu_{\alpha,\alpha+1}=\mu_{\alpha,\alpha+1}^{+}\upharpoonright M_{\alpha}$.
It is easy to see that the properties $(+.i)$,$(+.ii)$ are stable under taking
direct limits. ∎

The point is that

$\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\emptyset\rangle$

is a semantic certificate for $\emptyset$ in $M^{+}\coloneqq M_{\kappa}^{+}$
with respect to

$\mu^{+}(\mathbb{V}_{\mathrm{max}}),\mu^{+}(A),\left(H_{\omega_{2}}\right)^{M^{+}},\mu^{+}(T_{0}),\mu^{+}(\langle
A_{\nu}\mid\nu\in
C\cap\lambda\rangle),\mu^{+}(\langle\mathbb{P}^{\diamondsuit}_{\nu}\mid\nu\in
C\cap\lambda\rangle)$

for $\lambda=\min(C)$ and $\mu^{+}=\mu_{0,\kappa}^{+}$. By Proposition 5.28,

$M^{+}\models\emptyset\in\mu^{+}(\mathbb{P}^{\diamondsuit}_{\min(C)})$

so that $\emptyset\in\mathbb{P}^{\diamondsuit}_{\min(C)}$ in $V$ by
elementarity of $\mu^{+}$. ∎

######  Lemma 5.31.

Suppose $\lambda\in C\cup\\{\kappa\\}$ and
$g\subseteq\mathbb{P}^{\diamondsuit}_{\lambda}$ is a filter with

  1. $(i)$

$g\cap E\neq\emptyset$ whenever
$E\subseteq\mathbb{P}^{\diamondsuit}_{\lambda}$ is dense and definable over

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda}),$

  2. $(ii)$

$g$ is an element of a generic extension of $V$ by a forcing of size $\leq
2^{\omega_{2}}$.

Then $\bigcup g$ is a semantic certificate.

###### Proof.

Read off the canonical candidate

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

from $g$. The proof of Lemma 3.7 in [AS21] shows that $\bigcup g$
$\lambda$-precertifies $\mathfrak{C}$. Note that the argument from Proposition
5.28 gives that $M_{0},N_{0}\in\mathbb{V}_{\mathrm{max}}$ and
$(\mathfrak{C}.3)$ follows from ($\Sigma$.1) and
$(\mathbb{V}_{\mathrm{max}}.ii)$. It remains to check 4. So suppose $\xi\in K$
and $Z$ is a $\lambda_{\xi}$-code for a dense subset of
$(I^{+})^{\dot{N}_{\omega_{1}}}$ definable over

$\mathcal{Q}_{\lambda}\coloneqq(Q_{\lambda_{\xi}};\in,\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}},A_{\lambda_{\xi}})$

from a parameter $x\in X_{\xi}$. Then there is $p\in g$ with

$\ulcorner\underline{\xi}\mapsto\underline{\lambda_{\xi}}\urcorner,\ulcorner\underline{x}\in\dot{X}_{\xi}\urcorner\in
p.$

Let $\Sigma^{\prime}$ be a syntactic certificate certifying $p$ (in some
extension of $V$ by $\mathrm{Col}(\omega,\omega_{2})$) and

$\mathfrak{C}^{\prime}=\langle\langle
M_{i}^{\prime},\mu_{i,j}^{\prime},N_{i}^{\prime},\sigma_{i,j}^{\prime}\mid
i\leq j\leq\omega_{1}\rangle,\langle(k_{n}^{\prime},\alpha_{n}^{\prime})\mid
n<\omega\rangle,\langle\lambda_{\rho}^{\prime},X_{\rho}^{\prime}\mid\rho\in
K^{\prime}\rangle\rangle$

the corresponding semantic certificate. We have $\xi\in K$ and
$\lambda_{\xi}^{\prime}=\lambda_{\xi}$ as well as $x\in X_{\xi}^{\prime}$.
Thus $Z$ is definable over $\mathcal{Q}_{\lambda}$ from parameters in
$X_{\xi}^{\prime}$. As $\Sigma^{\prime}$ satisfies 4, there is $S\in(Z\cap
X^{\prime}_{\xi})^{\Sigma^{\prime}}$ with $\xi\in S$. We may now find
$(q,i,n)\in Z\cap X_{\xi}^{\prime}$ so that

$S=\sigma_{i,\omega_{1}}([n]_{i}^{\Sigma^{\prime}}).$

Note that $i<\xi$ as $\delta^{X_{\xi}^{\prime}}=\xi$. Let
$\sigma_{i,\xi+1}([n]_{i}^{\Sigma^{\prime}}]=[m]_{\xi+1}^{\Sigma^{\prime}}$.
It follows that

$\ulcorner\dot{N}_{\xi+1}\models``\underline{\xi}\in\dot{m}"\urcorner,\ulcorner\dot{\sigma}_{i,\xi+1}(\dot{n})=\dot{m}\urcorner\in\Sigma^{\prime}.$

This is a density argument that shows: There are $s\geq r\in g$, $j<\xi$,
$l<\omega$ so that

  1. $(i)$

$(s,j,l)\in Z$,

  2. $(ii)$

$\ulcorner\underline{s}\in\dot{X}_{\xi}\urcorner\in r$ and

  3. $(iii)$

$\ulcorner\dot{N}_{\xi+1}\models``\underline{\xi}\in\dot{k}"\urcorner,\ulcorner\dot{\sigma}_{j,\xi+1}(\dot{l})=\dot{k}\urcorner\in
r$ for some $k<\omega$.

It follows that for $S=\sigma_{j,\omega_{1}}([l]_{j}^{\bigcup g})$, we have
$S\in(Z\cap X_{\xi})^{\bigcup g}$ and $\xi\in S$. ∎

######  Lemma 5.32.

Suppose $g$ is generic for $\mathbb{P}^{\diamondsuit}$ and

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

is the resulting semantic certificate. Then in $V[g]$,

$\langle N_{i},\sigma_{i,j}\mid i\leq j\leq\omega_{1}\rangle$

is a $\diamondsuit$-iteration.

###### Proof.

Let $\dot{S},\dot{C}$ be $\mathbb{P}^{\diamondsuit}$-names with

$p\Vdash``\dot{C}\subseteq\omega_{1}\text{ is club and
}\dot{S}\in(\dot{I}^{+})^{\dot{N}_{\omega_{1}}}"$

for some $p\in\mathbb{P}^{\diamondsuit}$. Further suppose
$\langle\dot{D}_{\alpha}\mid\alpha<\omega_{1}\rangle$ is a sequence of
$\mathbb{P}^{\diamondsuit}$-names for dense subsets of
$(I^{+})^{\dot{N}_{\omega_{1}}}$. We may suppose that

$\displaystyle
p\Vdash\dot{S}=\dot{\sigma}_{i_{0},\omega_{1}}([\check{n}]_{\check{i}_{0}}^{\bigcup\dot{G}})$

for some $i_{0}<\omega_{1}$ and $n<\omega$ where
$\dot{\sigma}_{i_{0},\omega_{1}}$ is a name for $\sigma_{i_{0},\omega_{1}}$
which arises in the semantic certificate corresponding to the generic filter.
It is our duty to find $\xi<\omega_{1}$ and $q\leq p$ with

$q\Vdash\check{\xi}\in\dot{S}\cap\dot{C}\wedge\forall\alpha<\check{\xi}\
\dot{g}_{\xi}\cap\dot{\sigma}_{\xi,\omega_{1}}^{-1}[\dot{D}_{\alpha}]\neq\emptyset$
($\spadesuit$)

where $\dot{g}_{\xi}$ is a name for the generic ultrafilter applied to
$\dot{N}_{\xi}$ along the iteration to $\dot{N}_{\omega_{1}}$. We will replace
the $\dot{D}_{\alpha}$ with codes for them: For $\alpha<\omega_{1}$, let
$Z_{\alpha}$ be defined by $(q,j,m)\in Z_{\alpha}$ iff

  1. $(Z.i)$

$(q,j,m)\in\mathbb{P}^{\diamondsuit}\times\omega_{1}\times\omega$,

  2. $(Z.ii)$

$\ulcorner\dot{N}_{j}\models``\dot{m}\in\dot{I}_{j}"\urcorner\in q$ and

  3. $(Z.iii)$

$q\Vdash\dot{\sigma}_{j,\omega_{1}}\left([m]_{j}^{\bigcup\dot{G}}\right)\in\dot{D}_{\alpha}$.

Further, for $\alpha<\omega_{1}$, we let

$E_{\alpha}=\\{q\leq p\mid\exists\beta\ \alpha\leq\beta\wedge
q\Vdash\check{\beta}\in\dot{C}\\}$

and

$E=\\{(q,\alpha)\in\mathbb{P}^{\diamondsuit}\times\omega_{1}\mid
q\Vdash\check{\alpha}\in\dot{C}\\}.$

Finally we define

$\tau=\left(\bigoplus_{\alpha<\omega_{1}}Z_{\alpha}\right)\oplus\left(\bigoplus_{\alpha<\omega_{1}}E_{\alpha}\right)\oplus
E.$

We may now find $\lambda\in C$ so that
$p\in\mathbb{P}^{\diamondsuit}_{\lambda}$ and

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda})\prec(H_{\kappa};\in,\mathbb{P}^{\diamondsuit},\tau).$

Here, $\oplus$ denotes some canonical way of coding at most $\omega_{1}$-many
subsets of $H_{\kappa}$ into a subset of $H_{\kappa}$. Let $h$ be
$\mathrm{Col}(\omega,\omega_{2})$-generic over $V$.

######  Claim 5.33.

In $V[h]$, there are filters $g,G$ that satisfy the following properties
$(i)$-$(iii)$:

  1. $(i)$

$g$ meets every dense subset of $\mathbb{P}^{\diamondsuit}_{\lambda}$ that is
definable (with parameters) in

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda}).$

Let

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

denote the semantic certificate corresponding to $g$.

  1. $(ii)$

$G$ is $(I^{+})^{N_{\omega_{1}}}$-generic over $N_{\omega_{1}}$ with
$\dot{S}^{g}=[n]_{i_{0}}^{\bigcup g}\in G$.

  2. $(iii)$

$G$ meets $Z^{\bigcup g}$ whenever $Z$ is a $\lambda$-code for a dense subset
of $(\dot{I}^{+})^{\dot{N}_{\omega_{1}}}$ definable (with parameters) over

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda}).$

###### Proof.

Let $g^{\prime}\subseteq\mathbb{P}^{\diamondsuit}_{\lambda}$ be generic over
$V$ and let

$\mathfrak{C}^{\prime}=\langle\langle
M_{i}^{\prime},\mu_{i,j}^{\prime},N_{i}^{\prime},\sigma_{i,j}^{\prime}\mid
i\leq j\leq\omega_{1}\rangle,\langle(k_{n}^{\prime},\alpha_{n}^{\prime})\mid
n<\omega\rangle,\langle\lambda_{\rho}^{\prime},X_{\rho}^{\prime}\mid\rho\in
K^{\prime}\rangle\rangle$

be the semantic certificate corresponding to $\bigcup g^{\prime}$. Let further
$G^{\prime}$ be $(I^{+})^{N_{\omega_{1}}^{\prime}}$-generic over
$V[g^{\prime}]$ (so in particular over $N_{\omega_{1}}^{\prime}$) with
$\dot{S}^{g^{\prime}}=[n]_{i_{0}}^{\bigcup g^{\prime}}\in G^{\prime}$. It is
clear that $g^{\prime},G^{\prime}$ satisfy $(i)$-$(iii)$ above. The existence
of such filters is $\Sigma^{1}_{1}$ in a real code for
$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda})$ so that
there are $g,G\in V[h]$ with $(i)$-$(iii)$ by Shoenfield-absoluteness. ∎

We now work in $V[h]$. Let $G,g$ be the filters given by the claim above and
let

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

be the semantic certificate that comes from $g$. Let

$\sigma_{\omega_{1},\omega_{1}+1}\colon N_{\omega_{1}}\rightarrow
N_{\omega_{1}+1}=\mathrm{Ult}(N_{\omega_{1}},G)$

be the generic ultrapower. We can further extend the generic iteration

$\langle N_{i},\sigma_{i,j}\mid i\leq j\leq\omega_{1}+1\rangle$

to one of length $\kappa+1$, say

$\langle N_{i},\sigma_{i,j}\mid i\leq j\leq\kappa\rangle.$

Further, set

$\vec{M}=\langle M_{i},\mu_{i,j}\mid i\leq
j\leq\kappa\rangle\coloneqq\sigma_{\omega_{1},\kappa}(\langle
M_{i},\mu_{i,j}\mid i\leq j\leq\omega_{1}\rangle).$

As $\mathfrak{C}$ is certified, $M_{\omega_{1}}=\mathcal{H}$ and as in Claim
5.30, we can extend the tail of $\vec{M}$ that is an iteration of
$M_{\omega_{1}}$ to a generic iteration of
$M_{\omega_{1}}^{+}\coloneqq(V,\mathrm{NS}_{\omega_{1}}^{V},A)$, say

$\langle M_{i}^{+},\mu_{i,j}^{+}\mid\omega_{1}\leq i\leq j\leq\kappa\rangle$

and have all $M_{i}^{+}$, $i\in[\omega_{1},\kappa]$, wellfounded. Let us write

$\mu^{+}\coloneqq\mu_{\omega_{1},\kappa}^{+}\colon V\rightarrow
M_{\omega_{1}}^{+}=\mathrel{\mathop{\mathchar 58\relax}}M^{+}.$

Work in $M^{+}$. We will now use

$\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq j\leq\kappa\rangle$

as part of a certificate. Set

$q\coloneqq\mu^{+}(p)\cup\\{\ulcorner\underline{\omega_{1}}\mapsto\mu^{+}(\lambda)\urcorner,\ulcorner\dot{\sigma}_{i_{0},\omega_{1}+1}(\dot{n})=\dot{m}\urcorner,\ulcorner\dot{N}_{\omega_{1}+1}\models``\underline{\omega_{1}}\in\dot{m}"\urcorner\\}$

where $\dot{m}$ represents $\sigma_{\omega_{1},\omega_{1}+1}(S)$ in the term
model for $N_{\omega_{1}+1}$.

######  Claim 5.34.

$q\in\mu^{+}(\mathbb{P}^{\diamondsuit})$.

###### Proof.

Set

$\mathfrak{C}^{\ast}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid
i\leq j\leq\kappa\rangle,\langle(k_{n},\mu^{+}(\alpha_{n}))\mid
n<\omega\rangle,\langle\lambda_{\xi}^{\ast},X_{\xi}^{\ast}\mid\xi\in
K^{\ast}\rangle\rangle$

where

  * •

$K^{\ast}=K\cup\\{\omega_{1}\\}$,

  * •

for $\xi\in K$, $\lambda_{\xi}^{\ast}=\mu^{+}(\lambda_{\xi})$ and
$X_{\xi}^{\ast}=\mu^{+}[X_{\xi}]$ and

  * •

$\lambda_{\omega_{1}}=\mu^{+}(\lambda)$,
$X_{\omega_{1}}^{\ast}=\mu^{+}[Q_{\lambda}]$.

We show that $\mathfrak{C}^{\ast}$ is a semantic certificate for $q$ in
$M^{+}$. Note that we have to show that $\mathfrak{C}^{\ast}$ is a certificate
relative to

$\mu^{+}(\mathbb{V}_{\mathrm{max}}),\mu^{+}(A),\mu^{+}(H_{\omega_{2}})=(H_{\omega_{2}})^{M^{+}},\mu^{+}(T_{0}),\mu^{+}(\langle
A_{\nu}\mid\nu\in C\rangle),\mu^{+}(\langle\mathbb{P}_{\nu}\mid\nu\in
C\rangle).$

Observe that we can find a corresponding set of formulae $\Sigma^{+}$ that
corresponds to $\mathfrak{C}^{\ast}$ with $\mu^{+}[\bigcup
g]\subseteq\Sigma^{+}$ which we aim to prove to be a syntactic certificate.
We have $M_{\kappa}=\left(H_{\omega_{2}}\right)^{M^{+}}$. Notice also that

$\langle(k_{n},\mu^{+}(\alpha_{n}))\mid n<\omega\rangle\in[\mu^{+}(T_{0})]$

and that $(k_{n})_{n<\omega}$ is still a real code for $N_{0}$. Next, we prove
4. First assume $\xi\in K$. Then

$X_{\xi}^{\ast}=\mu^{+}[X_{\xi}]\prec(\mu^{+}(Q_{\lambda_{\xi}});\in,\mu^{+}(\mathbb{P}^{\diamondsuit}_{\lambda_{\xi}}),\mu^{+}(A_{\lambda_{\xi}}))$

and $\delta^{X_{\xi}^{\ast}}=\delta^{X_{\xi}}=\xi$ as
$\mathrm{crit}(\mu)=\omega_{1}>\xi$. As $\mu^{+}[X_{\xi}]=X_{\xi}^{\ast}$, 4
holds for $\xi$ in $\mathfrak{C}^{\ast}$, since it holds for $\xi$ in
$\mathfrak{C}$.
Finally, let us consider the case $\xi=\omega_{1}$. We have

$X_{\omega_{1}}^{\ast}=\mu^{+}[Q_{\lambda}]\prec(\mu^{+}(Q_{\lambda});\in,\mu^{+}(\mathbb{P}^{\diamondsuit}_{\lambda}),\mu^{+}(A_{\lambda}))$

and $\delta^{X_{\omega_{1}}^{\ast}}=\omega_{1}$ as $\mu^{+}$ has critical
point $\omega_{1}$. Clearly $X_{\omega_{1}}^{\ast}$ collapses to
$Q_{\lambda}$. So if $x\in X_{\omega_{1}}^{\ast}$ and

$\displaystyle M^{+}\models``$ $\displaystyle\hat{Z}\text{ is a
}\mu^{+}(\lambda)\text{-code for a dense subset of
}(\dot{I}^{+})^{N_{\kappa}}\text{ definable over}$ $\displaystyle\hskip
60.0pt(\mu^{+}(Q_{\lambda});\in,\mu^{+}(\mathbb{P}^{\diamondsuit}_{\lambda}),\mu^{+}(A_{\lambda}))$
$\displaystyle\text{with parameter }x"$

for some $x\in X_{\omega_{1}}^{\ast}$, then by elementarity, the same
definition defines a $\lambda$-code $Z$ for a dense subset of
$(\dot{I}^{+})^{\dot{N}_{\omega_{1}}}$ over

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda})$

with parameter $(\mu^{+})^{-1}(x)$ and we have $\mu^{+}(Z)=\hat{Z}$. Our
properties of $g,G$ imply that there is $R\in G\cap Z^{\bigcup g}$. It is not
difficult to see

$(\hat{Z}\cap
X_{\omega_{1}}^{\ast})^{\Sigma^{+}}=\sigma_{\omega_{1},\kappa}[Z^{\bigcup g}]$

and hence $\omega_{1}\in\sigma_{\omega_{1},\kappa}(R)\in(\hat{Z}\cap
X_{\omega_{1}}^{\ast})^{\Sigma^{+}}$. This shows 4 at $\omega_{1}$.
We conclude that indeed, $\mathfrak{C}^{\ast}$ is a semantic certificate for
$q$ which exists in some outer model of $M^{+}$. This gives
$q\in\mu^{+}(\mathbb{P}^{\diamondsuit})$ by Proposition 5.28. ∎

Thus we have

$\displaystyle M^{+}\models$ $\displaystyle``\exists\xi<\mu^{+}(\omega_{1})$
$\displaystyle\left(\mu^{+}(p)\cup\\{\ulcorner\underline{\xi}\mapsto\underline{\mu^{+}(\lambda)}\urcorner,\ulcorner\dot{\sigma}_{i_{0},\xi+1}(\dot{n})=\dot{m}\urcorner,\ulcorner\dot{N}_{\xi+1}\models``\underline{\xi}\in\dot{m}"\urcorner\\}\in\mu^{+}(\mathbb{P}^{\diamondsuit})\right)".$

By elementarity of $\mu^{+}$, we conclude

$V\models``\exists\xi<\omega_{1}\
\left(p\cup\\{\ulcorner\underline{\xi}\mapsto\underline{\lambda}\urcorner,\ulcorner\dot{\sigma}_{i_{0},\xi+1}(\dot{n})=\dot{m}\urcorner,\ulcorner\dot{N}_{\xi+1}\models``\underline{\xi}\in\dot{m}"\urcorner\\}\in\mathbb{P}^{\diamondsuit}\right)".$

Let $\xi$ witness this and set

$q=p\cup\\{\ulcorner\underline{\xi}\mapsto\underline{\lambda}\urcorner,\ulcorner\dot{\sigma}_{i_{0},\xi+1}(\dot{n})=\dot{m}\urcorner,\ulcorner\dot{N}_{\xi+1}\models``\underline{\xi}\in\dot{m}"\urcorner\\}.$

We will show that $q,\xi$ witness ($\spadesuit$ ‣ 5.4). From this point on, we
work in $V$ again and forget about $h,g,\mathfrak{C}$, etc.

######  Claim 5.35.

$q\Vdash\check{\xi}\in\dot{C}\cap\dot{S}$.

###### Proof.

As in Claim 3.17 in [AS21], exploit the components of $\tau$ made up from $E$
as well as $E_{\alpha}$, $\alpha<\omega_{1}$. ∎

######  Claim 5.36.

$q\Vdash\forall\alpha<\check{\xi}\
\dot{g}_{\xi}\cap\dot{\sigma}_{\xi,\omega_{1}}^{-1}[\dot{D}_{\alpha}]\neq\emptyset$.

###### Proof.

Let $g$ be $\mathbb{P}^{\diamondsuit}$-generic with $q\in g$ and let

$\mathfrak{C}=\langle\langle M_{i},\mu_{i,j},N_{i},\sigma_{i,j}\mid i\leq
j\leq\omega_{1}\rangle,\langle(k_{n},\alpha_{n})\mid
n<\omega\rangle,\langle\lambda_{\xi},X_{\xi}\mid\xi\in K\rangle\rangle$

be the resulting semantic certificate. We have $\xi\in K$ and
$\lambda_{\xi}=\lambda$ as $q\in g$. Fix some $\alpha<\xi$. Clearly,

$\bar{Z}_{\alpha}=Z_{\alpha}\cap Q_{\lambda}$

is a $\lambda$-code for a dense subset of
$(\dot{I}^{+})^{\dot{N}_{\omega_{1}}}$ which is definable over

$(Q_{\lambda};\in,\mathbb{P}^{\diamondsuit}_{\lambda},A_{\lambda})$

from a parameter in $X_{\xi}$, namely $\alpha$. Recall that
$\delta^{X_{\xi}}=\xi$. Using 4, we find that there is

$R\in(\bar{Z}_{\alpha}\cap X_{\xi})^{\bigcup g}$

with $\xi\in R$. Note that there are $r\in g$, $j<\xi=\delta^{X_{\xi}}$ as
well as $k<\omega$ with

  1. $(i)$

$(r,j,k)\in\bar{Z}_{\alpha}\subseteq Z_{\alpha}$ and

  2. $(ii)$

$R=\sigma_{j,\omega_{1}}([k]_{j}^{\bigcup g})$.

By definition of $Z_{\alpha}$, and as $r\in g$, $R\in D_{\alpha}$ and since
$\xi\in R$, $R\in g_{\xi}$, where $g_{\xi}$ is the generic ultrafilter
generating $\sigma_{\xi,\xi+1}\colon N_{\xi}\rightarrow N_{\xi+1}$. ∎

($\spadesuit$ ‣ 5.4) follows from Claim 5.35 together with Claim 5.36. ∎

This completes the proof of Theorem 5.20. We denote the forcing
$\mathbb{P}^{\diamondsuit}$ constructed above in the instance of a
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$, the
set $A\in H_{\omega_{2}}$ and appropriate dense
$D\subseteq\mathbb{V}_{\mathrm{max}}$ by
$\mathbb{P}^{\diamondsuit}(\mathbb{V}_{\mathrm{max}},A,D)$ (and forget that
$\mathbb{P}^{\diamondsuit}$ also depends on the choice of $T,T_{0},$ etc.).

###  5.5 The first blueprint

We will formulate a general theorem that will allow us to prove a variety of
instances of $\mathrm{MM}^{++}\Rightarrow(\ast)$. In order to formulate the
relevant forcing axioms, we use that in practice
$\varphi^{\mathbb{V}_{\mathrm{max}}}$ has a specific form.

######  Definition 5.37.

A $\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$ is
typical if $\varphi^{\mathbb{V}_{\mathrm{max}}}$ can be chosen to be the form

$\displaystyle\varphi^{\mathbb{V}_{\mathrm{max}}}(x)=``$ $\displaystyle\exists
M,I,a_{0},\dots,a_{n}\ x=(M,I,a_{0},\dots,a_{n})$ $\displaystyle\wedge\forall
y\in
M\bigwedge_{\psi\in\Psi}\left[\psi(y)\leftrightarrow(M;\in,I,a_{0},\dots,a_{n})\models\psi(y)\right]"$

for $n=n^{\mathbb{V}_{\mathrm{max}}}$ and a finite set $\Psi$ of formulae
$\psi(y)$ in the language $\\{\in,\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$.
Moreover, $\Psi$ contains the formulae $\psi(x)=``x\in\dot{I}"$ and
$\psi_{i}(x)=``x=\dot{a}_{i}"$ for all $i\leq n^{\mathbb{V}_{\mathrm{max}}}$.
We say that $\Psi$ witnesses the typicality of $\mathbb{V}_{\mathrm{max}}$.
This means that $q<_{\mathbb{V}_{\mathrm{max}}}p$ iff there is a generic
iteration $\mu\colon p\rightarrow p^{\ast}$ of $p$ in $q$ of length
$\omega_{1}^{q}+1$ so that the formulae in $\Psi$ are absolute between
$q,p^{\ast}$.

######  Remark 5.38.

For example, $\mathbb{P}_{\mathrm{max}}$ is (or can be construed as) a typical
$\mathbb{P}_{\mathrm{max}}\text{-variation}$. We have that typicality of
$\mathbb{P}_{\mathrm{max}}$ is witnessed by
$\\{\psi_{0}^{\mathbb{P}_{\mathrm{max}}},\psi_{1}^{\mathbb{P}_{\mathrm{max}}}\\}$
where

  * •

$\psi_{0}^{\mathbb{P}_{\mathrm{max}}}(y)=``y\in\dot{I}"$ and

  * •

$\psi_{1}^{\mathbb{P}_{\mathrm{max}}}(y)=``y=\dot{a}_{0}"$.

All $\mathbb{P}_{\mathrm{max}}\text{-variations}$ we will encounter, except
for $\mathbb{Q}_{\mathrm{max}}^{-}$, are typical
$\mathbb{P}_{\mathrm{max}}\text{-variations}$.

Next, we formulate the relevant bounded and unbounded forcing axioms as
general as possible.

######  Definition 5.39.

Suppose $\psi(x)$ is a formula in the language
$\\{\in,\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$ and
$\vec{A}=(A_{0},\dots,A_{n})\in H_{\omega_{2}}$.

  1. $(i)$

We define $R^{\psi}_{\vec{A}}$ via

$R^{\psi}_{\vec{A}}\coloneqq\\{x\in
H_{\omega_{2}}\mid(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A_{0},\dots,A_{n})\models\psi(x)\\}.$

  2. $(ii)$

For $x\in H_{\omega_{2}}$, we say that $C\subseteq\omega_{1}$ is a code for
$x$ if: Let $l\colon\omega_{1}\rightarrow\omega_{1}\times\omega_{1}$ denote
Gödels pairing function and $E=l[C]$. Then $(\omega_{1}\times\omega_{1},E)$ is
wellfounded and $(\mathrm{tc}(\\{x\\}),\in)$ is the transitive
isomorph161616$\mathrm{tc}$ denotes transitive closure..

  3. $(iii)$

$C\subseteq\omega_{1}$ is a code for an element of $R^{\psi}_{\vec{A}}$ if $C$
is a code for some $x\in R^{\psi}_{\vec{A}}$.

######  Definition 5.40.

Suppose that

  * •

$\Gamma$ is a class of forcings,

  * •

$\vec{A}=(A_{0},\dots,A_{n})\in H_{\omega_{2}}$ and

  * •

$\Psi$ is a set of formulae $\psi(x)$ in the language
$\\{\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$.

  1. $(i)$

$D\text{-}\mathrm{BFA}^{\Psi}_{\vec{A}}(\Gamma)$ states that
$D\subseteq\mathbb{R}$ is $\infty$-universally Baire and whenever
$\mathbb{P}\in\Gamma$ and $g$ is $\mathbb{P}$-generic then

$\left(H_{\omega_{2}};\in,D,R_{\vec{A}}^{\psi}\mid\psi\in\Psi\right)^{V}\prec_{\Sigma_{1}}\left(H_{\omega_{2}};\in,D^{\ast},R_{\vec{A}}^{\psi}\mid\psi\in\Psi\right)^{V[g]}.$

For $\Delta\subseteq\mathcal{P}(\mathbb{R})$,
$\Delta\text{-}\mathrm{BFA}^{\Psi}_{\vec{A}}(\Gamma)$ means
$D\text{-}\mathrm{BFA}^{\Psi}_{\vec{A}}(\Gamma)$ for all $D\in\Delta$.

  2. $(ii)$

$\mathrm{FA}^{\Psi}_{\vec{A}}(\Gamma)$ states that whenever
$\mathbb{P}\in\Gamma$ and

    1. $(\mathrm{FA}.i)$

$\mathcal{D}$ is a set of at most $\omega_{1}$-many dense subsets of
$\mathbb{P}$,

    2. $(\mathrm{FA}.ii)$

$\mathcal{N}_{\psi}$ is a set of at most $\omega_{1}$-many $\mathbb{P}$-names
for codes of elements of $(R^{\psi}_{\vec{A}})^{V^{\mathbb{P}}}$ for
$\psi\in\Psi$

then there is a filter $g\subseteq\mathbb{P}$ so that

    1. $(g.i)$

$g\cap D\neq\emptyset$ for all $D\in\mathcal{D}$ and

    2. $(g.ii)$

$\dot{S}^{g}=\\{\alpha<\omega_{1}\mid\exists p\in g\
p\Vdash\check{\alpha}\in\dot{S}\\}$ is a code for an element of
$R^{\psi}_{\vec{A}}$ for all $\dot{S}\in\mathcal{N}_{\psi}$, $\psi\in\Psi$.

We note that the methods of Bagaria in [Bag00] readily yield the following.

######  Lemma 5.41.

Suppose that

  1. $(i)$

$\Gamma$ is a class of forcings,

  2. $(ii)$

$\vec{A}=(A_{0},\dots,A_{n})\in H_{\omega_{2}}$ and

  3. $(iii)$

$\Psi$ is a set of formulae $\psi(x)$ in the language
$\\{\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$.

If $\mathrm{FA}^{\Psi}_{\vec{A}}(\Gamma)$ holds then so does
$\mathrm{uB}\text{-}\mathrm{BFA}^{\Psi}_{\vec{A}}(\Gamma)$.

######  Definition 5.42.

Let $\Psi$ be a set of formulae in the language
$\\{\dot{I},\dot{a}_{0},\dots,\dot{a}_{n}\\}$ for some $n$. For
$\vec{A}=(A_{0},\dots,A_{n})$, we say that a forcing $\mathbb{P}$ is
$(\Psi,\vec{A})$-preserving iff

$R^{\psi}_{\vec{A}}=\left(R^{\psi}_{\vec{A}}\right)^{V^{\mathbb{P}}}\cap V$

for all $\psi\in\Psi$. $\Gamma^{\Psi}_{\vec{A}}$ denotes the class of
$(\Psi,\vec{A})$-preserving forcings.

######  Definition 5.43.

A $\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$
accepts $\diamondsuit$-iterations if

$\displaystyle``$ $\displaystyle\text{If }p\in\mathbb{V}_{\mathrm{max}}\text{
and }p\rightarrow
p^{\ast}=(M,I,a_{0},\dots,a_{n^{\mathbb{V}_{\mathrm{max}}}})$
$\displaystyle\text{ is a }\diamondsuit\text{-iteration }\text{then
}\mathcal{H}_{(a_{0},\dots,a_{n^{\mathbb{V}_{\mathrm{max}}}})}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(p^{\ast})"$

is provable in $\mathrm{ZFC}^{-}+``\omega_{1}\text{ exists}"$ (that is, from
sufficiently much of $\mathrm{ZFC}$).

######  First Blueprint Theorem 5.44.

Suppose that

  1. $(i)$

$\mathbb{V}_{\mathrm{max}}$ is a typical
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ with typicality witnessed by
$\Psi$,

  2. $(ii)$

$\mathbb{V}_{\mathrm{max}}$ has unique iterations and accepts
$\diamondsuit$-iterations,

  3. $(iii)$

$\vec{A}\in H_{\omega_{2}}$ and $\mathcal{H}_{\vec{A}}$ is almost a
$\mathbb{V}_{\mathrm{max}}$-condition,

  4. $(iv)$

$\mathrm{SRP}$ holds and

  5. $(v)$

$\mathrm{FA}^{\Psi}_{\vec{A}}(\Gamma^{\Psi}_{\vec{A}})$ holds.

Then $\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ holds as witnessed by
$g_{\vec{A}}$.

###### Proof.

Let us assume $n^{\mathbb{V}_{\mathrm{max}}}=0$, so $\vec{A}=A$.
$\mathrm{SRP}$ entails “$\mathrm{NS}_{\omega_{1}}$ is saturated” as well as
$\forall\kappa\geq\omega_{2}\neg\Box_{\kappa}$. Results of Steel [Ste05] show
that the latter implies that $V$ is closed under $X\mapsto
M_{\omega}^{\sharp}(X)$. As a consequence

  * •

$\mathrm{AD}^{L(\mathbb{R})}$,

  * •

all sets of reals in $L(\mathbb{R})$ are $\infty$-universally Baire and

  * •

$(L(\mathbb{R})^{V};\in,D)\equiv(L(\mathbb{R})^{V[G]};\in,D^{\ast})$ for all
sets $D\subseteq\mathbb{R}$ in $L(\mathbb{R})$ and any generic extension
$V[G]$ of $V$.

Thus generic projective absoluteness holds in $V$ and if $D\in L(\mathbb{R})$
is a dense subset of $\mathbb{V}_{\mathrm{max}}$, then $D^{\ast}$ is a dense
subset of $\mathbb{V}_{\mathrm{max}}$ in any generic extension. Thus
$\mathbb{P}^{\diamondsuit}(\mathbb{V}_{\mathrm{max}},A,D)$ exists for any such
$D$.

######  Claim 5.45.

For any dense $D\subseteq\mathbb{V}_{\mathrm{max}}$, $D\in L(\mathbb{R})$,
$\mathbb{P}^{\diamondsuit}(\mathbb{V}_{\mathrm{max}},A,D)$ is
$(\Psi,A)$-preserving.

###### Proof.

Let $g$ be $\mathbb{P}^{\diamondsuit}(\mathbb{V}_{\mathrm{max}},A,D)$-generic.
By Theorem 5.20, in $V[g]$ we have

$D^{\ast}$$q_{0}$$q_{\omega_{1}}=(N^{*},I^{*},b^{\ast})$$p_{0}$$p_{\omega_{1}^{N}}$$p_{\omega_{1}}$$((H_{\omega_{2}})^{V},\mathrm{NS}_{\omega_{1}}^{V},A)=\mathcal{H}_{A}$$\mathbb{V}_{\mathrm{max}}$$\in$$\sigma_{0,\omega_{1}}$$\in$$\in$$\mu_{0,\omega_{1}^{N}}$$\mu_{\omega_{1}^{N},\omega_{1}}$$=$$\in$

where

  1. $(\mathbb{P}^{\diamondsuit}.i)$

$\mu_{0,\omega_{1}},\sigma_{0,\omega_{1}}$ are generic iterations of $p_{0}$,
$q_{0}$ respectively,

  2. $(\mathbb{P}^{\diamondsuit}.ii)$

$\mu_{0,\omega_{1}^{N}}$ witnesses $q_{0}<_{\mathbb{V}_{\mathrm{max}}}p_{0}$,

  3. $(\mathbb{P}^{\diamondsuit}.iii)$

$\mu_{0,\omega_{1}}=\sigma_{0,\omega_{1}}(\mu_{0,\omega_{1}^{N}})$ and

  4. $(\mathbb{P}^{\diamondsuit}.iv)$

the generic iteration $\sigma_{0,\omega_{1}}\colon q_{0}\rightarrow
q_{\omega_{1}}$ is a $\diamondsuit$-iteration.

Note that

$(N^{\ast};\in,I^{\ast},b^{\ast})\models\varphi^{\mathbb{V}_{\mathrm{max}}}(\mathcal{H}_{A}).$

As $\mathbb{V}_{\mathrm{max}}$ is typical, we must have $b^{\ast}=A$. As
$\mathbb{V}_{\mathrm{max}}$ accepts $\diamondsuit$-iterations,

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A)^{V[g]}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(q_{\omega_{1}})$

and finally it follows from typicality that

$(H_{\omega_{2}};\in,\mathrm{NS}_{\omega_{1}},A)^{V[g]}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(\mathcal{H}_{A}).$

As $\Psi$ witnesses the typicality of $\mathbb{V}_{\mathrm{max}}$, it follows
that $\mathbb{P}^{\diamondsuit}(\mathbb{V}_{\mathrm{max}},A,D)$ is
$(\Psi,A)$-preserving. ∎

It follows from Theorem 5.20, Lemma 5.41 and Lemma 5.17 that

  * •

$g_{\vec{A}}\cap D\neq\emptyset$ for all dense
$D\subseteq\mathbb{V}_{\mathrm{max}}$, $D\in L(\mathbb{R})$ and

  * •

$\mathcal{P}(\omega_{1})=\bigcup\\{\mathcal{P}(\omega_{1})\cap p^{\ast}\mid
p\in g_{\vec{A}}\wedge\mu\colon p\rightarrow p^{\ast}\text{ is guided by
}g_{\vec{A}}\\}$.

By Corollary 5.16, $g_{\vec{A}}$ witnesses
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$. ∎

######  Remark 5.46.

If additionally there are a proper class of Woodin cardinals, then
$g_{\vec{A}}$ meets all $\infty$-universally Baire dense subsets of
$\mathbb{V}_{\mathrm{max}}$.

###  5.6 The second blueprint

From the right perspective, $\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ is a
forcing axiom. As noted before, Asperó-Schindler show that if there is a
proper class of Woodin cardinals, then $(\ast)$ is equivalent to
$(\mathcal{P}(\mathbb{R})\cap L(\mathbb{R}))$-$\mathrm{BMM}^{++}$. Some
additional assumption like large cardinals is necessary as $\mathrm{BMM}$
implies closure of $V$ under sharps while $(\ast)$ holds in the
$\mathbb{P}_{\mathrm{max}}$-extension of $L(\mathbb{R})$. We try to generalize
this result roughly to all natural
$\mathbb{P}_{\mathrm{max}}\text{-variations}$ for which the
$\mathbb{P}^{\diamondsuit}$-method can prove them from some forcing axiom. We
will have to restrict to better behaved
$\mathbb{P}_{\mathrm{max}}\text{-variations}$.

######  Definition 5.47.

Let $\mathbb{V}_{\mathrm{max}}$ be a
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ with unique iterations and $g$ be
$\mathbb{V}_{\mathrm{max}}$-generic over $L(\mathbb{R})$.

  1. $(i)$

We say that $g$ produces $(A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})$ if
there is $p\in g$ so that if

$\mu\colon p\rightarrow
p^{\ast}=(M,I,a_{0},\dots,a_{n^{\mathbb{V}_{\mathrm{max}}}})$

is the $g$-iteration of $p$ then $a_{i}=A_{i}$ for all $i\leq
n^{\mathbb{V}_{\mathrm{max}}}$.

  2. $(ii)$

If $\mathbb{V}_{\mathrm{max}}$ is typical, we set

$\mathcal{H}_{g}\coloneqq(H_{\omega_{2}},\mathrm{NS}_{\omega_{1}},A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})^{L(\mathbb{R})[g]}$

where $(A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})$ is the unique sequence
produced by $g$.

######  Definition 5.48.

A $\mathbb{P}_{\mathrm{max}}\text{-variation}$ $\mathbb{V}_{\mathrm{max}}$
with unique iterations is self-assembling if: Whenever $g$ is
$\mathbb{V}_{\mathrm{max}}$-generic over $L(\mathbb{R})$ then

  1. $(i)$

$\mathcal{H}_{g}$ is almost a $\mathbb{V}_{\mathrm{max}}$-condition and

  2. $(ii)$

$(H_{\omega_{2}})^{L(\mathbb{R})[g]}=\bigcup\\{p^{\ast}\mid p\in g,\mu\colon
p\rightarrow p^{\ast}\text{ guided by }g\\}$.

All $\mathbb{P}_{\mathrm{max}}\text{-variation}$ we will work with are self-
assembling (assuming $\mathrm{AD}$ in $L(\mathbb{R})$). For example,
$\mathbb{P}_{\mathrm{max}}$ is self-assembling. The relevance of this property
for us is partly explained by the following result.

######  Lemma 5.49.

Suppose $\mathbb{V}_{\mathrm{max}}$ is a self-assembling
$\mathbb{P}_{\mathrm{max}}\text{-variation}$ with unique iterations and
typicality of $\mathbb{V}_{\mathrm{max}}$ is witnessed by a set $\Psi$ of
$(\Sigma_{1}\cup\Pi_{1})$-formulae. If
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ holds as witnessed by $g$ then

  1. $(i)$

$\mathcal{H}_{\vec{A}}$ is almost a $\mathbb{V}_{\mathrm{max}}$-condition and

  2. $(ii)$

$g=g_{\vec{A}}$

where $g$ produces $\vec{A}$.

###### Proof.

As $\mathbb{V}_{\mathrm{max}}$ is self-assembling, $\mathcal{H}_{g}$ is almost
a $\mathbb{V}_{\mathrm{max}}$-condition. Moreover,
$\mathcal{P}(\omega_{1})\subseteq L(\mathbb{R})[g]$ as $g$ witnesses
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$. It follows that
$\mathcal{H}_{g}=\mathcal{H}_{\vec{A}}$ and thus $(i)$ holds.
Let us now prove $(ii)$, note that it suffices to show $g\subseteq
g_{\vec{A}}$.

######  Claim 5.50.

If $q\in g$ and

$\sigma\colon q\rightarrow
q^{\ast}=(M^{\ast},I^{\ast},a_{0}^{\ast},\dots,a_{n^{\mathbb{V}_{\mathrm{max}}}}^{\ast})$

is the $g$-iteration of $q$ then $I^{\ast}=\mathrm{NS}_{\omega_{1}}\cap
M^{\ast}$ and $a_{i}^{\ast}=A_{i}$ for $i\leq n^{\mathbb{V}_{\mathrm{max}}}$.

###### Proof.

$a_{i}^{\ast}=A_{i}$ for $i\leq n^{\mathbb{V}_{\mathrm{max}}}$ follows easily
from typicality, we show $I^{\ast}=\mathrm{NS}_{\omega_{1}}\cap M^{\ast}$. It
is clear that $I^{\ast}\subseteq\mathrm{NS}_{\omega_{1}}$ since if $S\in
I^{\ast}$, then a tail of the iteration points of the iteration $\sigma\colon
q\rightarrow q^{\ast}$ is missing from $S$. On the other hand, suppose
$S\in\mathcal{P}(\omega_{1})^{M^{\ast}}-I^{\ast}$. We may assume
$S=\mu(\bar{S})$ for some $\bar{S}\in q$. If $C\subseteq\omega_{1}$ is club
then as $\mathbb{V}_{\mathrm{max}}$ is self-assembling, there is $r\in g$,
such that if $\nu\colon r\rightarrow r^{\ast}$ is the $g$-iteration of $r$,
then $C\in\mathrm{ran}(\nu)$, say $C=\nu(\bar{C})$. Note that we may assume
$r<_{\mathbb{V}_{\mathrm{max}}}q$, say this is witnessed by

$\bar{\sigma}\colon q\rightarrow\bar{q}=(\bar{M},\bar{I},\bar{a}).$

Write $r=(N,J,b)$. As $\mathbb{V}_{\mathrm{max}}$ is typical,
$\bar{I}=J\cap\bar{M}$ and hence
$\bar{\sigma}(\bar{S})\cap\bar{C}\neq\emptyset$ which gives

$\nu\circ\bar{\sigma}(\bar{S})\cap C\neq\emptyset.$

Clearly, $\nu(\bar{\sigma})$ is an iteration of $q$ of length $\omega_{1}+1$
guided by $g$. Thus, by Lemma 5.11, $\nu(\bar{\sigma})=\sigma$. $S\cap
C\neq\emptyset$ follows. ∎

Let $p\in g$ and let $\mu\colon p\rightarrow p^{\ast}$ be the $g$-iteration of
$p$.

######  Claim 5.51.

$\mathcal{H}_{\vec{A}}\models\varphi^{\mathbb{V}_{\mathrm{max}}}(p^{\ast})$.

###### Proof.

Let $\psi\in\Psi$ and assume $\psi$ is $\Sigma_{1}$, so write $\psi(x)=\exists
y\ \theta(x,y)$ where $\theta$ is $\Sigma_{0}$. So suppose for some $x\in p$
and $y\in H_{\omega_{2}}$ we have

$\mathcal{H}_{\vec{A}}\models\exists y\ \theta(x,y).$

As $\mathbb{V}_{\mathrm{max}}$ is self-assembling, we can find $q\in g$ with

  1. $(q.i)$

$q<_{\mathbb{V}_{\mathrm{max}}}p$ as witnessed by $\bar{\mu}\colon
p\rightarrow\bar{p}$ and

  2. $(q.ii)$

$\mathcal{H}_{\vec{A}}\models\theta(x,\sigma(y))$ for some $y\in q$

where $\sigma\colon q\rightarrow q^{\ast}$ is the $g$-iteration of $q$. By
Claim 5.50,

$q^{\ast}\prec_{\Sigma_{0}}\mathcal{H}_{\vec{A}}$

and as $\sigma(\bar{\mu})=\mu$ by Lemma 5.11 as well as elementarity of
$\sigma$ we find

$q\models\theta(\bar{\mu}(x),y).$

Finally, $q\models(\varphi^{\mathbb{V}_{\mathrm{max}}}(\bar{p}))$ so that

$\bar{p}\models\exists z\ \theta(\bar{\mu}(x),z)$

and hence $p\models\exists z\ \theta(x,z)$ by elemntarity of $\bar{\mu}$.
The “dual argument” works if $\psi$ is $\Pi_{1}$ instead. ∎

Now if $G$ is $\mathrm{Col}(\omega,2^{\omega_{1}})$-generic then the above
shows that $\mu\colon p\rightarrow p^{\ast}$ witnesses
$\mathcal{H}_{\vec{A}}<_{\mathbb{V}_{\mathrm{max}}}p$ in $V[G]$. Thus $p\in
g_{\vec{A}}$. ∎

Theorem 5.44 gives a hint how the forcing axiom equivalent to
$\mathbb{V}_{\mathrm{max}}\text{-}(\ast)$ should look like. However,
$\Gamma^{\Psi}_{\vec{A}}$ is not the right class of forcings, for example one
can construe two $\mathbb{P}_{\mathrm{max}}\text{-variations}$ which are the
same as forcings, but for which the resulting classes
$\Gamma^{\Psi}_{\vec{A}}$ are fundamentally different for reasonable
$\vec{A}$. Instead, we should look at the class of forcings which roughly lie
on the way to the good extensions highlighted in the
$\mathbb{V}_{\mathrm{max}}$-Multiverse View.

######  Definition 5.52.

Suppose that

  1. $(i)$

$\mathbb{V}_{\mathrm{max}}$ is a typical
$\mathbb{P}_{\mathrm{max}}\text{-variation}$,

  2. $(ii)$

typicality of $\mathbb{V}_{\mathrm{max}}$ is witnessed by $\Psi$ and

  3. $(iii)$

$\vec{A}=(A_{0},\dots,A_{n^{\mathbb{V}_{\mathrm{max}}}})\in H_{\omega_{2}}$.

The class $\Gamma^{\mathbb{V}_{\mathrm{max}}}_{\vec{A}}(\Psi)$ consists of all
$(\Psi,\vec{A})$-preserving forcings $\mathbb{P}$ so that if $g$ is
$\mathbb{P}$-generic, then there is a forcing $\mathbb{Q}\in V[g]$ with

$V[g]\models``\mathbb{Q}\text{ is }(\Psi,\vec{A})\text{-preserving}"$

and if further $h$ is $\mathbb{Q}$-generic over $V[g]$, then in $V[g][h]$ both

  1. $(h.i)$

$\mathcal{H}_{\vec{A}}$ is almost a $\mathbb{V}_{\mathrm{max}}$-condition and

  2. $(h.ii)$

$\mathrm{NS}_{\omega_{1}}$ is saturated.

It just so happens that, maybe by accident, for the