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# Marginalia to a Theorem of Asperó and Schindler

Obrad Kasum, Boban Veličković

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Obrad Kasum, Boban Veličković. Marginalia to a Theorem of Asperó and Schindler. 2023. hal-04460606

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Marginalia to a Theorem of Asperó and
Schindler

Obrad Kasum,\* Boban Veličković

August 17, 2023

Abstract

We describe a general method for adding a model of an infinitary propositional formula by stationary set preserving forcing. As a consequence we obtain a somewhat different proof of the Asperó-Schindler theorem stating that MM<sup>++</sup> implies Woodin's axiom (*).

Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Overview</td><td>2</td></tr><tr><td>1.2</td><td>Consistency</td><td>3</td></tr><tr><td>2</td><td>SSP Consistency</td><td>6</td></tr><tr><td>2.1</td><td>Putative Poset</td><td>6</td></tr><tr><td>2.2</td><td>AS goodness</td><td>8</td></tr><tr><td>3</td><td>Forcing P<sub>max</sub> conditions</td><td>14</td></tr></table>

<footnote>\*Mr. Kasum has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 945322</footnote>
# 1 Introduction

## 1.1 Overview

An infinitary propositional formula differs from a finitary one in that one can use conjunctions and disjunctions of arbitrary length. One can then go on to adjust the Natural Deduction rules to get a proof theory for these longer formulas. Proofs will now be infinite wellfounded trees. In this way, we achieve a straightforward generalization of the notion of coherence: a formula is coherent iff it does not prove the contradiction. However, the usual completeness theorem requires a slight adjustment: a formula is coherent if and only if it has a model in some generic extension of the universe. We will not need this syntactic notion of coherence, so we will just focus on the notion of “having a model in a generic extension”.

**Definition 1.1.** Suppose that $w$ is an infinitary propositional formula or a set thereof. Then $w$ is consistent iff there exists a poset $\mathbb{P}$ such that $V^\mathbb{P} \models$ “there exists a model for $w$”. $\square$

What we want to study here is a strengthening of this notion, which we call the ssp<fnref n="1" /> consistency. The idea is that even if a model of a formula can be added to the universe by some forcing, it might not be possible to do so by a stationary set preserving poset.

**Definition 1.2.** Suppose that $\phi$ is an infinitary propositional formula. Then $\phi$ is ssp consistent iff there exists a stationary set preserving poset $\mathbb{P}$ such that $V^\mathbb{P} \models$ “there exists a model for $\phi$”. $\square$

In this paper, we give a sufficient condition for a formula to be ssp consistent (cf. Proposition 2.16). This condition is inspired by [AS21] and we use it to reprove<fnref n="2" /> their main result. In the cited paper, the authors prove that the forcing axiom $MM^{++}$ of [FMS88] implies the axiom (*) of [Woo99]. We will briefly describe what they do and how that relates to our result. Let us assume $MM^{++}$. By [Ste05], we have that $L(\mathbb{R}) \models \mathrm{AD}$, so to verify (*), we need to find (in $V$) an $L(\mathbb{R})$-generic filter $g$ for $\mathbb{P}_{\max}$ such that $L(\mathcal{P}(\omega_1)) = L(\mathbb{R})[g]$. By standard facts from [Woo99], we know that if this is true, then for any $A \subseteq \omega_1$ satisfying $\omega_1^{L[A]} = \omega_1$, the set

$$g_A := \{p \in \mathbb{P}_{\max} : \exists \mathcal{I}, I^{M_{\omega_1}^{\mathcal{I}}} = \mathrm{NS}_{\omega_1} \cap M_{\omega_1}^{\mathcal{I}}, a^{M_{\omega_1}^{\mathcal{I}}} = A\}$$

is an $L(\mathbb{R})$-generic filter for $\mathbb{P}_{\max}$ and satisfies $L(\mathbb{R})[g_A] = L(\mathcal{P}(\omega_1))$. Here, $\mathcal{I}$ is intended to denote a generic iteration of $p$ of length $\omega_1 + 1$, $M_\alpha^\mathcal{I}$ denotes the $\alpha$th iterate in $\mathcal{I}$, $I^{M_\alpha^\mathcal{I}}$ its ideal, and $a^{M_\alpha^\mathcal{I}}$ its distinguished set of ordinals (for all $\alpha \le \omega_1$).

<footnote n="1">¹“stationary set preserving”</footnote>

<footnote n="2">²Strictly speaking, our construction uses an inaccessible cardinal, while theirs does not. We believe that this additional assumption is not essential.</footnote>
This means that we may immediately fix a set $A \subseteq \omega_1$ satisfying $\omega_1^{L[A]} = \omega_1$ and work towards showing that $g_A$ is an $L(\mathbb{R})$-generic filter. It is again more or less standard to verify that $g_A$ is a filter, which leaves the $L(\mathbb{R})$-genericity as a hard part. To see this, we pick an arbitrary $D \in L(\mathbb{R})$ which is a dense subset of $P_{\max}$ and take upon ourselves to find $p \in D$ so that there exists an $(\omega_1 + 1)$-iteration $\mathcal{I}$ thereof satisfying

$$I^{M_{\omega_1}^{\mathcal{I}}}=\mathbf{NS}_{\omega_1} \cap M_{\omega_1}^{\mathcal{I}} \text{ and } a^{M_{\omega_1}^{\mathcal{I}}}=A.$$

By generic absoluteness of the type of [Bag00], we know that it suffices to find such a $P_{\max}$ condition in a stationary set preserving generic extension<fnref n="3" /> of $V$. The point of this in our case is that we can approach the problem by first figuring out how to code what we need by an infinitary propositional formula and then figure out how to add a model for that formula by a stationary set preserving poset. We will give more details in Section 3.

Our method is most likely sufficiently general to cover not only the construction of [AS21], but also its variations that have already appeared in the literature (for example, [Lie23]). Ben De Bondt plans to include an application of the method in his forthcoming thesis [DBar]. In addition to this, the method has the advantage of using real models as side conditions for the poset, as opposed to “potential” models of [AS21]. By virtue of this, it is possible to describe a local club of models for which the poset is semiproper (cf. Propositions 2.14 and 2.15). This information is relevant if one tries to iterate posets of this kind (cf. the iteration construction of [KV23]).

## 1.2 Consistency

The consistency was defined as having a model in a generic extension and we observed that this is equivalent to not proving the contradiction. It is also possible to characterize the consistency in terms of a game. This characterization inspires the proof of our result on the ssp consistency.

**Notation 1.3.** Suppose that $w$ is as set of infinitary propositional formulas. Then $\mathcal{L}(w)$ denotes the set of propositional letters appearing in $w$ and $w \downarrow$ denotes the set of subformulas of all formulas in $w$. $\square$

The following definition is based on [Vää11, Section 8.3]. We remind the reader that “negation normal form” of a formula is the form in which negations appear next to propositional letters. This is easily achieved in general since negations can always be pushed “all the way in”.

**Definition 1.4.** Suppose that $w$ is a set of infinitary propositional formulas in negation normal form. Then $\mathcal{G}(w)$ is length $\omega$ game played as follows.

a. Set first $w_{-1} := w$.

<footnote n="3">³The set $D$ is coded by universally Baire set of reals, so it has a canonical interpretation in any generic extension.</footnote>
b. In the round $n < \omega$, Player I plays $Q_n$ and Player II answers by $w_n \supseteq w_{n-1}$.

c. If $Q_n = \phi \in w_{n-1}$ where $\phi = \bigvee_{i \in I} \psi_i$, then for some $i \in I$, $\psi_i \in w_n$.

d. If $Q_n = (\phi, i) \in w_{n-1} \times I$, where $\phi = \bigwedge_{i \in I} \psi_i$, then $\psi_i \in w_n$.

e. The first player to break the rules loses.

f. Infinite plays according to the rules where

$$ \bigcup_{n < \omega} w_n $$

does not contain both an atomic formula and its negation, are won by Player II, while all other infinite plays are won by Player I. □

Note that the above game is closed for Player II. In particular, which player wins the game is absolute between $V$ and its generic extensions.

**Proposition 1.5.** Suppose that $w$ is a set of infinitary propositional formulas in negation normal form. Then the following are equivalent.

a. $w$ is consistent.

b. $V^{\text{Col}(\omega, w \downarrow)} \models$ “there exists a model for $w$”.

c. Player II wins $\mathcal{G}(w)$.

*Proof.* This follows by the proof of [Vääl11, Theorem 8.12]. The point is that once $w \downarrow$ is collapsed to be countable, we are in the scenario of the quoted theorem. Since the game is closed, it is generically absolute whether or not Player II wins. □

There is a canonical poset that adds a model of a consistent formula. We will later transform this poset into a stationary set preserving poset which adds a model of a formula (satisfying certain requirements).

**Definition 1.6.** Suppose that $\phi$ is an infinitary propositional formula in negation normal form. The poset $\mathbb{H}(\phi)$ is defined as follows.

a. $w \in \mathbb{H}(\phi)$ iff $\phi \in w \subseteq \phi \downarrow$ and $w$ is finite and consistent.

b. $w \leq_{\mathbb{H}(\phi)} z$ iff $w \supseteq z$.

**Proposition 1.7.** Suppose that $\phi$ is an infinitary propositional formula in negation normal form. Then $\phi$ is consistent if and only if $\mathbb{H}(\phi) \neq \emptyset$. In that case, if $g$ is $V$-generic for $\mathbb{H}(\phi)$ and

$$ \mu_g : \mathcal{L}(\{\phi\}) \to 2 : p \mapsto \begin{cases} 1, & p \in \bigcup g \\ 0, & p \notin \bigcup g \end{cases}, $$

then $\mu_g$ is a model for $\phi$.
*Proof*. The first part is obvious. The second part follows by induction on the complexity of $\phi$. We show that for a subformula $\psi$ of $\phi$, we have that $\mu_g(\psi) = 1$ if and only if $\psi \in \bigcup g$. $\square$
## 2 SSP Consistency

### 2.1 Putative Poset

Let us immediately fix the scenario in which we work.

**Declaration 2.1.** Suppose that $\phi$ is a consistent infinitary propositional formula in negation normal form and that $\kappa > \text{rank}(\phi)$ is inaccessible. $\square$

We want to give a sufficient condition for $\phi$ to be ssp consistent. Furthermore, we want to find an ssp poset which adds a model for $\phi$. A poset that might work was proposed by Ben De Bondt and Boban Veličković. For the rest of this subsection, we essentially reproduce their definition.

**Notation 2.2.**

a. $\mathbb{H} := \mathbb{H}(\phi)$

b. $\mathcal{E} := \{\lambda < \kappa : \phi \in V_\lambda \prec V_\kappa\}$

c. For all $\lambda \in \mathcal{E}$, $C_\lambda$ denotes the set of all countable $M$ satisfying$^4$

i. $M \prec \widehat{M} \models \text{ZFC}^{-}$,

ii. $\widehat{M} = \text{Hull}(M, V_\lambda)$,

iii. $\phi, V_\lambda \in M$.

d. For all $S \subseteq \text{Ord}$, we denote $\mathcal{C}_S := \bigcup_{\lambda \in S \cap \mathcal{E}} \mathcal{C}_\lambda$.

e. For all $M \in \mathcal{C}_{[0,\kappa]}$, we denote by $\lambda_M$ the unique $\lambda$ such that $M \in \mathcal{C}_\lambda$.

f. For all $M \prec \widehat{M} \models \text{ZFC}^{-}$ and for all $\lambda \in \text{Ord}$, we denote by $M \downarrow_\lambda$ the image of $M$ under the transitive collapse of $\text{Hull}(M, V_\lambda)$. $\square$

**Remark 2.3.** If $M \in \mathcal{C}_\lambda$ and if $\xi \in \lambda \cap \mathcal{E}$, then we do not know a priori that $M \downarrow_\xi \in \mathcal{C}_\xi$. $\square$

The poset will be defined in two steps. The first step is to isolate the set $\mathbb{P}_\lambda^*$ of pre-conditions (for all $\lambda \in \mathcal{E} \cup \{\kappa\}$).

**Definition 2.4.** Suppose that $p, q \in \mathbb{H} \times \mathcal{C}_{[0,\kappa]}$ and $\lambda \in \mathcal{E} \cup \{\kappa\}$. Then we define the following.

a. $(w_p, \mathcal{M}_p) := p$

b. $p \upharpoonright \lambda := (w_p, \{M \in \mathcal{M}_p : \lambda_M < \lambda\})$

c. $p \in \mathbb{P}_\lambda^*$ iff for all $M, N \in \mathcal{M}_p$,

i. $\lambda_M < \lambda$,

<footnote>$^4$We use $\widehat{M}$ to denote the transitive collapse of $M$. Models $M$ satisfying $M \prec \widehat{M} \models \text{ZFC}^{-}$ were called *virtual models* by Veličković when he used them to iterate forcing (see e.g. [KV23])</footnote>
ii. $\delta_M = \delta_N \implies M = N,$

iii. $\delta_M < \delta_N \implies M \in N \wedge \lambda_M < \lambda_N.$

d. $p \le q$ iff $w_p \supseteq w_q$ and for all $N \in \mathcal{M}_q$, there exists $M \in \mathcal{M}_p$ such that $\delta_M = \delta_N$, $\lambda_M = \lambda_N$, and $M \supseteq N$. $\square$

Now that we have the sets of pre-conditions, we recursively define the family $(\mathbb{P}_\lambda : \lambda \in \mathcal{E} \cup \{\kappa\})$ of posets. The final poset $\mathbb{P}_\kappa$ is the one that we are interested in.

**Recursive Hypothesis 2.5.** Suppose that

1. $\lambda \in \mathcal{E} \cup \{\kappa\},$

2. $(\mathbb{P}_\xi : \xi \in \lambda \cap \mathcal{E})$ is defined,

3. for all $\xi \in \lambda \cap \mathcal{E}$, $\mathbb{P}_\xi$ is a suborder of $(\mathbb{P}_\xi^*, \le)$ and for all $p \in \mathbb{P}_\xi$, $\mathcal{M}_p \subseteq \mathcal{C}_{<\xi}$. $\square$

The following game is a modification of the game in Definition 1.4. It is used to pick the conditions among pre-conditions.

**Definition 2.6.** Suppose that $p \in \mathbb{P}_\lambda^*$. Then $\mathcal{G}_\lambda(p)$ is the game of length $\omega$ played as follows.

a. Set first $p_{-1} := p$.

b. In the round $n < \omega$, Player I plays $Q_n$ and Player II answers by $p_n \in \mathbb{P}_\lambda^*$ satisfying $p_n \le p_{n-1}$.

c. If $Q_n = \phi \in w_{p_{n-1}}$ where $\phi = \bigvee_{i \in I} \psi_i$, then for some $i \in I$, $\psi_i \in w_{p_n}$.

d. If $Q_n = (\phi, i) \in w_{p_{n-1}} \times I$, where $\phi = \bigwedge_{i \in I} \psi_i$, then $\psi_i \in w_{p_n}$.

e. If $Q_n = (M, D)$ where $M \in \mathcal{M}_{p_{n-1}}$ and $D \in M$ is dense in $\mathbb{P}_{\lambda_M}$, then there exists $q \in D$ such that $\delta(\mathrm{Hull}(M, q)) = \delta(M)$ and $p_n \le p_{n-1}, q$.

f. The first player to break the rules loses.

g. Infinite plays according to the rules where

$$ \bigcup_{n<\omega} w_{p_n} $$

does not contain both an atomic formula and its negation, are won by Player II, while all other infinite plays are won by Player I. $\square$

**Definition 2.7.** $\mathbb{P}_\lambda$ is the suborder of $(\mathbb{P}_\lambda^*, \le)$ consisting of those $p \in \mathbb{P}_\lambda^*$ for which Player II wins $\mathcal{G}_\lambda(p)$. $\square$

**Proposition 2.8.** Recursive Hypothesis 2.5. is verified at $\lambda$. $\square$
## 2.2 AS goodness

The following criterion is implicit in [AS21] (hence the name AS). In stating it, recall that we have already fixed a formula $\phi$ and an inaccessible $\kappa > \text{rank}(\phi)$.

**Definition 2.9.** $\phi$ is AS good at $\kappa$ iff for all stationary $S \subseteq \omega_1$, it holds in $V^{\text{Col}(\omega,<\kappa)}$ that for all $\mu \models \phi$, there exist elementary $\tau: V \to W$ and $\hat{\mu} \models \tau(\phi)$ such that $\text{crit}(\tau) = \omega_1^V \in \tau(S)$ and for all $\psi \in \phi \downarrow, \hat{\mu}(\tau(\psi)) = \mu(\psi)$. $\square$

We want to show that if $\phi$ is AS good, then it is ssp consistent. In fact, we will show that if $\phi$ is AS good, then the poset $\mathbb{P}_\kappa$ is a stationary set preserving poset that adds a model for $\phi$. We first list some basic properties of the posets introduced in the previous subsection. They are verified in a routine manner.

**Proposition 2.10.** *Suppose that $\lambda \in \mathcal{E} \cup \{\kappa\}$ and $\xi \in \lambda \cap \mathcal{E}$. Then $\mathbb{P}_\lambda \cap \mathbb{P}_\xi^* = \mathbb{P}_\xi$. $\square$*

**Proposition 2.11.** *Suppose that*

1. $\lambda \in \mathcal{E} \cup \{\kappa\}$,

2. $p \in \mathbb{P}_\lambda$,

3. $q \in \mathbb{P}_\lambda^*$,

4. $q \ge p$.

Then $q \in \mathbb{P}_\lambda$.

<figure><img src="image_2.png" /></figure>

**Proposition 2.12.** *Suppose that*

1. $\lambda \in \mathcal{E} \cup \{\kappa\}$,

2. $g$ is $V$-generic for $\mathbb{P}_\lambda$,

3. $\mu_g: \mathcal{L}(\phi) \to 2$,

4. for all $l \in \mathcal{L}(\phi)$, $\mu_g(l) = 1$ iff there exists $p \in g$ such that $l \in w_p$.

Then $\mu_g \models \phi$.

*Proof.* By induction on complexity of subformulas $\psi$ of $\phi$, we show that $\mu_g(\psi) = 1$ if and only if there exists $p \in g$ such that $\psi \in w_p$. $\square$

We will now identify elementary submodels of $H_\theta$ for which the poset $\mathbb{P}_\kappa$ is semiproper. This is important because saying that $\mathbb{P}_\kappa$ is semiproper is equivalent to saying that there is a local club of countable elementary submodels of $H_\theta$ for which this poset is semiproper.

**Definition 2.13** (De Bondt, Veličković). Suppose that

1. $\theta \gg \kappa$ is regular,

2. $M \prec (H_\theta, \in, \kappa, \phi)$ is countable.
Then $M$ is good iff for all $p \in \mathbb{P}_\kappa \cap M$, there exist $q \in \mathbb{P}_\kappa$ and $\lambda \in \mathcal{E}$ such that

a. $q \le p$,

b. $\kappa \cap M \subseteq \lambda$,

c. $M \downarrow \lambda \in \mathcal{M}_q$.

☐

**Proposition 2.14** (De Bondt, Veličković). Suppose that

1. $\theta \gg \kappa$ is regular,

2. $M \prec H_\theta$ is good.

Then $\mathbb{P}_\kappa$ is semiproper for $M$.

*Proof.*

1° Let $p \in \mathbb{P}_\kappa \cap M$. We want to find $q \in \mathbb{P}_\kappa$ such that $q \le p$ and that $q$ is semigeneric for $(M, \mathbb{P}_\kappa)$.

2° There exist $\lambda \in \mathcal{E}$ and $q \in \mathbb{P}_\kappa$ such that

a. $q \le p$,

b. $\kappa \cap M \subseteq \lambda$,

c. $M \downarrow \lambda \in \mathcal{M}_q$.

We want to show that $q$ is semigeneric for $(M, \mathbb{P}_\kappa)$.

3° Let $r \in \mathbb{P}_\kappa$ and $D \subseteq \mathbb{P}_\kappa$ be such that

1. $r \le q$,

2. $D \in M$,

3. $D$ is dense in $\mathbb{P}_\kappa$,

4. $r \in D$.

We want to find $s \in D$ such that

a. $s \parallel r$ is $\mathbb{P}_\kappa$,

b. $\delta(\mathrm{Hull}(M, s)) = \delta(M)$.

4° Let $\rho : \mathrm{Hull}(M, V_\lambda) \to \widehat{M \downarrow \lambda}$ be the transitive collapse. We have that

a. $\rho(D) = D \cap V_\lambda = D \cap \mathbb{P}_\lambda$,

b. $\rho(D)$ is dense in $\mathbb{P}_\lambda$,

c. $\rho(D) \in M \downarrow \lambda$.

5° Let Player I play $Q_0 := (M \downarrow \lambda, \rho(D))$ in $\mathcal{G}_\kappa(r)$ and let Player II answer according to his strategy by $p_0$. We have that for some $s \in \rho(D)$,
a. $p_0 \in \mathbb{P}_\kappa$,

b. $p_0 \le r$,

c. $p_0 \le s$,

d. $\delta(\mathrm{Hull}(M\downarrow\lambda, s)) = \delta(M\downarrow\lambda)$.

6° This means that

a. $s \in D$,

b. $s \parallel p$ in $\mathbb{P}_\kappa$ (as witnessed by $p_0$),

c. $\delta(\mathrm{Hull}(M, s)) = \delta(\mathrm{Hull}(M\downarrow\lambda, s)) = \delta(M\downarrow\lambda) = \delta(M)$.

Thus, $s$ is as required in 3°.

□

To verify that $\mathbb{P}_\kappa$ is semiproper, it suffices to check that there exists a local club of good models (cf. [FJ89, Theorem 2.1]). This is where AS goodness comes in.

**Proposition 2.15.** Suppose that $\phi$ is AS good at $\kappa$ and that $\theta \gg \kappa$ is regular. Then the set of all good models $M \prec H_\theta$ is a local club.

*Proof.*

1° Let $R$ be a wellordering of $H_\theta$ and let $\mathcal{H} := (H_\theta, \in, R, \kappa, \phi)$. We want to show that for all $X \prec \mathcal{H}$ satisfying $\omega_1 \subseteq X$ and $|X| = \omega_1$, we have that club many countable $M \prec X$ are good.

2° Let $X \prec \mathcal{H}$ satisfying $\omega_1 \subseteq X$ and $|X| = \omega_1$ be arbitrary. We denote by $\mathcal{X}$ the structure on $X$ inherited from $\mathcal{H}$.

3° Let us assume towards contradiction that there exists a stationary $S \subseteq [X]^\omega$ such that for all $M \in S$, it holds that $M \prec \mathcal{X}$ and that there exists $p_M \in \mathbb{P}_\kappa \cap M$ witnessing that $M$ is not good.

4° By pressing down applied to $S \ni M \mapsto p_M$, there exist $p \in \mathbb{P}_\kappa \cap X$ and stationary $S' \subseteq S$ such that for all $M \in S'$, $p$ witnesses that $M$ is not good.

5° Let $e : \omega_1 \to X$ be a bijection and let

$$T := \{\alpha < \omega_1 : e[\alpha] \in S', \delta(e[\alpha]) = \alpha\}.$$

Then $T$ is stationary in $\omega_1$.

6° Let

1. $\lambda \in \mathcal{E}$ be such that for all $\eta < \lambda$, $\kappa \cap \mathrm{Hull}^{\mathcal{H}}(V_\eta) \subseteq \lambda$,

2. $h$ be $V$-generic for $\mathrm{Col}(\omega, < \kappa)$,
3. $g \in V[h]$ be $V$-generic for $\mathbb{P}_\lambda$ containing $p$,

4. $\mu := \mu_g \models \phi$.

We work in $V[h]$.

7° Since $\phi$ is AS good, there exist elementary $\tau: V \to W$ and $\hat{\mu} \models \tau(\phi)$ such that $\mathrm{crit}(\tau) = \omega_1^V \in \tau(T)$ and for all $\psi \in \phi\downarrow, \hat{\mu}(\tau(\psi)) = \mu(\psi)$.

8° Let $N := \mathrm{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup \{\tau(p)\}) \prec \tau(\mathcal{H})$. It holds that

a. $N \in W$,

b. $|N|^W = \omega$,

c. $N \prec \tau[X] \prec \tau(\mathcal{X})$,

d. $\delta_N = \omega_1^V \in \tau(T)$.

9° **Claim.** In $W$, there exists $q \in \tau(\mathbb{P}_\kappa)$ such that $q \le \tau(p)$ and $N\downarrow\tau(\lambda) \in \mathcal{M}_q$.

*Proof.*

1' We will verify that one can take

$$q := (\tau(w_p), \tau(\mathcal{M}_p) \cup \{N\downarrow\tau(\lambda)\}).$$

To that end, it suffices to show that $q \in \tau(\mathbb{P}_\lambda)$.

2' Assume towards contradiction that $q \not\in \tau(\mathbb{P}_\lambda)$. Then there exists a winning strategy $\sigma$ for Player I in $\mathcal{G}_{\tau(\lambda)}^W(q)$. We will defeat this strategy in $V[g]$, reaching a contradiction. (We use in this step that the game $\mathcal{G}_{\tau(\lambda)}^W(q)$ is closed for Player II.)

3' Let Player I play according to $\sigma$ and let us show that Player II can play by maintaining that for all $n \in [-1, \omega)$, there exist $r, w, P$ such that

a. $r \in g$,

b. there exists finite $F \subseteq \tau[V_\lambda]$ such that $P = \mathrm{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup F)$,

c. $w \subseteq \tau(\phi)\downarrow$ is finite, $\hat{\mu} \models w$, and $w \cap \tau[V] = \emptyset$,

d. $p_n = (\tau(w_r) \cup w, \tau(\mathcal{M}_r) \cup \{P\downarrow\tau(\lambda)\})$.

4' The above conditions are satisfied for $n = -1$ and $p_{-1} = q$. Let us consider $n \ge 0$ and the move $Q_n$ of Player I.

5' Let $r, w, P$ be such that

a. $r \in g$,

b. there exists finite $F \subseteq \tau[V_\lambda]$ such that $P = \mathrm{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup F)$,

c. $w \subseteq \tau(\phi)\downarrow$ is finite, $\hat{\mu} \models w$, and $w \cap \tau[V] = \emptyset$,

d. $p_{n-1} = (\tau(w_r) \cup w, \tau(\mathcal{M}_r) \cup \{P\downarrow\tau(\lambda)\})$.
6' **Case.** $Q_n = \psi \in w_{p_{n-1}}$ where $\psi \equiv \bigvee_{i \in I} \psi_i$.

*Proof.*

1" We have that $\hat{\mu} \models \psi$, so there exists $i \in I$ such that $\hat{\mu} \models \psi_i$.

2" We set $p_n := (w_{p_{n-1}} \cup \{\psi_i\}, \mathcal{M}_{p_{n-1}})$.

☐

7' **Case.** $Q_n = (\psi, i) \in w_{p_{n-1}} \times I$ where $\psi \equiv \bigwedge_{i \in I} \psi_i$.

*Proof.*

1" We can take $p_n := (w_{p_{n-1}} \cup \{\psi_i\}, \mathcal{M}_{p_{n-1}})$.

☐

8' **Case.** $Q_n = (P \downarrow \tau(\lambda), D)$ where $D \in P \downarrow \tau(\lambda)$ is dense $P_{\tau(\lambda)}$.

*Proof.*

1" Let $\rho : \text{Hull}(P, \tau(V_\lambda)) \to \widehat{P \downarrow \tau(\lambda)}$ be the transitive collapse and
let $D^+ := \rho^{-1}(D) \in P$. Then $D^+$ is dense in $\tau(\mathbb{P}_\kappa)$.

2" Since $D^+ \in P$, there exists $\bar{D}^+ \in V$ such that $\tau(\bar{D}^+) = D^+$.

3" Let $\bar{D} := \bar{D}^+ \cap V_\lambda$. By elementarity, we have that $\tau(\bar{D}) = D$ and $\bar{D}$ is dense in $\mathbb{P}_\lambda$.

4" There exists $s \in \bar{D} \cap g$ such that $s \le r$.

5" We now have that

a. $\tau(s) \in \tau(\bar{D}) = D$,

b. $P' := \text{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup F \cup \{\tau(s)\})$ satisfies $\delta(P') = \delta(P)$,

c. $p_n := (\tau(s) \cup w, \tau(\mathcal{M}_s) \cup \{P' \downarrow \tau(\lambda)\}) \in \tau(\mathbb{P}_\lambda^*)$,

d. $p_n \le p_{n-1}, \tau(s)$.

☐

9' **Case.** $Q_n = (M, D)$ where $M \in \tau(\mathcal{M}_r)$ and $D \in M$ is dense $\mathbb{P}_{\lambda_M}$.

*Proof.*

1" We have that $D \in P$, so there exists $\bar{D} \in V$ such that $\tau(\bar{D}) = D$.

2" Let $\bar{M} := \tau^{-1}(M) \in \mathcal{M}_r$. By elementarity of $\tau$, we have that $\bar{D} \in \bar{M} \in \mathcal{M}_r$ and that $\bar{D}$ is dense in $\mathbb{P}_{\lambda_{\bar{M}}}$.

3" **Claim.** There exist $s \in g$ and $t \in \bar{D}$ such that $s \le r, t$ and $\delta(\text{Hull}(\bar{M}, t)) = \delta(\bar{M})$.

*Proof.*

1''' It suffices to show that the set of all $s \in \mathbb{P}_{\lambda}$ satisfying that $s \le r$ and that there exists $t \in \bar{D}$ such that $s \le t$ and $\delta(\text{Hull}(\bar{M}, t)) = \delta(\bar{M})$, is dense below $r$.

2''' Let $r' \in \mathbb{P}_{\lambda}$ be such that $r' \le r$. Then there exists $\bar{M}' \in \mathcal{M}_{r'}$ such that $\delta(\bar{M}') = \delta(\bar{M})$, $\lambda_{\bar{M}'} = \lambda_{\bar{M}}$, and $\bar{M} \subseteq \bar{M}'$.
3'''' By the existence of the appropriate strategy for $G_\lambda(r')$, there exists $s \in \mathbb{P}_\lambda$ such that $s \le r'$ and such that there exists $t \in \bar{D}$ satisfying $s \le t$ and $\delta(\mathrm{Hull}(\bar{M}', t)) = \delta(\bar{M}')$.

4'''' We have that

$$ \delta(\bar{M}) \le \delta(\mathrm{Hull}(\bar{M}, t)) \le \delta(\mathrm{Hull}(\bar{M}', t)) \le \delta(\bar{M}') = \delta(\bar{M}), $$

which shows that all of the above values are equal and consequently, $s \le r'$ is as required.

4'' Let $P' := \mathrm{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup F \cup \{\tau(s)\})$ and let

$$ p_n := (\tau(w_s) \cup w, \tau(\mathcal{M}_s) \cup \{P' \downarrow \tau(\lambda)\}) \in \tau(\mathbb{P}_\lambda^*). $$

Note that $\delta_{P'} = \omega_1^V$.

5'' We have $\tau(t) \in D$ and $p_n \le \tau(t)$, while

$$ \begin{aligned} \delta(\mathrm{Hull}(M, \tau(t))) &= \delta(\tau(\mathrm{Hull}(\bar{M}, t))) = \delta(\mathrm{Hull}(\bar{M}, t)) = \\ &= \delta(\bar{M}) = \delta(M). \end{aligned} $$

10' We have shown that Player II can survive $\omega$ steps against the strategy $\sigma$, which is a contradiction.

10° Having proven the claim, we conclude that in $W$, $\tau(p)$ cannot be a witness that $N$ is not good.

11° Since $\delta_N = \omega_1^V \in \tau(T)$, we have that $N \in \tau(S')$, which contradicts the previous point.

**Corollary 2.16.** Suppose that $\phi$ is AS good at $\kappa$. Then $\mathbb{P}_\kappa$ is spp and for all $V$-generic $g \subseteq \mathbb{P}_\kappa$, we have that $\mu_g \models \phi$. In particular, $\phi$ is ssp consistent. □
### 3 Forcing $\mathbb{P}_{\max}$ conditions

In this section, we aim to describe how our general method gives rise to another solution to the main problem of [AS21]. In Declaration 3.2, we fix the background assumptions.

**Definition 3.1.** Suppose that $\theta$ is a cardinal and $\Gamma \subseteq \mathcal{P}(\mathbb{R})$. Then $\Gamma$ is $\theta$-well behaved iff

a. for all $A \in \Gamma$, $A$ is $\theta$-universally Baire,

b. for all $A \in \Gamma$, for all formulas $\phi(x, \dot{A})$ projective in $\dot{A}$, we have

$$B := \{x \in \mathbb{R} : \phi(x, A)\} \in \Gamma$$

$$\text{and } V^{\text{Col}(\omega, \theta)} \models B_{\dot{G}} = \{x \in \mathbb{R}_{\dot{G}} : \phi(x, A_{\dot{G}})\}. \quad \square$$

**Declaration 3.2.** Suppose that

1. $\mathbb{N}_{\omega_1}$ is saturated,

2. $2^{\omega_1} = \omega_2$,

3. $\mathbb{M}_{\omega_1}$ holds,

4. $D$ is dense in $\mathbb{P}_{\max}$,

5. for all $\theta$, there exists $\theta$-well behaved $\Gamma$ such that $D$ is coded by a set in $\Gamma$,

6. $A \subseteq \omega_1$ satisfies $\omega_1^{[A]} = \omega_1$. $\square$

Our goal is to construct an ssp poset $\mathbb{P}$ such that in $V^{\mathbb{P}}$, there exist generic iterations $\mathcal{I}$ and $\mathcal{J}$ of $\mathbb{P}_{\max}$ conditions of length $\omega_1 + 1$ such that

a. $M_0^{\mathcal{J}} \in D_{\dot{G}}$,

b. $\mathcal{I}|_{\omega_1^{M_0^{\mathcal{J}}}}$ witnesses that $M_0^{\mathcal{J}} < M_0^{\mathcal{I}}$,

c. $\pi^{\mathcal{J}}(\mathcal{I}|_{\omega_1^{M_0^{\mathcal{J}}}}) = \mathcal{I}$,

d. $M_{\omega_1}^{\mathcal{I}} = (H_{\omega_2}, \in, \mathbb{N}_{\omega_1}, A)^{\mathbb{V}}$,

e. $\mathbb{N}_{\omega_1} \cap \mathcal{P}(\omega_1)^{M_{\omega_1}^{\mathcal{J}}} = I^{M_{\omega_1}^{\mathcal{J}}}.$

We will start by forgetting for a moment about the requirement e and recasting the rest of the problem as adding a model to an infinitary propositional formula by an ssp forcing. It will turn out that we get e for free.
**Notation 3.3.** Suppose $\mathcal{M} = (\omega, \dots)$ is a structure in a recursively enumerable language. Then $\vartheta(\mathcal{M}) : \omega \to 2$ is defined in such a way that for all $n < \omega$, for $\psi$ denoting the formula in the language of

$$ (\mathcal{M}, m : m < \omega) $$

satisfying $\lceil \psi \rceil = n$, it holds that

$$ \vartheta(\mathcal{M})(n) = 1 \iff (\mathcal{M}, m : m < \omega) \models \psi. $$

□

**Notation 3.4.**

a. If $E \subseteq X \times Y$ and $x \in X$, we denote $E_x := \{y \in Y : (x, y) \in E\}$.

b. If $a : X \to Y$ and $x \in X$, then $a_x := a(x)$.

□

We also need to fix two trees before proceeding.

**Declaration 3.5.** Let $T, U$ be $\omega_2$-absolutely complemented trees on $\omega \times \omega_2$ such that

a. $p[T]$ is the set of all $\vartheta(\omega, E, I, a)$ for which there exists $p \in \mathbb{P}_{\max}$ satisfying $(\omega, E, I, a) \simeq p$,

b. $p[U]$ is the set of all $\vartheta(\omega, E, I, a)$ for which there exists $p \in D$ satisfying $(\omega, E, I, a) \simeq p$. □

We are now ready to restate the goal in a way that makes it clear that we actually want to add a model to an infinitary propositional formula.

**Goal 3.6.** Construct an ssp poset $\mathbb{P}$ such that in $V^{\mathbb{P}}$, there exist $E, F, I, J, a, b, u, v, \pi, \sigma, \jmath, r$ satisfying

a. $E, F \subseteq \omega_1 \times (\omega \times \omega)$,

b. $I, J \subseteq \omega_1 \times \omega$,

c. $a, b : \omega_1 \to \omega$,

d. $u, v : \omega \to \omega_2$,

e. for all $n < \omega$, $(\vartheta(\omega, E_0, I_0, a_0) \upharpoonright n, u \upharpoonright n) \in T$,

f. for all $n < \omega$, $(\vartheta(\omega, F_0, J_0, b_0) \upharpoonright n, v \upharpoonright n) \in U$,

g. $\pi, \sigma \subseteq \{(\alpha, \beta) : \alpha \le \beta < \omega_1\} \times (\omega \times \omega)$,

h. for all $\alpha \le \beta < \omega_1$, $\pi_{\alpha, \beta} : (\omega, E_\alpha, I_\alpha, a_\alpha) \underset{\Sigma_\omega}{\longrightarrow} (\omega, E_\beta, I_\beta, a_\beta)$,

i. for all $\alpha \le \beta \le \gamma < \omega_1$, $\pi_{\beta, \gamma} \circ \pi_{\alpha, \beta} = \pi_{\alpha, \gamma}$,
j. for all $\alpha < \omega_1$, there exists $k < \omega$ such that

i. $(\omega, E_{\alpha+1}) \models \dot{k} < \dot{\omega}_1$,

ii. for all $\iota < \omega$, there exists $\jmath < \omega$ such that $(\omega, E_\alpha) \models \dot{\jmath} : \dot{\omega}_1 \to \dot{V}$
and such that for $w := \pi_{\alpha, \alpha+1}(\jmath)$, $(\omega, E_{\alpha+1}) \models \dot{w}(k) = \dot{i}$,

iii. for $g_\alpha := \{w < \omega : (\omega, E_\alpha, I_\alpha) \models \dot{w} \in I_\alpha^+, k E_{\alpha+1} \pi_{\alpha,\alpha+1}(w)\}$, we
have that $(\omega, E_\alpha, I_\alpha, g_\alpha) \models$ "$g_\alpha$ is generic for $I_\alpha^+$",

k. for all limit $\gamma < \omega_1$, for all $n < \omega$, there exist $\alpha < \gamma$ and $m < \omega$ such that
$\pi_{\alpha,\gamma}(m) = n$,

l. for all $\alpha \le \beta < \omega_1$, $\sigma_{\alpha,\beta} : (\omega, F_\alpha, J_\alpha, b_\alpha) \xrightarrow{\Sigma_\omega} (\omega, F_\beta, J_\beta, b_\beta)$,

m. for all $\alpha \le \beta \le \gamma < \omega_1$, $\sigma_{\beta,\gamma} \circ \sigma_{\alpha,\beta} = \sigma_{\alpha,\gamma}$,

n. for all $\alpha < \omega_1$, there exists $k < \omega$ such that

i. $(\omega, F_{\alpha+1}) \models \dot{k} < \dot{\omega}_1$,

ii. for all $\iota < \omega$, there exists $\jmath < \omega$ such that $(\omega, F_\alpha) \models \dot{\jmath} : \dot{\omega}_1 \to \dot{V}$
and such that for $w := \sigma_{\alpha, \alpha+1}(\jmath)$, $(\omega, F_{\alpha+1}) \models \dot{w}(k) = \dot{i}$,

iii. for $h_\alpha := \{w < \omega : (\omega, F_\alpha, J_\alpha) \models \dot{w} \in J_\alpha^+, k F_{\alpha+1} \sigma_{\alpha,\alpha+1}(w)\}$, we
have that $(\omega, F_\alpha, J_\alpha, h_\alpha) \models$ "$h_\alpha$ is generic for $J_\alpha^+$",

o. for all limit $\gamma < \omega_1$, for all $n < \omega$, there exist $\alpha < \gamma$ and $m < \omega$ such that
$\sigma_{\alpha,\gamma}(m) = n$,

p. $\exists c \subseteq \omega_1 \times (\omega \times H_{\omega_2}^V)$,

q. for all $\alpha < \omega_1$, $\exists c_\alpha : (\omega, E_\alpha, I_\alpha, a_\alpha) \xrightarrow{\Sigma_\omega} (H_{\omega_2}, \in, \text{NS}_{\omega_1}, A)^V,$

r. for all $\alpha \le \beta < \omega_1$, $\exists c_\alpha = \exists c_\beta \circ \pi_{\alpha,\beta}$,

s. for all $x \in H_{\omega_2}^V$, there exist $\alpha < \omega_1$ and $\iota < \omega$ such that $\exists c_\alpha(\iota) = x$,

t. $r : \omega_1 \to (\omega \to \omega_1)$,

u. for all $\alpha < \omega_1$, $r_\alpha$ is the ranking function of $(\omega, F_\alpha)$. $\square$

Note that if $|\mathbb{P}| > \omega_2$, then every element of $p[T]$ in $V^{\mathbb{P}}$ will be of the form
$\vartheta(M)$ for some $M$ isomorphic to a $\mathbb{P}_{\max}$-condition, but the converse might not
hold. A similar remark holds for $U$. Now, we said that Goal 3.6 can be expressed
as adding via an ssp poset a model of a certain infinitary propositional formula
$\phi$. Here are some examples of how to go about obtaining this $\phi$.

a. Let $\dot{E}_{\alpha,m,n}, \dot{F}_{\alpha,m,n}, \dot{I}_{\alpha,m}, \dot{J}_{\alpha,m}, \dot{a}_{\alpha,m}, \dot{b}_{\alpha,m}, \dot{u}_{m,\xi}, \dot{v}_{m,\xi}, \dot{\pi}_{\alpha,\beta,m,n}, \dot{\sigma}_{\alpha,\beta,m,n},$
$\exists c_{\alpha,m,x}, \dot{r}_{\alpha,m,\gamma}$ be propositional letters, for all $\alpha \le \beta < \omega_1$, $\gamma < \omega_1$,
$m, n < \omega$, $\xi < \omega_2$, $x \in H_{\omega_2}$.
b. If $\psi$ is the language of some $(\omega, E_\alpha, I_\alpha, a_\alpha)$, we can express

$$ (\omega, E_\alpha, I_\alpha, a_\alpha) \models \psi $$

as a propositional formula by induction on complexity of $\psi$. Similarly for $(\omega, F_\alpha, J_\alpha, b_\alpha)$.

c. We can now express requirement e as

$$ \bigwedge_{n<\omega} \bigvee_{s \in T, i<n} \bigwedge_{|s|=n} \theta_{s,i} $$

where

i. for $s(i) = (1, \xi)$ and for $\psi_i$ being the formula with code $i$, $\theta_{s,i}$ is defined as the propositional formula asserting

$$ \text{"}(\omega, E_\alpha, I_\alpha, a_\alpha) \models \psi_i\text{"} \wedge \dot{u}_{i,\xi}, $$

ii. for $s(i) = (0, \xi)$ and for $\psi_i$ being the formula with code $i$, $\theta_{s,i}$ is defined as the propositional formula asserting

$$ \text{"}(\omega, E_\alpha, I_\alpha, a_\alpha) \models \neg \psi_i\text{"} \wedge \dot{u}_{i,\xi}. $$

Similarly for the requirement f.

d. One can proceed to successfully code all of the requirements of Goal 3.6 and obtain the formula $\phi$, which will have $\omega_2$ propositional letters and whose every conjuction and disjunction will have length at most $\omega_2$.

**Definition 3.7.** We denote by $\phi_{AS}$ the negation normal form of the formula sketched just above. $\square$

The following lemma simply states that the formula $\phi_{AS}$ codes that which it is designed to code.

**Lemma 3.8.** Suppose that $\mu \models \phi_{AS}$ belongs to an outer model. Then in the outer model, there exist unique generic iterations $\mathcal{I}$ and $\mathcal{J}$ of $\mathbb{P}_{\max}$ conditions of length $\omega_1^V + 1$ such that

a. $\mathcal{I} \models \omega_1^{M_0^J}$ witnesses that $M_0^J < M_0^I$,

b. $\pi^J(\mathcal{I} \models \omega_1^{M_0^J}) = \mathcal{I}$,

c. $M_{\omega_1}^I = (H_{\omega_2}, \in, \mathrm{NS}_{\omega_1}, A)^V$,

d. $M_0^J \in D_g$,

e. for all $\alpha < \omega_1^V$, $(\omega, E_\alpha, I_\alpha, a_\alpha) \simeq M_\alpha^I$ and $(\omega, F_\alpha, J_\alpha, b_\alpha) \simeq M_\alpha^J$,
f. for all $\alpha \le \beta < \omega_1^V$, the diagrams

$$ (\omega, E_\alpha, I_\alpha, a_\alpha) \xrightarrow{\pi_{\alpha,\beta}} (\omega, E_\beta, I_\beta, a_\beta) \\ \downarrow \cong \qquad \downarrow \cong \\ M_\alpha^I \xrightarrow{\pi_{\alpha,\beta}^I} M_\beta^I $$

$$ (\omega, E_\alpha, I_\alpha, a_\alpha) \xrightarrow{\omega c_\alpha} (H_{\omega_2}, \in, \mathrm{NS}_{\omega_1}, A)^V \\ \downarrow \cong \qquad \downarrow = \\ M_\alpha^I \xrightarrow{\pi_{\alpha,\omega_1}^V} (H_{\omega_2}, \in, \mathrm{NS}_{\omega_1}, A)^V $$

$$ (\omega, F_\alpha, J_\alpha, b_\alpha) \xrightarrow{\sigma_{\alpha,\beta}} (\omega, F_\beta, J_\beta, b_\beta) \\ \downarrow \cong \qquad \downarrow \cong \\ M_\alpha^J \xrightarrow{\pi_{\alpha,\beta}^J} M_\beta^J $$

commute.

<figure><img src="image_3.png" /></figure>

**Notation 3.9.** In the case of the above proposition, we denote $\mathcal{I}^\mu := \mathcal{I}$ and $\mathcal{J}^\mu := \mathcal{J}$. We also denote

$$ E^\mu := \{(\alpha, m, n) \in \omega_1^V \times (\omega \times \omega) : \mu(\dot{E}_{\alpha,m,n}) = 1\}, $$

$$ F^\mu := \{(\alpha, m, n) \in \omega_1^V \times (\omega \times \omega) : \mu(\dot{F}_{\alpha,m,n}) = 1\}, $$

$$ I^\mu := \{(\alpha, m) \in \omega_1^V \times \omega : \mu(\dot{I}_{\alpha,m}) = 1\}, $$

and so on.

<figure><img src="image_4.png" /></figure>

We now want to show that $\phi_{AS}$ is AS good. By the results of Section 2, this will give us the ssp consistency of $\phi_{AS}$. In fact, it will give us the actual poset witnessing this ssp consistency, thus allowing us to achieve Goal 3.6.

**Lemma 3.10.** $\phi_{AS}$ is consistent.

*Proof.* This is shown as [AS21, Lemma 3.6].

<figure><img src="image_5.png" /></figure>

**Proposition 3.11.** $\phi_{AS}$ is AS good.

*Proof.*

1.° Let $h$ be $V$-generic for $\mathrm{Col}(\omega, < \kappa)$ and let us work in $V[h]$. We fix an arbitrary valuation $\mu \models \phi$ and an arbitrary stationary $S \subseteq \omega_1^V$ and we are going to produce elementary $\tau: V \to W$ and $\hat{\mu} \models \tau(\phi)$ such that $\mathrm{crit}(\tau) = \omega_1^V \in \tau(S)$ and for all $\psi \in \phi \downarrow, \hat{\mu}(\tau(\psi)) = \mu(\psi)$.
2° Let $\hat{\mathcal{J}}$ be a generic iteration of length $\omega_1 + 1$ extending $\mathcal{J}^\mu$ and satisfying $S \in g_{\omega_1}^{\hat{\mathcal{J}}}$ and let $\hat{\mathcal{I}} := \pi_{\omega_1}^{\hat{\mathcal{J}}}, \omega_1(\mathcal{I})$. Since

$$M_{\omega_1}^{\hat{\mathcal{I}}} = (H_{\omega_2}, \in, \mathrm{NS}_{\omega_1}, A)^V,$$

there exists a unique generic iteration $\mathcal{K}$ of $V$ of length $\omega_1 + 1$ such that for all $\alpha < \omega_1$, $g_\alpha^\mathcal{K} = g_{\omega_1+\alpha}^{\hat{\mathcal{I}}}$.

3° We let $\tau := \pi^\mathcal{K} : V \to M_{\omega_1}^\mathcal{K} :=: W$.

4° For all $\alpha < \omega_1$, let $e_\alpha : \omega \to M_\alpha^{\hat{\mathcal{I}}}$ be a bijection such that if $\alpha < \omega_1^V$, then $e_\alpha$ is the transitive collapse of $(\omega, E_\alpha)$. We define for all $\alpha < \omega_1$,

$$\hat{E}_\alpha := \{(m, n) : e_\alpha(m) \in e_\alpha(n)\}.$$

In the case $\alpha < \omega_1^V$, we get the old $\hat{E}_\alpha = E_\alpha$.

5° In the similar fashion, we extended other sets from Goal 3.6, producing the “hat versions” of them. Then the valuation $\hat{\mu}$ is defined according to these “hat versions”.

6° **Claim.** For $\psi \in \phi \downarrow, \hat{\mu}(\tau(\psi)) = \mu(\psi)$.

*Proof.* This requires a tedious verification. Suppose for example that $\psi$ comes from the part s. of Goal 3.6. Then $\psi$ might of the formula

$$\bigwedge_{x \in H_{\omega_2}^V} \bigvee_{\alpha < \omega_1^V} \bigvee_{u < \omega} \partial c_{\alpha, u, x}.$$

In this case, $\mu(\psi) = 1$, $\tau(\psi)$ is the formula

$$\bigwedge_{x \in H_{\omega_2}^W} \bigvee_{\alpha < \omega_1} \bigvee_{u < \omega} \partial c_{\alpha, u, x},$$

and we need to verify that $\tau(\psi)$ is evaluated as true by $\hat{\mu}$. However, this is trivial, since

$$(H_{\omega_2}^W, \partial c_\alpha : \alpha < \omega_1)$$

is the direct limit

$$(M_\alpha^{\hat{\mathcal{I}}}, \pi_{\alpha, \beta}^{\hat{\mathcal{I}}} : \alpha \le \beta < \omega_1).$$

We leave it to the reader to convince herself of the rest. $\square$

7° Thus, $\tau$ and $\hat{\mu}$ are as required. $\square$

<figure><img src="image_6.png" /></figure>
We now know that the poset $\mathbb{P}_\kappa$ (of Section 2) is stationary set preserving and adds a model for $\phi_{AS}$. Let $g$ be a $V$-generic for $\mathbb{P}_\kappa$ and let $\mu_g := \mu$, $\mathcal{I} := \mathcal{I}^\mu$, and $\mathcal{J} := \mathcal{J}^\mu$. The $\mathbb{P}_{\max}$-iterations $\mathcal{I}$ and $\mathcal{J}$ are like what we wanted, except the thing that we forgot about. Namely, recall that we did not code into $\phi_{AS}$ the requirement that

$$NS_{\omega_1} \cap \mathcal{P}(\omega_1)^{M_{\omega_1}^{\mathcal{J}}} = I^{M_{\omega_1}^{\mathcal{J}}}.$$

However, we get this requirement for free, which we show in the following proposition.

**Proposition 3.12.** It holds in $V^{\mathbb{P}_\kappa}$ that

$$NS_{\omega_1} \cap \mathcal{P}(\omega_1)^{M_{\omega_1}^{\mathcal{J}^{\mu\dot{g}}}} = I^{M_{\omega_1}^{\mathcal{J}^{\mu\dot{g}}}}.$$

*Proof.*

1.° Let us assume otherwise. Since $(\supseteq)$ clearly holds, the other inclusion must fail. More precisely, there exist $p \in \mathbb{P}_\kappa$, $\dot{C} \in V^{\mathbb{P}_\kappa}$, $\alpha < \omega_1$, and $m < \omega$ such that $p$ forces that

a. $(\omega, F_\alpha^{\mu\dot{g}}) \models$ “$\dot{m}$ is a subset of $\omega_1$ which is not in $J_\alpha^{\mu\dot{g}}$”,

b. $\dot{C}$ is a club in $\omega_1^V$,

c. the image of $m$ in the final iterate of $\mathcal{J}^{\mu\dot{g}}$ has an empty intersection with $\dot{C} \subseteq V_\kappa$.

2.° Let

1. $\theta \gg \kappa$ be regular,

2. $R$ be a wellordering of $H_\theta$,

3. $\mathcal{H} := (H_\theta, \in, R, \kappa, \dot{C}, T, U, A)$,

4. $\lambda \in \mathcal{E}$ be such that

a. $p \in \mathbb{P}_\lambda$,

b. for all $\eta < \lambda$, $\kappa \cap \text{Hull}^\mathcal{H}(V_\eta) \subseteq \lambda$,

c. $V_\lambda < (V_\kappa, \in, \mathbb{P}_\kappa, \dot{C})$.

5. $h$ be $V$-generic for $\text{Col}(\omega, < \kappa)$,

6. $g \in V[h]$ be $V$-generic for $\mathbb{P}_\lambda$ containing $p$,

7. $\mu := \mu_g \models \phi$.

We work in $V[h]$.

3.° Let $S_0$ be the image of $m$ under the transitive collapse of $(\omega, F_\alpha^\mu)$ and let $S := \pi_{\alpha, \omega_1^V}^{\mathcal{J}^\mu}(S_0)$.

4.° Let $\hat{\mathcal{I}}$, $\hat{\mathcal{J}}$, $\mathcal{K}$, $\tau$, $W$, and $\hat{\mu}$ be obtained as in the proof of Proposition 3.11. (The reason why we cannot simply quote this proposition is because now $S$ is not necessarily in $V$.) We have that
a. $\tau: V \to W$ elementarily,

b. $\text{crit}(\tau) = \omega_1^V,$

c. $\omega_1^V \in \pi_{\omega_1^V, \omega_1}^{\mathcal{J}}(S) = \pi_{\alpha, \omega_1}^{\mathcal{J}}(S_0),$

d. for all $\psi \in \phi \downarrow, \hat{\mu}(\tau(\psi)) = \mu(\psi)$.

5° Let

1. $\hbar_{\delta}$ be the formula coding the assertion that "$\omega_1^{M_{\delta}^{\mathcal{J}}}$ = $\delta$ and this ordinal belongs the image of the set coded by $m$ under the $\alpha$-to-($\delta$ + 1) iteration embedding",

2. $N := \text{Hull}^{\tau(\mathcal{H})}(\omega_1^V \cup \{\tau(p), \tau(\dot{C})\}),$

3. $q := (\tau(w_p) \cup \{\bar{\omega}_1^V\}, \tau(M_p) \cup \{N \downarrow \tau(\lambda)\}) \in \tau(\mathbb{P}_{\lambda}^*).$

6° **Claim.** $q \in \tau(\mathbb{P}_{\lambda})$

*Proof.* This is shown like Claim 9° of the proof of Proposition 2.15. □

7° By elementarity, it holds in $V$ that there exist $r \in \mathbb{P}_{\lambda}$, $\delta < \omega_1$, and $P \prec \mathcal{H}$ such that

a. $r = (w_p \cup \{\bar{\delta}\}, M_p \cup \{P \downarrow \lambda\}) \in \mathbb{P}_{\lambda},$

b. $p, \dot{C} \in P,$

c. $\delta_P = \delta > \alpha.$

8° **Claim.** $r \Vdash_{\mathbb{P}_{\kappa}}^{V}$ "$\delta$ belongs to both the image of the set coded by $m$ along the $\mathcal{J}$-iteration embedding and to the club $\dot{C}$".

*Proof.*

1' Since $\bar{\delta} \in w_r$, it is immediate that $r \Vdash$ "$\delta$ belongs to both the image of the set coded by $m$ along the $\mathcal{J}$-iteration embedding". It remains to verify that $r \Vdash \delta \in \dot{C}$.

2' The proof of Proposition 2.14. shows that in $V$, $r$ is semigeneric for $(P, \mathbb{P}_{\kappa})$.

3' Let $g$ be an arbitrary $V$-generic for $\mathbb{P}_{\kappa}$ containing $r$. We have that $\delta(P[g]) = \delta(P) = \delta$.

4' Since $\dot{C}^g \in P[g]$ is a club in $\omega_1$, we have that $\delta \in C$, as required. □

9° The previous claim is in contradiction with 1°c. □
Finally, we arrive at the conclusion.

**Corollary 3.13.** The poset $\mathbb{P}_\kappa$ is stationary set preserving. In $V^{\mathbb{P}_\kappa}$, there exist generic iterations $\mathcal{I}$ and $\mathcal{J}$ of $\mathbb{P}_{\max}$ conditions of length $\omega_1 + 1$ such that

a. $M_0^{\mathcal{J}} \in D_{\dot{G}}$,

b. $\mathcal{I} \upharpoonright \omega_1^{M_0^{\mathcal{J}}}$ witnesses that $M_0^{\mathcal{J}} < M_0^{\mathcal{I}}$,

c. $\pi^{\mathcal{J}}(\mathcal{I} \upharpoonright \omega_1^{M_0^{\mathcal{J}}}) = \mathcal{I}$,

d. $M_{\omega_1}^{\mathcal{I}} = (H_{\omega_2}, \in, \mathbf{NS}_{\omega_1}, A)^V$,

e. $\mathbf{NS}_{\omega_1} \cap \mathcal{P}(\omega_1)^{M_{\omega_1}^{\mathcal{J}}} = I^{M_{\omega_1}^{\mathcal{J}}}$. $\square$
References

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