Proof For part 1: $\mathbb {Q}^{\omega _1-\mathrm{SSP}}_{T,\theta }$ is defined to be the poset

(4) $$ \begin{align} \mathrm{Add}(\omega) \ * \ \dot{\mathbb{C}}^{\mathrm{fin}}_{\mathrm{dec}} (\dot{X}^\theta_T) \ * \ \dot{\mathbb{S}}(H^V_\theta), \end{align} $$

where, letting $\dot {\sigma }$ be the $\mathrm{Add}(\omega )$ -name for its generic object,

$$ \begin{align*} \dot{X}^\theta_T:= & \mathrm{Lift}^{H^V_\theta[\dot{\sigma}]} (([H^V_\theta]^\omega \searrow T )^V \cup ( [\omega_2]^\omega \searrow T^c) \setminus V) \\ = & \{ z \in [H_\theta[\dot{\sigma}]]^\omega\,\! :\,\! (z \cap H^V_\theta \in V \mathrm{\ and\ } z \cap \omega_1 \in T ) \mathrm{\ or }\\ & \quad (z \cap \omega_2 \notin V \mathrm{\ and\ } z \cap \omega_1 \in T^c)\}, \end{align*} $$

$\dot {\mathbb {C}}^{\mathrm{fin}}_{\mathrm{dec}}(\dot {X}^\theta _T)$ is the poset from Definition 2.12, and $\mathbb {S}(H^V_\theta )$ is the poset given by Theorem 2.9, assuming that the assumptions of that theorem hold, which we verify first. For Theorem 2.9 to be applicable, we need to check that the first two steps of (4) force $H^V_\theta \in \mathrm{IU}$ . This is part of the following claim:

Proof of Claim 3.2 the proofs of parts (a) and (b) break into cases, depending on whether or not T is stationary in V. Though the proofs of these parts could be combined, we find it conceptually simpler to prove each part separately, at the expense of some slight repetition.

For part (a): the poset $\mathrm{Add}(\omega )$ is strongly proper, and the second step is forced to be strongly proper on a stationary set by Fact 2.13, provided that the set $\dot {X}_T^\theta $ is forced by $\mathrm{Add}(\omega )$ to be a stationary subset of $ [ H_\theta [\dot {\sigma }] ]^\omega $ . We verify this now. Let $\sigma $ be $\mathrm{Add}(\omega )$ -generic; we break into cases depending on whether T is stationary in V. If T is stationary in V, then $[H_\theta ]^\omega \searrow T$ is stationary, so properness of $\mathrm{Add}(\omega )$ implies that the set

$$ \begin{align*} ( [H^V_\theta]^\omega \searrow T)^V \end{align*} $$

remains stationary in $V[\sigma ]$ , and hence its lifting to $H^V_\theta [\sigma ]$ —which is a subset of $(\dot {X}^\theta _T)_\sigma $ —is stationary from the point of view of $V[\sigma ]$ . On the other hand, if T is nonstationary in V, then $T^c$ contains a club subset of $\omega _1$ . By Gitik–Velickovic’s Theorem 2.15 (noting that $\omega _2^V = \omega _2^{V[\sigma ]}$ ), the set $ ( [\omega _2]^\omega ) \setminus V$ is (projective) stationary in $V[\sigma ]$ . So in particular $ ([\omega _2]^\omega \searrow T^c ) \setminus V$ is stationary, and hence its lifting to $ H^V_\theta [\sigma ]$ is stationary. And this lifting is a subset of $(\dot {X}^\theta _T)\sigma $ .

In summary, whether T is stationary or not, the set $\dot {X}^\theta _T$ is forced by $\mathrm{Add}(\omega )$ to be stationary, and hence the first two steps of (4) are of the form “strongly proper, followed by strongly proper on a stationary set.” So this two-step iteration has the $\omega _1$ approximation and cover properties by Facts 2.11 and 2.8. This, in turn, ensures that $H^V_\mu $ will be $\omega _1$ -guessing and internally unbounded after the first two steps of the iteration (4), for all V-regular $\mu \in [\omega _2,\theta ]$ . This completes the proof of part (a) of the claim.

Next we verify part (b) of the claim. Let R be a stationary subset of $\theta \cap \mathrm{cof}(\omega )$ from the ground model. Consider two cases, depending on whether or not T is stationary in V. If T is stationary in V, then

$$ \begin{align*} V \models \ \widetilde{R}^V_T:= \{ z \in ([H_\theta]^\omega)^V : \mathrm{sup}(z) \in R \mathrm{\ and\ } z \cap \omega_1 \in T \} \mathrm{\ is\ stationary\ in\ } [\theta]^\omega. \end{align*} $$

By properness of $\mathrm{Add}(\omega )$ , $\widetilde {R}^V_T$ remains stationary in $V[\sigma ]$ , and the lifting of $\widetilde {R}^V_T$ to $H_\theta [\sigma ]$ is a stationary subset of $(\dot {X}^\theta _T)_\sigma $ . By Fact 2.13, the second step of the forcing (4) preserves the stationarity of $\widetilde {R}^V_T$ . Since the supremum of every element of $\widetilde {R}^V_T$ lies in R, it follows that R is still stationary as well.

Now suppose T is nonstationary in V. By Fact 2.13 and the definition of $\dot {X}^\theta _T$ , it suffices to prove that

$$ \begin{align*} V[\sigma] \models \left\{ z \in [\theta]^\omega : \mathrm{sup}(z) \in R, \ z \cap \omega_2 \notin V \mathrm{\ and\ } z \cap \omega_1 \in T^c \right\} \mathrm{\ is\ stationary\ in\ } [\theta]^\omega. \end{align*} $$

Now $\omega _2^V = \omega _2^{V[\sigma ]}$ , and $T^c$ contains a club subset of $\omega _1$ by our case. So, in turn, it suffices to show that

(5) $$ \begin{align} V[\sigma] \models \left\{ z \in [\theta]^\omega : \mathrm{sup}(z) \in R \mathrm{\ and\ } z \cap \omega_2 \in [\omega_2]^\omega \setminus V \right\} \mathrm{\ is\ stationary\ in\ } [\theta]^\omega. \end{align} $$

Since $\omega _2^V = \omega _2^{V[\sigma ]}$ , Theorem 2.15 implies that $[\omega _2]^\omega \setminus V$ is stationary in $V[\sigma ]$ (where the $\lambda $ from the statement of that theorem is taken to be $\omega _2$ ). By properness of $\mathrm{Add}(\omega )$ , R is still a stationary subset of $\theta \cap \mathrm{cof}(\omega )$ in $V[\sigma ]$ . If $\theta = \omega _2$ then (5) follows from Theorem 2.15 directly; if $\theta> \omega _2$ then (5) follows from Theorem 2.15 (again with $\lambda = \omega _2$ ) together with Lemma 2.16.

Next we verify part (c) of the claim. Let S be a stationary subset of $\omega _1$ , and let $\sigma $ be $(V,\mathrm{Add}(\omega ))$ -generic. Then at least one of $S \cap T$ or $S \cap T^c$ is stationary. Suppose first that $S \cap T$ is stationary. Then $([H^V_\theta ]^\omega )^V \searrow (S \cap T)$ is stationary in V, and remains so in $V[\sigma ]$ by properness of $\mathrm{Add}(\omega )$ ; then its lifting to $H^V_\theta [\sigma ]$ is a stationary subset of $X^\theta _T:=(\dot {X}^\theta _T)_\sigma $ . So by Fact 2.13 (viewing $V[\sigma ]$ as the ground model), $\mathbb {C}^{\mathrm{fin}}_{\mathrm{dec}}(X^\theta _T)$ preserves the stationarity of $\mathrm{Lift}^{H^V_\theta [\sigma ]}(([H^V_\theta ]^\omega )^V \searrow (S \cap T) )$ , and in particular the stationarity of $S \cap T$ . Now suppose $S \cap T^c$ is stationary. Then it is still stationary in $V[\sigma ]$ , so by Gitik–Velickovic’s Theorem 2.15, $([\omega _2]^\omega \searrow (S \cap T^c) ) \setminus V$ is stationary, and hence its lifting to $H^V_\theta [\sigma ]$ is stationary. Moreover, this lifting is a subset of $X^\theta _T$ . Then by Fact 2.13 (again viewing $V[\sigma ]$ as the ground model), the poset $\mathbb {C}^{\mathrm{fin}}_{\mathrm{dec}}(X^\theta _T)$ preserves stationarity of $\mathrm{Lift}^\theta ( ([\omega _2]^\omega \searrow (S \cap T^c) ) \setminus V )$ , and hence stationarity of $S \cap T^c$ .

Finally we verify part (d) of the claim. The poset $\mathbb {C}(X^\theta _T)$ adds a filtration $\langle Q_i : i < \omega _1 \rangle $ of $H^V_\theta [\sigma ]$ with the property that whenever i is in the club $C := \{ i < \omega _1 : Q_i \cap \omega _1 = i \}$ , then:

• • if $i \in T$ then $Q_i \cap H^V_\theta \in V$ , which implies that $Q_i \cap H^V_\mu \in V$ for all $\mu \in [\omega _2,\theta ]$ ; and

• • if $i \in T^c$ then $Q_i \cap \omega _2 \notin V$ , which implies that $Q_i \cap H^V_\mu \notin V$ for all $\mu \in [\omega _2,\theta ]$ .

It follows that for each V-regular $\mu \in [\omega _2,\theta ]$ , the filtration $\langle Q_i \cap H^V_\mu : i < \omega _1 \rangle $ has the property that for every $i \in C$ , $Q_i \cap H_\mu ^V$ is an element of $H^V_\mu $ if and only if $i \in T$ . This shows that the internal part of $H^V_\mu $ is forced to be exactly T.

Then by Claim 3.2 and Theorem 2.9, the third step $\dot {\mathbb {S}}(H^V_\theta )$ of the poset (4) forces $H^V_\theta $ to be indestructibly guessing (and still internally unbounded). Also, Claim 3.2 and Lemma 2.1 imply that for every V-regular $\mu \in [\omega _2^V,\theta ]$ , $H^V_\mu $ still has internal part exactly T after forcing with $\dot {\mathbb {S}}(H^V_\theta )$ , since the latter preserves $\omega _1$ (it is c.c.c.). Since $\dot {\mathbb {S}}(H^V_\theta )$ is forced to be c.c.c., in particular it preserves all stationary subsets of $\omega _1$ and of $\theta \cap \mathrm{cof}(\omega )$ that lie in $V^{\mathrm{Add}(\omega ) \ * \ \dot {\mathbb {C}}^{\mathrm{fin}}_{\mathrm{dec}} (\dot {X}^\theta _T)}$ . Together with Claim 3.2, it follows that all stationary subsets of $\theta \cap \mathrm{cof}(\omega )$ , and of $\omega _1$ , from V are preserved by the three-step iteration (4). This completes the proof of part 1 of the theorem.

Part 2 is similar: $\mathbb {Q}^{\mathrm{proper}}_{T,\theta }$ is defined similarly to the poset $\mathbb {Q}^{\omega _1\hbox{-}\mathrm{SSP}}_{T,\theta }$ defined in (4), except that instead of the $\mathrm{Add}(\omega )$ -name $\dot {X}^\theta _T$ , we use the $\mathrm{Add}(\omega )$ -name

$$ \begin{align*} \dot{Y}^\theta_T:= & \mathrm{Lift}^{H^V_\theta[\sigma]}(([H^V_\theta]^\omega)^V \cup ([\omega_2]^\omega \searrow T^c) \setminus V) \\ & = \{ z \in [H_\theta[\sigma]]^\omega : ( z \cap H^V_\theta \in V) \mathrm{\ or\ } ( z \cap \omega_1 \in T^c \mathrm{\ and\ } z \cap \omega_2 \notin V) \}. \end{align*} $$

The proof is similar to the proof of part 1, so we only briefly sketch it. Roughly, the fact that $\dot {Y}^\theta _T$ contains the lifting of all of $([H^V_\theta ]^\omega )^V$ —rather than just of $([H^V_\theta ]^\omega )^V \searrow T$ as was the case with the club shooting portion of $\mathbb {Q}^{\omega _1\hbox{-}\mathrm{SSP}}_{T,\theta }$ —ensures that the iteration $\mathbb {Q}^{\mathrm{proper}}_{T,\theta }$ will indeed be proper. Now suppose $\sigma $ is $(V,\mathrm{Add}(\omega ))$ -generic and $Y^\theta _T:=(\dot {Y}^\theta _T)_\sigma $ . Let $\langle Q_i : i < \omega _1 \rangle $ be a generic filtration of $H^V_\theta [\sigma ]$ added by $\mathbb {C}^{\mathrm{fin}}_{\mathrm{dec}}(Y^\theta _T)$ over $V[\sigma ]$ . By definition of $Y^\theta _T$ , if i is such that $Q_i \cap \omega _1 = i$ and $i \in T$ , then $Q_i \cap H^V_\theta \in V$ , and hence $Q_i \cap H^V_\mu \in V$ for every V-regular $\mu \in [\omega _2^V,\theta ]$ . It follows that the internal part of any such $H^V_\mu $ contains (mod $\mathrm{NS}_{\omega _1}$ ) the set T. In particular, if T is stationary, then $H^V_\mu \in \mathrm{IS}^{V[\sigma *\vec {Q}]}$ . Now for the “moreover” clause of part 2, assume that T is costationary; we want to show that both the internal and external parts of each $H^V_\mu $ have stationary intersection with $T^c$ . That the internal part of each $H^V_\mu $ has stationary intersection with $T^c$ follows from Fact 2.13 together with the fact that $([H^V_\theta ]^\omega )^V \searrow T^c$ is, in $V[\sigma ]$ , a stationary subset of $Y^\theta _T$ (more precisely, its lifting to $H_\theta [\sigma ]$ is a stationary subset of $Y^\theta _T$ ). For the external part, the Gitik–Velickovic Theorem 2.15 ensures that, in $V[\sigma ]$ , the lifting of $( [\omega _2]^\omega \searrow T^c ) \setminus V$ is a stationary subset of $Y^\theta _T$ , and hence by Fact 2.13 there are stationarily many $i < \omega _1$ such that $Q_i \cap \omega _1 =i \in T^c$ and $Q_i \cap \omega _2^V \notin V$ . For such i, $Q_i \cap H^V_\mu \notin H^V_\mu $ for every V-regular $\mu \in [\omega _2^V,\theta ]$ .

The proof that $H^V_\theta $ is forced by $\mathbb {Q}^{\mathrm{proper}}_{T,\theta }$ to be indestructibly guessing is almost identical to the proof from part 1.⊣

Proof For part 2, the fact that $\dot {S}^{\omega _2}_{\mathrm{external}}$ is forced to be stationary follows immediately from Theorem 3.1. Also, Observation 2.2 implies that

(6) $$ \begin{align} \Vdash \ \ \forall \mu \in \mathrm{REG} \cap [\omega_2,\theta] \ \ \ \mathrm{ext}(H^V_\mu) \supseteq \mathrm{ext}(H^V_{\omega_2}) = \dot{S}^{\omega_2}_{\mathrm{external}}. \end{align} $$

Now suppose W is as in the statement of part 2, and that g is a $(W,\mathbb {Q}^{\mathrm{proper}}_{T, \theta })$ -generic filter such that $(\dot {S}^{\omega _2}_{\mathrm{external}})_g$ is a stationary subset of $\omega _1$ (in V). Let $\sigma _W:H_W \to W \prec (H_{(2^\theta )^+}, \in , T) $ be the inverse of the transitive collapse of W, and $b_W := \sigma ^{-1}_W(b)$ for every $b \in W$ . Let $\bar {g}$ be the pointwise image of g under the collapsing map. Note that $T \in W$ and T is not moved by the collapsing map. It can also be easily verified that $(\dot {S}^{\omega _2}_{\mathrm{external}})_g$ is the same as $ (\sigma _W^{-1}(\dot {S}^{\omega _2}_{\mathrm{external}}) )_{\bar {g}}$ ; let S denote this set. Fix any regular $\mu \in W \cap [\omega _2,\theta ]$ . Then by (6) and Theorem 3.1, $H_W[\bar {g}] \models $ “ $\sigma ^{-1}(H_\mu )$ is indestructibly guessing, size $\omega _1$ , its internal part contains $\sigma ^{-1}(T) = T$ , and its external part contains S.” Since $H_W[\bar {g}]$ is a transitive $\mathrm{ZF}^-$ model and V is an outer model of $H_W[\bar {g}]$ with the same $\omega _1$ , those statements are upward absolute to V. Finally, notice that the statement “X is indestructibly guessing, its internal part contains T, and its external part contains S”—viewing T and S as fixed parameters—is invariant across sets that are $\in $ -isomorphic to X. In particular, V believes the same statement about $\sigma _W [ \sigma _W^{-1}(H_\mu )]$ , which is just $W \cap H_\mu $ . This completes the proof of part 2.

The proof of 1 is similar, using instead part 1 of Theorem 3.1. The only new thing to check is that if the W-generic filter g is stationary correct, then W is diagonally reflecting for stationary subsets of $\theta \cap \mathrm{cof}(\omega )$ lying in W. To see this, note that $\theta $ is collapsed to $\omega _1$ and still has uncountable cofinality. Let $\dot {f}$ be a $\mathbb {Q}^{\omega _1-\mathrm{SSP}}_{T,\theta }$ -name for a strictly increasing, continuous, cofinal function from $\omega _1 \to \theta $ . Suppose $R \in W$ and R is stationary in $\theta \cap \mathrm{cof}(\omega )$ . Let $\dot {T}_R$ be the name for $\{ i < \omega _1 : \dot {f}(i) \in \check {R} \}$ . Since R remains stationary by Theorem 3.1, it follows that $\dot {T}_R$ is forced to be a stationary subset of $\omega _1$ , and hence its interpretation by g is really stationary in $\omega _1$ . By Viale and Weiss [Reference Viale and Weiß34], $W \cap \theta $ is an $\omega $ -closed set of ordinals. This, together with W-genericity of g, implies that $(\dot {f})_g$ maps $\omega _1$ continuously and cofinally into $\mathrm{sup}(W \cap \theta )$ . Then the pointwise image of $(\dot {T}_R)_g$ under $(\dot {f})_g$ is a stationary subset of $\mathrm{sup}(W \cap \theta )$ , and also contained in R.⊣

Observation 4.2. If $\mathrm{FA}(\Gamma )$ holds and $\mathbb {P}$ is a regular suborder of some member of $\Gamma ($ but possibly $\mathbb {P} \notin \Gamma )$ , then $\mathbb {P}$ preserves stationary subsets of $\omega _1$ . This is because, by Foreman et al. [Reference Foreman, Magidor and Shelah12], every member of $\Gamma $ must be $\omega _1$ -SSP, and this property clearly is inherited by regular suborders.

Proof of Theorem 4.5 Since this is very similar to the proof of Theorem 2.53 of Woodin [Reference Hugh Woodin18], we only briefly sketch the proof.

First assume $\mathrm{FA}^{+\mu }(\Gamma )$ . Fix $\mathbb {Q} \in \Gamma $ , $q \in \mathbb {Q}$ , a sequence $\dot {\vec {S}}=\langle \dot {S}_i : i < \mu \rangle $ of $\mathbb {Q}$ -names for stationary subsets of $\omega _1$ , and a large regular $\theta $ . Since $\Gamma $ is closed under restrictions, $\mathbb {Q} \restriction q$ is an element of $\Gamma $ , and hence the forcing axiom holds for it. Let R be the set of $W \in \wp _{\omega _2}(H_\theta )$ such that $\omega _1 \subset W$ , $\dot {\vec {S}} \in W$ , and there exists an $\dot {\vec {S}}$ -correct, $(W,\mathbb {Q} \restriction q)$ -generic filter $h_W$ . By Lemma 4.3, R is stationary. For each $W \in R$ , let $\bar {W}$ be the transitive collapse of W, and $\sigma _W: \bar {W} \to W \prec H_\theta $ be the inverse of the Mostowski collapsing map. Then $\bar {h}_W:=\sigma _W^{-1}[h_W]$ is a $(\bar {W}, \sigma _W^{-1}(\mathbb {Q}))$ -generic filter, $\bar {q}_W :=\sigma _W^{-1}(q) \in \bar {h}_W$ , $\mathrm{crit}(\sigma _W) = \omega _2^{\bar {W}}$ , and the evaluation of $\sigma _W^{-1}(\dot {S}_i)$ by $\bar {h}_W$ is a stationary subset of $\omega _1$ (in V), for all $i < \mu $ . Let $\mathcal {J}$ be the restriction of the nonstationary ideal to R. Let U be $(V,\mathcal {J}^+)$ -generic and $j: V \to _U N$ be the resulting generic elementary embedding. Then standard applications of Los’ Theorem (see Foreman [Reference Foreman, Foreman and Kanamori9]) ensure that j has the desired properties mentioned above. In particular, $[\mathrm{id}_R]_U = j[H^V_\theta ]$ , $j \restriction H^V_\theta =[W \mapsto \sigma _W]_U$ , $H^V_\theta =[W \mapsto \bar {W}]_U$ and is an element of the (transitivized) wellfounded part of N, $\mathrm{crit}(j) = \omega _2^V$ , and $H:=[ W \mapsto \bar {h}_W ]_U$ is an $(H^V_\theta ,\mathbb {Q} \restriction q)$ -generic filter such that from the point of view of the generic ultrapower N, $(\dot {S}_i)_H$ is a stationary subset of $\omega _1$ for all $i < \mu = j(\mu )$ . Since $\theta $ was chosen sufficiently large from the start, H is also V-generic.

Now we prove the converse. Assume 2, and that $\mathbb {Q} \in \Gamma $ . Fix any $\mu $ -sequence $\dot {\vec {S}} = \langle \dot {S}_i : i < \mu \rangle $ of $\mathbb {Q}$ -names for stationary subsets of $\omega _1$ . By Woodin’s Theorem 4.5 it suffices to show that if $F: [H_\theta ]^{<\omega } \to H_\theta $ with $F \in V$ , then there is a $W \in \wp _{\omega _2}(H_\theta )$ that is closed under F such that $\omega _1 \subset W$ and there exists an $\dot {\vec {S}}$ -correct, $(W,\mathbb {Q})$ -generic, $\omega _1$ -stationary correct filter. Let $j: V \to N$ and $H \in N$ be as in assumption 2. Now $j[H^V_\theta ]$ is closed under $j(F)$ , and $j[H^V_\theta ] \in N$ by assumption. Also note that $j[H] \in N$ , since $H \in N$ , $j \restriction H^V_\theta \in N$ , and $\mathbb {Q} \in H^V_\theta $ . It is routine to check that $j[H^V_\theta ]$ and $j[H]$ witness that $N \models $ “there is a model containing $\omega _1$ and closed under $j(F)$ for which there exists a $j(\dot {\vec {S}})=\langle j(\dot {S}_i) : i < \mu = j(\mu ) \rangle $ -correct, generic filter for $j(\mathbb {Q}).$ ” Then elementarity of j yields the analogous statement in V, completing the proof.⊣

Proof of Theorem 1.20 In order to show that $\mathrm{FA}^{+\mu }(\dot {\Delta })$ holds in $V^{\mathbb {P}}$ , it suffices to verify that clause 2 of Theorem 4.5 holds in $V^{\mathbb {P}}$ (with respect to the class $\dot {\Delta }$ ). More precisely, we prove that $1_{\mathbb {P}}$ forces that “whenever $\mathbb {Q}$ is a poset in $\dot {\Delta }$ , $q \in \mathbb {Q}$ , and $\langle \dot {S}_i : i < \mu \rangle $ is a $\mu $ -sequence of $\mathbb {Q}$ -names for stationary subsets of $\omega _1$ , then there exists a generic elementary embedding

$$ \begin{align*} \pi: V[\dot{G}] \to M \end{align*} $$

(where $\dot {G}$ is the $\mathbb {P}$ -name for its generic) such that $\pi \restriction H^{V[\dot {G}]}_\theta \in M$ , $\mathrm{crit}(\pi ) = \omega _2^{V[\dot {G}]}$ , $H^{V[\dot {G}]}_\theta $ is in the wellfounded part of M, and there is an $H \in M$ that is $(V[\dot {G}],\mathbb {Q})$ -generic such that $q \in H$ and for every $i < \mu $ , $M \models (\dot {S}_i)_{H}$ is a stationary subset of $\omega _1$ .”

So let p be an arbitrary condition in $\mathbb {P}$ , $\dot {\mathbb {Q}}$ a $\mathbb {P}$ -name for a poset in $\dot {\Delta }$ , $\dot {q}$ a $\mathbb {P}$ -name for a condition in $\dot {\mathbb {Q}}$ , and $\dot {\vec {S}} = \langle \dot {S}_i : i < \mu \rangle $ a $\mu $ -sequence of $\mathbb {P}*\dot {\mathbb {Q}}$ -names for stationary subsets of $\omega _1$ . Let $\dot {\mathbb {R}}$ be as in the hypotheses of Theorem 1.20. Fix a regular $\theta $ such that all objects mentioned so far are in $H_\theta $ . Note that by the assumptions about $\dot {\mathbb {R}}$ , each $\dot {S}_i$ remains stationary in $V^{\mathbb {P}*\dot {\mathbb {Q}}*\dot {\mathbb {R}}}$ ; so $\dot {\vec {S}}$ can be regarded as a sequence of $\mathbb {P}*\dot {\mathbb {Q}}*\dot {\mathbb {R}}$ -names of stationary subsets of $\omega _1$ . Since $\mathbb {P}*\dot {\mathbb {Q}}*\dot {\mathbb {R}}$ is an element of $\Gamma $ by assumption, Theorem 4.5 ensures that there is, in some generic extension W of V, a generic elementary embedding $j: V \to N$ such that

1. (1) $H^V_{(2^\theta )^+}$ is in the wellfounded part of N, and has size $\omega _1$ in N;

2. (2) $j \restriction H^V_{(2^\theta )^+} \in N$ ;

3. (3) $\mathrm{crit}(j) = \omega _2^V$ ;

4. (4) There is a $G*H*K \in N$ that is V-generic for $\mathbb {P}*\dot {\mathbb {Q}}*\dot {\mathbb {R}}$ , and such that $(p,\dot {q}, \dot {1}) \in G*H*K$ and $N \models $ “ $(\dot {S}_i)_{G*H*K}=(\dot {S}_i)_{G*H}$ is a stationary subset of $\omega _1$ , for all $i < \mu $ .”

Note that $j[G] \in N$ because $G \in N$ and $j \restriction H^V_{(2^\theta )^+} \in N$ . So $j:V \to N$ and $G*H*K$ satisfy all the hypothesis in the “if-then” clause 3 of the theorem we are currently proving. So by that clause, there is some $p' \in j(\mathbb {P})$ that N believes is a lower bound of $j[G]$ . Now $p'$ may be in the illfounded part of N, but this will not matter. Recall that W is the generic extension of V where the embedding $j: V \to N$ resides. Work in W for the moment. Then even if N is illfounded, the fact that $j(\mathbb {P})$ is a partial order is upward absolute to W, and W-generics are also N-generics. More precisely, in W consider the $\in _N$ -extension $\mathcal {P}'$ of $j(\mathbb {P})$ —i.e., $\mathcal {P}':= \{ x \in N : x \in ^N j(\mathbb {P}) \}$ —and order $\mathcal {P}'$ by $x \le y$ iff $N \models x \le _{j(\mathbb {P})} y$ ; it is routine to check that $\mathcal {P}'$ is a partial order in W. Choose a $(W,\mathcal {P}')$ -generic $G'$ such that $p' \in G'$ . Then again it is routine to see that $G'$ is also $ ( W, j(\mathbb {P}) )$ -generic, in the sense that for every $D \in N$ such that $N \models $ “D is dense in $j(\mathbb {P})$ ” there is some $q \in G'$ such that $q \in ^N D$ .

Now consider the generic extension $N[G']$ . A word is in order about what is meant by $N[G']$ , since the standard recursive definition of name-evaluation by a generic does not make sense when the ground model is illfounded. One way to make sense of $N[G']$ is via quotients of Boolean-valued models.Footnote ¹¹ In the current context, let $\mathbb {B}:= \mathrm{ro}^N ( j(\mathbb {P}) )$ and consider the quotient $N^{\mathbb {B}}/ G'$ of the $\mathbb {B}$ -valued model $N^{\mathbb {B}}$ (see Hamkins and Seabold [Reference Hamkins and Seabold17]). Then since $G'$ is generic over N, Lemma 13 and Theorem 1.16 of [Reference Hamkins and Seabold17] ensure that there is a definable isomorphic copy of N inside $N^{\mathbb {B}}/ G'$ —which we identify with N—such that $N^{\mathbb {B}}/G' \models $ “I am the forcing extension of N by $G'$ .” Viewed in this way, $N[G']$ is simply $N^{\mathbb {B}}/G'$ , and if $\tau $ is a $j(\mathbb {P})$ -name in N, then the name evaluation $\tau _{G'}$ makes sense if computed from the point of view of $N^{\mathbb {B}}/ G'$ (which believes that N is wellfounded).

Since $p' \in G'$ and $p'$ was stronger than every element of $j [G]$ , elementarity of $j:V \to N$ ensures that $j[G] \subseteq G'$ . Then the standard argument shows that the map

$$ \begin{align*} \hat{j}:V[G] \to N[G'] \end{align*} $$

defined by $\sigma _G \mapsto j(\sigma )_{G'}$ is well-defined and elementary (here $j(\sigma )_{G'}$ is computed in $N^{\mathbb {B}}/ G' =N[G']$ , which makes sense as described above). Note that $\hat {j}$ is definable in the extension $W[G']$ , which (since $G \in W$ and W was a generic extension of V) is a generic extension of $V[G]$ .

Next we observe that $\mathbb {P}$ must have been $< \omega _2$ -distributive; this will guarantee that $\mathrm{crit}(\hat {j}) = \mathrm{crit}(j) = \omega _2^V = \omega _2^{V[G]}$ . If $\mathbb {P}$ were not $<\omega _2$ -distributive, then there would be some name $\dot {\vec {\alpha }} = \langle \dot {\alpha }_i : i < \omega _1^V \rangle \in H^V_\theta $ for a new $\omega _1^V$ -length sequence of ordinals, and hence $j(\dot {\vec {\alpha }}) = \langle j(\dot {\alpha }_i) : i < \omega _1^V \rangle $ would be a $j(\mathbb {P})$ -name for a new $j(\omega _1^V) = \omega _1^V$ -length sequence of ordinals. Now N sees that the condition $p'$ extends the $ (j[H^V_\theta ], j(\mathbb {P}) )$ -generic filter $j[G]$ , and hence that $p'$ is a total master condition for the model $j[H^V_\theta ]$ (i.e., for every $D \in j[H^V_\theta ]$ such that D is dense in $j(\mathbb {P})$ , there is a condition weaker than $p'$ in $D \cap j[H^V_\theta ]$ ). It follows that $p'$ already decides all values from the sequence $\langle j(\dot {\alpha }_i) : i < \omega _1^V \rangle $ . Since $p' \in G'$ , $N[G']$ sees that $\hat {j}(\dot {\vec {\alpha }}_G) = (j(\dot {\vec {\alpha }}))_{G'}$ is already an element of the ground model N. By elementarity of $\hat {j}$ , $V[G]$ sees that $\dot {\vec {\alpha }}_G$ is already an element of the ground model V, which is a contradiction.

Let $\mathbb {Q}:=\dot {\mathbb {Q}}_G$ . Since $j \restriction H^V_\theta \in N$ and $G*H \in N$ , it follows that $H \in N[G']$ and

$$ \begin{align*} \hat{j} \restriction ( H^V_\theta[G]) = \langle \sigma_G \mapsto j(\sigma)_{G'} : \sigma \in H^V_\theta \cap N^{j(\mathbb{P})} \rangle \end{align*} $$

is an element of $N[G']$ . Furthermore, $q:=\dot {q}_G$ is an element of H, by (4).

Finally, we need to verify that from the point of view of $N[G']$ , each $(\dot {S}_i)_{G*H}$ is stationary. Indeed, each $(\dot {S}_i)_{G*H}$ is stationary from the point of view of N, by (4). Furthermore, since $\mathbb {P}$ is a regular suborder of $\mathbb {P}*\dot {\mathbb {Q}}*\dot {\mathbb {R}}$ and the latter is in $\Gamma $ , Observation 4.2 ensures that $\mathbb {P}$ is $\omega _1$ -SSP from the point of view of V. By elementarity of j, $j(\mathbb {P})$ is $\omega _1$ -SSP from the point of view of N. Hence, each $(\dot {S}_i)_{G*H}$ remains stationary in $N[G']$ .⊣