2.1 Special trees

Let $T=(T,\leq _T)$ be a tree of height $\omega _1$ . Recall that T is called special if there is a function $f:T\to \omega $ , such that for every $s<_T t$ in T, we have $f(s)\neq f(t)$ . If T is a tree of height $\omega _1$ , the standard forcing $\mathbb S(T)$ to generically specialize T consists of finite partial specializing functions, ordering by reverse inclusion. If T has no cofinal branches then $\mathbb S(T)$ has the countable chain condition, see [Reference Baumgartner, Malitz and Reinhardt2]. We say that T is weakly special if there is a function $f:T \to \omega $ such that whenever $s,t,u\in T$ and $f(s)=f(t)=f(u)$ , if $s<_T t$ and $s<_Tu$ then t and u are comparable. Baumgartner [Reference Baumgartner and Mathias3, Theorem 7.3] proved that for a tree T of height $\omega _1$ which has no branches of length $\omega _1$ , T is special iff T is weakly special. He also showed [Reference Baumgartner and Mathias3, Theorem 7.5] that if every tree of height and size $\omega _1$ that has no branches of length $\omega _1$ is special, then every tree of height and size $\omega _1$ that has at most $\omega _1$ branches is weakly special. We recall the following well-known fact, see [Reference Cox and Krueger6, Proposition 4.3] for a proof.

2.2 Guessing models

Throughout this paper by a model M, we mean a set or a class such that $(M,\in )$ satisfies a sufficient fragment of $\mathrm {ZFC}$ . For a set or class M, we say that a set $x\subseteq M$ is bounded in M if there is $y\in M$ such that $x\subseteq y$ . Let us call a transitive model R powerful if it includes every set bounded in R. We say that x is guessed in M if there is $x^*\in M$ with $x^*\cap M=x\cap M$ . Suppose that $\gamma $ is an M-regular cardinal. We say x is $\gamma $ -approximated in M if for every $a\in M$ with $|a|^M<\gamma $ , we have $a\cap x\in M$ . We say that M is $\gamma $ -guessing in a model N if for every $x\in N$ bounded in M, if x is $\gamma $ -approximated in M, then x is guessed in M.

Recall from [Reference Hamkins9] that a pair $(M,N)$ of transitive models with $\gamma \in M\subseteq N$ has the $\gamma $ -approximation property, if M is $\gamma $ -guessing in N.

2.3 Indestructible and special guessing models

The notion of an indestructible guessing model was introduced by Cox and Krueger [Reference Cox and Krueger6].

Let us first recall that a set X of size $\omega _1$ is internally unbounded if $X\cap \mathcal P_{\omega _1}(X)$ is cofinal in $\mathcal P_{\omega _1}(X)$ , and is internally club if $X\cap \mathcal P_{\omega _1}(X)$ contains a closed unbounded subset of $\mathcal P_{\omega _1}(X)$ . It is easily seen that if M is an $\omega _1$ -sized elementary submodel of a powerful model R, then M is internally unbounded (I.U.) if and only if there is an I.U. sequence for M, i.e., a $\subseteq $ -increasing and $\in $ -sequence $\langle M_\alpha :\alpha <\omega _1\rangle $ of countable sets with union M, and that M is internally club (I.C.) if and only if there is an I.C. sequence for M, i.e., a $\subseteq $ -continuous I.U. sequence for M.

It was shown by Krueger [Reference Krueger10] that if R is a powerful model and $M\prec R$ with $\omega _1\subseteq M$ is an $\omega _1$ -guessing model of size $\omega _1$ , then M is internally unbounded.

Let us fix an I.U. sequence $\langle M_\alpha :\alpha <\omega _1\rangle $ for an I.U. $\omega _1$ -guessing model M of size $\omega _1$ . For every $\alpha <\omega _1$ , let

Let also

$$\begin{align*}T(M)=\bigcup_{\alpha<\omega_1}T_\alpha(M). \end{align*}$$

We order $T(M)$ by $(z,f)\leq (z',f')$ if and only if $z=z'$ and $f\subseteq f'$ .

Notice that $T(M)\subseteq M$ , and hence $|T(M)|\leq \omega _1$ . Clearly $T(M)$ is a tree of height $\omega _1$ . Assume that b is an uncountable branch through $T(M)$ . Then there is $z\in M$ such that for all $(z',f')\in b$ , we have $z'=z$ . Let

$$\begin{align*}F= \bigcup \{ f : (z,f) \in b \}. \end{align*}$$

Then $F:z \cap M \mapsto 2$ is a function which is $\omega _1$ -approximated in M. Since M is an $\omega _1$ -guessing model, there is $F^*\in M$ with $F\restriction (z\cap M)=F^*\restriction (z\cap M)$ . Given $F^*$ , we can recover the branch b, so $T(M)$ has at most $\omega _1$ branches. By Proposition 2.1 we now have the following [Reference Cox and Krueger6, Proposition 4.4].

2.4 Strong properness

A forcing notion $\mathbb P$ is called strongly proper, if for every sufficiently large regular cardinal $\theta $ there is a club $\mathcal C$ in $\mathcal P_{\omega _1}(H_\theta )$ such that $\mathbb P$ is strongly proper for $\mathcal C$ .

The following are standard.

2.5 Magidor models

The following definition is motivated by Magidor’s reformulation of a supercompact cardinal, see [Reference Magidor12].

2.6 Some lemmata

We need the following.

Lemma 2.15. Let $\mathbb P\in H_\theta $ be a forcing with the $\omega _1$ -approximation property. Assume that $\mathcal {P}(\mathbb P)\in H_\theta $ . Suppose $M\prec H_\theta $ is countable and $\mathbb P\in M$ . Let $Z\in M$ and let $f:M\cap Z\rightarrow 2$ be a function that is not guessed in M. If $p\in \mathbb P$ is $(M,\mathbb P)$ -generic, then

$$ \begin{align*}p\Vdash \text{"}\check{f} \text{ is not guessed in } M[\dot{G}].\text{"}\end{align*} $$

Proof. Suppose that $h\in M$ is a function so that $h\restriction M\cap Z=g$ , we will show that $p\Vdash "\dot f=\check h"$ . Fix $q\leq p$ . We claim that there is a condition below q that forces $\dot f=\check h$ . If not, the set

$$ \begin{align*}D=\{r\in\mathbb P:\exists\zeta\in Z \text{ such that } r\Vdash \check h(\zeta)\neq \dot f(\zeta)\}\end{align*} $$

that belongs to M is pre-dense below q. Since q is $(M,\mathbb P)$ -generic, there is $r\in D\cap M$ compatible with q. So there exists $\zeta \in M\cap Z$ such that $r\Vdash "\dot f(\zeta )\neq \check h(\zeta )=\check g(\zeta )"$ . This is a contradiction, since r is compatible with q and q forces $\dot f(\zeta )=\check g(\zeta )$ .

Proof. For every $\zeta \in Z$ , let

$$ \begin{align*}A_\zeta^\epsilon=\{z\in B: f_z(\zeta)=\epsilon\}, \text{ where } \epsilon=0,1.\end{align*} $$

Note that the sequence

$$ \begin{align*}\langle A_\zeta^\epsilon:\zeta\in Z,\epsilon\in\{0,1\}\rangle\end{align*} $$

belongs to M. We are done if there is some $\zeta \in Z$ such that both $A_\zeta ^0$ and $A_\zeta ^1$ are cofinal in B, as then one can find by elementarity such $\zeta \in M\cap Z$ and let $B^*=A_\zeta ^{1-f(\zeta )}$ . Therefore, let us assume that for every $\zeta \in Z$ , there is a unique $\epsilon \in \{0,1\}$ such that $A_\zeta ^\epsilon $ is cofinal in B. Now define h on $ Z$ by letting $h(\zeta )$ be $\epsilon $ if and only if $A_\zeta ^\epsilon $ is cofinal in $[Z]^\omega $ . Obviously h belongs to M. But $h\restriction {M\cap Z}\neq f$ , since f is not guessed in M. So there must exist $\zeta \in M\cap Z$ such that $h(\zeta )\neq f(\zeta )$ , which in turn implies that $A_\zeta ^{1-f(\zeta )}$ is a cofinal subset of B and belongs to M. Let $B^*$ be $A_\zeta ^{1-f(\zeta )}$ .

It is well-known that if T has no uncountable branches, then $\mathbb S(T)$ has the ccc. We also need the following fact which was shown by Chodounský–Zapletal [Reference Chodounský and Zapletal4]. We provide a proof for completeness.

4.1 The general setting

We consider the language $\mathcal L$ obtained by adding a single constant symbol c and a predicate U to the standard language of set theory. Let us say that an $\mathcal L$ -structure $\mathcal A=(A,\in , \kappa , U)$ is suitable if A is a transitive set and $\mathcal A\models "{\mathrm {ZFC}}"$ in the expanded language, where $\kappa =c^{\mathcal A}$ is an uncountable regular in $\mathcal A$ . For a suitable structure $\mathcal A$ and an ordinal $\alpha \in A\setminus \kappa $ , let $\mathcal A_\alpha =(A\cap V_\alpha ,\in ,\kappa , U\cap V_\alpha )$ . Finally, we let

$$\begin{align*}E_{\mathcal A}=\{ \alpha \in \mathrm{{ORD}}^A : \mathcal A_\alpha \prec \mathcal A \}. \end{align*}$$

Note that $E_A$ is a closed subset of $\mathrm {{ORD}}^{\mathcal A}$ . It is not definable in $\mathcal A$ , but $E_{\mathcal A}\cap \alpha $ is uniformly definable in $\mathcal A$ with parameter $\alpha $ , for each $\alpha \in E_{\mathcal A}$ . For $\alpha \in E_{\mathcal A}$ , we let $\mathrm {next}_{\mathcal A}(\alpha )$ be the least ordinal in $E_{\mathcal A}$ above $\alpha $ , if such an ordinal exists. Otherwise, we leave $\mathrm {next}_{\mathcal A}(\alpha )$ undefined. We shall often abuse notation and refer to the structure $\mathcal A$ by A.

If M is a submodel of a suitable structure A and X is a subset of A, let the operation $\mathrm {Hull}$ be defined by

$$\begin{align*}\mathrm{Hull}(M,X) =\{ f(\bar{x}) : f\in M, \bar{x}\in X^{<\omega}, f \mbox{ is a function, and } \bar{x}\in \mathrm{{dom}} (f)\}. \end{align*}$$

For $\alpha \in E_A$ , we let ${\mathscr A}_\alpha $ denote the set of all transitive $\mathcal L$ -structures $\mathcal H\in A$ which are elementary extensions of $\mathcal A_\alpha $ and have the same cardinality as $A_\alpha $ . Note that if $\mathcal H \in \mathscr A_\alpha $ and $\alpha \in \mathcal H$ , then $E_{\mathcal H}\cap \alpha =E_A\cap \alpha $ . For $\mathcal H\in \mathscr A_\alpha $ , we will refer to $A_\alpha $ as the standard part of $\mathcal H$ . Note that if $\mathcal H$ has nonstandard elements, then $\alpha \in E_{\mathcal H}$ .

4.2 Virtual models in $V_\lambda $

Assume that $\kappa <\lambda $ are supercompact and inaccessible cardinals, respectively. Consider the $\mathcal L$ -structure

Let $E=E_{V_\lambda }$ and $\mathrm {next}(\alpha )=\mathrm {next}_{V_\lambda }(\alpha )$ .

We write $\mathscr {V}_{< \alpha }$ for $\bigcup \{\mathscr {V}_\gamma : \gamma \in E \cap \alpha \}$ and $\mathscr V$ for $\mathscr {V}_{< \lambda }$ . The collections $\mathscr {V}_{\leq \alpha }$ and $\mathscr {V}_{\geq \alpha }$ are defined in the obvious way. For $M\in \mathscr V$ , $\eta (M)$ denotes the unique ordinal $\alpha $ such that $M\in \mathscr V_\alpha $ . Note that if $M \in \mathscr V_\alpha $ , then $\sup (M \cap \mathrm {{ORD}})\geq \alpha $ . In general, M is not elementary in $V_\lambda $ , in fact, this only happens if $M \subseteq V_\alpha $ . In this case, we will say that M is a standard $\alpha $ -model. We refer to the members of $\mathscr V$ as virtual models. We also refer to members of $\mathscr V^A$ , for some suitable structure A with $A\subseteq V_\lambda $ , as general virtual models.

Suppose $M,N\in \mathscr V$ and $\alpha \in E$ . We say that an isomorphism $\sigma :M\rightarrow N$ is an $\alpha $ -isomorphism if it has an extension to an isomorphism $\bar {\sigma }:\mathrm {Hull}(M,V_\alpha )\rightarrow \mathrm {Hull}(N,V_\alpha )$ . We say that M and N are $\alpha $ -isomorphic and write $M\cong _{\alpha }N$ if there is an $\alpha $ -isomorphism between them. Note that $\sigma $ and $\bar \sigma $ , if they exist, are unique.

It is clear that $\cong _{\alpha }$ is an equivalence relation, for every $\alpha \in E$ . If $\alpha <\beta $ are in E, then for every $\beta $ -model M there is a canonical representative of the $\cong _\alpha $ -equivalence class of M which is an $\alpha $ -model.

Note that if $A\in \mathscr A_\alpha $ then $\mathscr V_\alpha ^A \subseteq \mathscr V_\alpha $ . Therefore, if $A,B\in \mathscr A_\alpha $ , $M \in \mathscr V^{A}$ , and $N \in \mathscr V^B$ , we can still write $M \cong _\alpha N$ if $M \restriction \alpha = N \restriction \alpha $ . It is straightforward to check that if $\alpha \leq \beta $ are in E, and $M\in \mathscr V$ . Then $(M \restriction \beta ) \restriction \alpha = M \restriction \alpha $ . For each $\alpha \in E$ , the virtual version of the membership relation is defined as follows.

Suppose $M,N\in \mathscr V$ and $\alpha \in E$ . We write $M\in _\alpha N$ if there is $M'\in N$ with $M' \in \mathscr V^N$ such that $M'\cong _\alpha M$ . If this is the case, we say that M is $\alpha $ -in N.

The collections $\mathscr C_{<\alpha }$ , $\mathscr C_{\leq \alpha }$ , $\mathscr C_{\geq \alpha }$ , $\mathscr U_{<\alpha }$ , $\mathscr U_{\leq \alpha }$ , and $\mathscr U_{\geq \alpha }$ are defined similarly. We write $\mathscr C$ for $\mathscr C_{<\lambda }$ , and $\mathscr U$ for $\mathscr U_{< \lambda }$ , $\mathscr C_{\mathrm {st}}$ for the collection of standard models in $\mathscr C$ , and $\mathscr U_{\mathrm {st}}$ for the standard models in $\mathscr U$ . Note that both classes $\mathscr C$ and $\mathscr U$ are closed under projections.

One can easily see that a Magidor model is active at $\alpha $ if and only if it is strongly active at $\alpha $ .

Note that $a(M)$ is a closed subset of E of size at most $|\mathrm {Hull}(M,V_{\kappa _M})|$ . We now recall the definition of a meet from [Reference Mohammadpour and Velicković15] which is the virtual version of intersection defined only for two models of different types. Suppose $N\in \mathscr U$ and $M\in \mathscr C$ . Let $\overline {N}$ be the transitive collapse of N, and let $\pi $ be the collapsing map. Note that if $\overline {N}\in M$ , then $\overline {N}\cap M$ is a countable elementary submodel of $\overline {N}$ . Then $\overline {N}\cap M \in \overline {N}$ since $\overline {N}$ is closed under countable sequence. Note that $\pi ^{-1}(\overline {N}\cap M)= \pi ^{-1}[\overline {N}\cap M]$ , and this model is elementary in N.

Definition 4.10. Suppose $N\in \mathscr U$ and $M \in \mathscr C$ . Let $\alpha = \max (a(N)\cap a(M))$ . We will define $N \land M$ if $N\in _\alpha M$ . Let $\overline {N}$ be the transitive collapse of N, and let $\pi $ be the collapsing map. Set

$$\begin{align*}\eta = \sup (\sup (\pi^{-1}[\overline{N}\cap M]\cap \mathrm{{ORD}})\cap E\cap (\alpha +1)). \end{align*}$$

We define the meet of N and M to be $N\land M=\pi ^{-1}[\overline {N}\cap M]\restriction \eta $ .

Notation 4.12. Let $\alpha \in E$ and let $\mathcal M$ be a set of virtual models. We let

$$\begin{align*}\mathcal M\restriction\alpha=\{M\restriction\alpha:M\in\mathcal M\} \mbox{ and } \mathcal M^\alpha=\{M\restriction\alpha:~M\in \mathcal M \text{ is active at } \alpha\}. \end{align*}$$

5.1 The iteration

For a condition $\mathcal M_p\in \mathbb M^\kappa _\lambda $ and an ordinal $\gamma \in E$ , we let

In other words, $\mathcal M_p(\gamma )$ consists of those models in $\mathcal M_p{\restriction {\mathrm {next}(\gamma )}}$ that are strongly active at $\mathrm { next}(\gamma )$ .

Let $E^+= E \cup \{ \alpha + 1: \alpha \in E\}$ . We will define by induction the poset $\mathbb Q^\kappa _\alpha $ , for $\alpha \in E^+\cup \{ \lambda \}$ . Let $\mathrm {Fn}(\omega _1,\omega )$ denote the poset of finite partial functions from $\omega _1$ to $\omega $ , ordered under reverse inclusion. Conditions in $\mathbb Q^\kappa _\alpha $ will be triples of the form $p= (\mathcal M_p,d_p,w_p)$ , where $(\mathcal M_p,d_p) \in \mathbb P^\kappa _\alpha $ , and $w_p$ is a finite function with $\mathrm {dom}(w_p)\subseteq E\cap \alpha $ , such that $w_p(\gamma )\in \mathrm {Fn}(\omega _1,\omega )$ , for all $\gamma \in \mathrm {dom}(w_p)$ . If p is such a triple and $\gamma <\alpha $ is in E, we let $p\restriction \gamma $ denote the triple $(\mathcal M_p\restriction \gamma , d_p \restriction \gamma , w_p \restriction \gamma )$ , where $(\mathcal M_p \restriction \gamma , d_p\restriction \gamma )$ is defined as in $\mathbb P^\kappa _\alpha $ and $w_p \restriction \gamma $ is the restriction of $w_p$ to $\mathrm {dom}(w_p)\cap \gamma $ .

Let us recall that we are working with a suitable structure $(V_\lambda ,\in ,\kappa ,U)$ . To be precise, $U:\lambda \rightarrow V_\lambda $ is a (bookkeeping) function that we regard it as a binary predicate. Thus let us assume that U is onto and for every $x\in V_\lambda $ , the set $\{\alpha <\lambda : U(\alpha )=x\}$ is unbounded.

Note that if $\alpha = \mathrm {min}(E)$ then $\mathbb Q^\kappa _\alpha = \mathbb P^\kappa _\alpha $ . Also, for $\alpha \in E$ , $\mathbb Q^\kappa _{\alpha +1}$ is isomorphic to $\mathbb Q^\kappa _\alpha \ast \dot {\mathbb S}(\dot {T}_\alpha )$ , if $U(\alpha )$ has the right form, otherwise $\mathbb Q^\kappa _{\alpha +1}\cong \mathbb Q^\kappa _\alpha $ . For $p \in \mathbb Q^\kappa _\lambda $ and $\alpha \in E$ , we let

$$\begin{align*}p \restriction (\alpha +1) = (\mathcal M_p \restriction \alpha, d_p\restriction \alpha, w_p \restriction (\alpha+1)). \end{align*}$$

The order is transitive and whenever $p\in \mathbb Q^\kappa _\lambda $ and $\alpha \in E^+$ , then $p \restriction \alpha \in \mathbb Q^\kappa _\alpha $ . Moreover, if $p \leq q$ then $p \restriction \alpha \leq q\restriction \alpha $ .

Proof. Let $(\mathcal M_r,d_r)$ be $(\mathcal M_p,d_p)\land (\mathcal M_q,d_q) $ as defined in Lemma 5.3. Let also $w_r$ be defined on $\mathrm {dom}(w_p)\cup \mathrm {dom}(w_q)$ by

$$ \begin{align*} w_r(\gamma)=\left\{ \begin{array}{cl} w_q(\gamma) & \textrm{if }\gamma<\alpha,\\ w_p(\gamma) & \textrm{if }\gamma\geq\alpha. \end{array}\right. \end{align*} $$

It is evident that r is a condition which extends p and q.

The following corollary is immediate.

We need the following lemma in several proofs.

Proof. First, by Proposition 5.12, we have that $(\mathcal M_q, d_q)$ and $(\mathcal M_p,d_p)$ are compatible in $\mathbb P^\kappa _\alpha $ and the meet $(\mathcal M_q, d_q) \wedge (\mathcal M_p,d_r)$ exists. Let us denote this meet by $(\mathcal M,d)$ . Let us also fix $r\in G_{\beta +1}$ extending $p\restriction {\beta +1}$ and $q\restriction {\beta +1}$ . Let $(\mathcal M_s,d_s)$ be the meet $(\mathcal M,d) \wedge (\mathcal M_r,d_r)$ as defined in Lemma 5.3. We now define $w_s$ on $\mathrm {dom}(w_r) \cup \mathrm {dom}(w_p) \cup \mathrm {dom}(w_q)$ by letting:

$$ \begin{align*} w_s(\gamma)=\begin{cases} w_r(\gamma) & \mbox{if } \gamma \in \mathrm{dom}(w_r) \\ w_q(\gamma) & \mbox{if } \gamma\in \mathrm{dom}(w_q)\setminus \mathrm{dom}(w_r) ,\\ w_p(\gamma) & \mbox{if } \gamma \in \mathrm{dom}(w_p) \setminus ( \mathrm{dom}(w_r) \cup \mathrm{dom}(w_q) ). \end{cases} \end{align*} $$

It is easy to see that $s=(\mathcal M_s,d_s,w_s)$ is a condition in $\mathbb Q^\kappa _\alpha $ extending p and q.

As in [Reference Mohammadpour and Velicković15, Lemma 5.2] we have the following.

Let $\alpha \in E^+$ . Assume that $G_\alpha $ is a V-generic filter on $\mathbb Q^\kappa _\alpha $ . One can form the quotient forcing $\mathbb Q^\kappa _\lambda /G_\alpha $ . Suppose that also $M\in \mathscr C_{>\alpha }$ , $M\restriction \alpha \in \mathcal M^\alpha _{G_\alpha }$ , and $p\in M\cap \mathbb Q^\kappa _\lambda /G_\alpha $ . It is then easily seen that $p^M$ belongs to $\mathbb Q^\kappa _\lambda /G_\alpha $ . By an argument, as in Proposition 5.24, we have the following.

Suppose $\alpha \in E^+$ and let $G_\alpha $ be V-generic over $\mathbb Q^\kappa _\alpha $ . In $V[G_\alpha ]$ fix a large regular cardinal $\lambda ^*$ and let $\mathscr S$ be the collection of all countable models of the form $M[G_\alpha ]$ , where $M\prec H_{\lambda ^*}^V$ contains the relevant objects, and $M \cap V_\lambda \in \mathscr C_{\mathrm {st}}[G_\alpha ]$ . Now, by Propositions 5.29 and 5.30 we have the following.

Putting everything together, we have the following.

One can use Proposition 5.30 and the proof of Proposition 5.34 to prove the following.

The following corollary is immediate.

5.2 Quotients by Magidor models

The first lemma of this subsection states that $\mathbb Q^\kappa _\lambda $ is strongly proper for $\mathscr U$ in a canonical way.

Definition 5.37. Let $p\in \mathbb Q^\kappa _\alpha $ . Assume $N\in \mathcal L(\mathcal M_p)$ is a Magidor model. We let

$$\begin{align*}p\restriction N=(\mathcal M_{p\restriction N},d_{p\restriction N},w_p\restriction N). \end{align*}$$

Proof. We prove both items simultaneously by induction on $\alpha $ . Note that both are true for the minimum of E. Thus let us assume that $\alpha>\min (E)$ . Note that $p\restriction N$ belongs to N. So all we have to show is that $p\restriction N$ is a condition in $\mathbb Q^\kappa _\alpha $ that satisfies the second item.

One needs first to show that for every $\gamma \in \mathrm {{dom}}(p)\cap N$ , $(p\restriction N)\restriction \gamma $ , which is a condition by the inductive hypothesis and Lemma 5.38, forces in ${\mathbb Q^\kappa _\gamma }$ that $"\check {w}_p(\gamma ) \in \dot {\mathbb S}(\dot {T}_\gamma )"$ . Assume towards a contradiction that this is not the case, so there is $q\in {\mathbb Q^\kappa _\gamma }$ with $q\leq (p\restriction N)\restriction \gamma $ which forces $\check {w}_p(\gamma )$ is not in $\dot {\mathbb S}(\dot {T}_\gamma )"$ . Since both $p\restriction N\restriction \gamma $ and $w_p(\gamma )$ belong to N, we can find such q in N by elementarity. Therefore, q is not compatible with $p\restriction \gamma $ . This contradicts the second item above applied to $p\restriction \gamma ,N\restriction \gamma $ , and q. Hence the first item follows.

To see the second item, we will define a common extension of $p,q$ , say r by letting $(\mathcal M_r,d_r)=(\mathcal M_p,d_p)\land (\mathcal M_q,d_q)$ , $\mathrm {{dom}}(w_r)=\mathrm {dom}(w_p)\cup \mathrm {dom}(w_q)$ , and that

$$ \begin{align*} w_r(\gamma)=\begin{cases} w_q(\gamma) & \mbox{if } \gamma\in N, \\ w_p(\gamma) & \mbox{if } \gamma\notin N. \end{cases} \end{align*} $$

It follows from Proposition 5.9 that r is a common extension of p and q.

Suppose that $N \in \mathscr U$ and $\alpha \in E$ . We set

By Lemma 5.39, the condition is $(N,\mathbb Q^\kappa _\alpha )$ -strongly generic, and therefore the mapping

$$ \begin{align*} \begin{aligned} \mathbb Q^\kappa_{\alpha,N}&\longrightarrow \mathbb Q^\kappa_\alpha\restriction N\\ p&\longmapsto p^{N\restriction \alpha} \end{aligned} \end{align*} $$

is a complete embedding. Moreover, if p is a condition in $\mathbb Q^\kappa _\alpha \restriction N$ , then $p\restriction N\in \mathbb Q^\kappa _{\alpha ,N}$ is such that if $q\in \mathbb Q^\kappa _{\alpha ,N}$ extends $p\restriction N$ , then p and q are compatible, and indeed the meet $p\land q$ exists.

For a V-generic filter $G_{\alpha ,N}\subseteq \mathbb Q^\kappa _{\alpha }\cap N$ we can form the following quotient forcing