Assumptions throughout the paper 3.1. Throughout the rest of the paper, suppose that

• • $2^{\aleph _0}=\aleph _1$ holds,

• • $\mathbb {R}$ stands for the set of real numbers, and $\vec {\mathbb {R}}$ is a fixed enumeration of $\mathbb {R}$ ,

• • X is a non-meager set of injective functions from $\omega $ into $2^{<\omega }$ (of size $\aleph _1$ ),

• • $\kappa $ is an uncountable regular cardinal such that $\kappa \geq \aleph _2$ and $2^{<\kappa }=\kappa $ ,

• • $\Phi $ is a surjection from $\kappa $ to $H_\kappa $ such that for every $x\in H_\kappa $ , $\Phi ^{-1}[\{x\}]$ is unbounded in $\kappa $ .

Assumptions throughout the paper 3.2.

• • $\vec {X} = \left \langle {f_\delta : \delta \in \omega _1}\right \rangle $ is an enumeration of X such that, for any countable elementary submodel M of $H_\kappa $ which contains the set $\left \{{\vec {\mathbb R}}\right \}$ , $f_{\omega _1\cap M}$ is Cohen over M.

Notation 3.7. For $\alpha \in \kappa +1$ , $n\in \omega $ , and a sequence $\left \langle {X_\xi ^i : i\in n, \xi \in \alpha }\right \rangle $ of subsets of $H_\kappa $ , we denote

$$\begin{align*}\langle\!\langle X_\xi^i : i\in n, \xi \in \alpha \rangle\!\rangle := \left\{{\left\langle{i, \xi, x}\right\rangle : i\in n, \xi \in \alpha, x\in X_\xi}\right\}. \end{align*}$$

Then the tuple $\langle \!\langle X_\xi ^i : i\in n, \xi \in \alpha \rangle \!\rangle $ is also a subset of $H_\kappa $ .

Proof. Let us only show the case that $\alpha $ is uncountable. By our assumption, $\alpha \in N_0\cap N_1$ and $N_0\cap \omega _1 =N_1\cap \omega _1$ . Then there exists a bijection $f:\omega _1\to \alpha $ which is in both $N_0$ and $N_1$ . Then

$$\begin{align*}N_0\cap \alpha = f[N_0\cap \omega_1] = f[N_1\cap \omega_1] = N_1\cap \alpha.\\[-35pt] \end{align*}$$

⊣

Proof. By definition, any condition of $\mathbb {P}_\beta $ is also a condition of $\mathbb {P}_\alpha $ . Suppose that $q\in \mathbb {P}_\beta $ , $p\in \mathbb {P}_\alpha $ , and $q\leq _{\mathbb {P}_\beta } p\restriction \beta $ . Then define

$$\begin{align*}r:= \left\langle{\mathcal{N}_q\cup \mathcal{N}_p, R_q\cup R_p, A_q\cup \left(A_p\restriction [\beta,\alpha)\right)}\right\rangle. \end{align*}$$

We can check that r is a condition of $\mathbb {P}_\zeta $ . So r is an extension of p in $\mathbb {P}_\alpha $ . Such an r is a canonical common extension of q and p in $\mathbb {P}_\alpha $ . Hence the identity map from $\mathbb {P}_\beta $ into $\mathbb {P}_\alpha $ is a complete embedding.⊣

Observation 4.4. It is proved in the next section that for each $\xi \in \omega _2$ , $\mathbb {P}_\xi $ is proper. Then, as in the case of $\mathbb {Q}(X,r)$ in Section 2.2, $\dot E_\xi $ is a $\mathbb {P}_{\xi +1}$ -name for a club subset of $\omega _1$ . By the definition, for each $\xi \in \omega _2$ , if $\Phi (\xi ) =\left \{{\dot r _\xi }\right \}$ and $\dot r_\xi $ is a $\mathbb {P}_\xi $ -name for a function from $\omega $ into $2$ , then $\mathbb {P}_{\xi +1}$ forces that $\dot E_\xi $ captures $\dot r_\xi $ relative to X.

Observation 4.5. The requirement (g-cl) is necessary in the definition of $\mathbb {P}_\kappa $ to show Lemma 5.2. To show properness of $\mathbb {P}_\alpha $ (for each $\alpha \leq \omega _2$ ) equipped with (g-cl), we want the requirements (el), (ho), (up), and (down) in Definition 4.1. This is the reason why we introduced a symmetric system of relational structures.

Proof. Suppose that $\alpha \in \kappa +1$ and $\left \{{p_\zeta : \zeta \in \omega _2}\right \}$ is a set of $\aleph _2$ -many conditions in $\mathbb {P}_\alpha $ . Recall that $\operatorname {\mathsf {CH}}$ holds (Assumption 3.1). By shrinking the set if necessary, we may assume that

• • the set $\left \{{\mathcal {N}_{p_\zeta } : \zeta \in \omega _2}\right \}$ forms a $\Delta $ -system,

• • the set $\left \{{\operatorname {dom} (A_{p_\zeta }) : \zeta \in \omega _2}\right \}$ forms a $\Delta $ -system with root D,

• (•) the set $\displaystyle \left \{{ \left ( \bigcup \mathcal {N}_{p_\zeta } \right ) \cap \kappa : \zeta \in \omega _2}\right \}$ forms a $\Delta $ -system with root K (which is a countable subset of $\kappa $ ),

• (•) for each $\xi \in K$ , the set

  $$\begin{align*}\left\{{ \overline M : M\in {R_{p_\zeta }}^{-1}\big[\, [\xi,\kappa) \,\big] \cap\mathfrak{M}_\xi^P }\right\} \end{align*}$$
  does not depend on $\zeta \in \omega _2$ ,

• (•) for each $\zeta \in \omega _2$ , $\left (\operatorname {dom}(A_{p_\zeta })\setminus D\right )\cap K=\emptyset $ ,

• (•) for each $\zeta , \zeta ' \in \omega _2$ , each $M\in \mathcal {N}_{p_\zeta }$ and each $M'\in \mathcal {N}_{p_{\zeta '}}$ , if $\overline M=\overline {M'}$ , then $M\cap \kappa $ and $M'\cap \kappa $ are order isomorphic and the corresponding isomorphism fixes $\kappa \cap M\cap M'$ (which is a subset of K)Footnote ² , and

• • for each $\xi \in D$ , the coordinate $A_{p_\zeta }(\xi ) \in \left [ \omega _1\times \omega _1\times \omega _1 \right ]^{<\aleph _0}$ does not depend on $\zeta \in \omega _2$ .

Then we claim that for each distinct $\zeta $ and $\zeta '$ , $p_\zeta $ and $p_{\zeta '}$ are compatible in $\mathbb {P}_\alpha $ . To see this, let $q \in U_\alpha $ such that

• • $\mathcal {N}_q:=\mathcal {N}_{p_\zeta }\cup \mathcal {N}_{p_{\zeta '}}$ ,

• • $R_q:=R_{p_\zeta }\cup R_{p_{\zeta '}}$ , and

• • $A_q$ is the function with domain $\operatorname {dom}(A_{p_\zeta })\cup \operatorname {dom}(A_{p_{\zeta '}})$ such that, for each $\xi \in \operatorname {dom}(A_{p_\zeta })\cup \operatorname {dom}(A_{p_{\zeta '}})$ ,

  $$\begin{align*}A_q(\xi):= A_{p_\zeta}(\xi)\cup A_{p_{\zeta'}}(\xi) \end{align*}$$
  (which is equal to $A_{p_\zeta }(\xi )$ or $A_{p_{\zeta '}}(\xi )$ ).

Such a q is a canonical amalgamation of $p_\zeta $ and $p_{\zeta '}$ . Then by the above items $(\boldsymbol \bullet )$ , $\mathcal {N}_q$ and $R_q$ satisfy Definition 4.1 (ob), (el), (ho), (up), and (down). Recall that for each $M\in \mathfrak {M}_0$ and each $\alpha \in \kappa $ , if $\alpha \not \in M$ , then $M\not \in \mathfrak {M}_\alpha ^P$ . So for any $\left \{{\zeta ,\zeta '}\right \} \in \left [\omega _2\right ]^2$ and any $\alpha \in \operatorname {dom}(A_{p_\zeta })\setminus D$ , $\operatorname {dom}(R_{p_{\zeta '}})\cap \mathfrak {M}_\alpha ^P =\emptyset $ , and hence ${R_q}^{-1}[\{\alpha \}] = {R_{p_\zeta }}^{-1}[\{\alpha \}]$ . Therefore q satisfies (g). Thus q is a condition of $\mathbb {P}_\alpha $ , and is a common extension of $p_\zeta $ and $p_{\zeta '}$ .⊣

Proof of  $\mathsf {(p)}_0$ .

This proof is a standard proof in the context of the side condition method (see, e.g., [Reference Todorčević12, Lemma 4]), and similar to the proof in [Reference Miyamoto and Yorioka10]. Suppose that $p\in \mathbb {P}_0$ (then $R_p=A_p=\emptyset $ ), $N\in \mathcal {N}_p \cap \mathfrak {M}^P_0$ , $\mathcal {D}\in N$ is a predense subset of $\mathbb {P}_0$ , and $q\leq _{\mathbb {P}_0} p$ . We notice that q is of the form $(\mathcal {N}_q,\emptyset , \emptyset )$ . It suffices to find $u'\in \mathcal {D} \cap N$ which is compatible with q in $\mathbb {P}_0$ .

By extending q if necessary, we may assume that there exists $u\in \mathcal {D}$ such that $q\leq _{\mathbb {P}_0} u$ . Define

$$ \begin{align*} { \begin{array}{r@{}l} \mathcal{E}:=\Big\{r\in\mathbb{P}_0:{\ } & \text{there are }u' \in\mathcal{D} \text{ and } M\in \mathcal{N}_r \text{ such that }r\leq_{\mathbb{P}_0} u' \text{ and }\\ &\mathcal{N}_r\cap M = \mathcal{N}_q\cap N \Big\}. \end{array} } \end{align*} $$

Since the set $\mathcal {N}_q\cap N$ belongs to N, $\mathcal {E}$ is a definable class in the expanded relational structure by $\mathbb {P}_0$ with parameters in N. Moreover, we note that $q\in \mathcal {E}$ . So by elementarity of N, there exist $r\in \mathcal {E}\cap N$ and $u'\in \mathcal {D}\cap N$ such that $r\leq _{\mathbb {P}_0} u'$ . Define $q' \in U_0$ such that

$$\begin{align*}\begin{array}{r@{\ }l} \mathcal{N}_{q'}:= & \mathcal{N}_q \cup \mathcal{N}_r \\[5pt] & \cup \Big\{ \left( {\Psi_{M'} }^{-1} \circ \Psi_{N} \right) (M) : M'\in \mathcal{N}_q \text{ with } \omega_1\cap M'=\omega_1\cap N, M\in \mathcal{N}_r \Big\}. \end{array} \end{align*}$$

$\mathcal {N}_{q'}$ forms a symmetric system. $q'$ is the canonical amalgamation of q and r. Therefore, $q'$ is a condition of $\mathbb {P}_0$ and a common extension of q and r in $\mathbb {P}_0$ . $\dashv$

Proof of  $\mathsf {(C)}_0$ .

Suppose that $p\in \mathbb {P}_0$ , $N \in \mathcal {N}_p \cap \mathfrak {M}^P_0$ , and $\left \{{\dot F_n: n\in \omega }\right \}$ is a set of $\mathbb {P}_0$ -names for nowhere dense subsets of $\left ({2^{<\omega }}\right )^\omega $ such that the sequence $\left \langle {\dot F_n: n\in \omega }\right \rangle $ belongs to $ N$ . Let us show that

$$\begin{align*}p \Vdash_{\mathbb{P}_0}\ \text{"} f_{\omega_1\cap N} \not \in \bigcup_{n\in\omega}\dot F_n \text{"}. \end{align*}$$

Suppose not, and let q be an extension of p in $\mathbb {P}_0$ such that, for some $n\in \omega $ ,

$$\begin{align*}q\Vdash_{\mathbb{P}_0}\ \text{"} f_{\omega_1\cap N } \in \dot F_n \text{"}. \end{align*}$$

For each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ , each $x\in \left [ \mathfrak {M}^P_0 \right ]^{<\aleph _0}$ , and each $r\in \mathbb {P}_0$ , define $\varphi _0(\nu , x, r)$ to be the assertion that there exists $K \in \mathcal {N}_r$ such that

• • $\left \{{\omega _1\cap M : M\in \mathcal {N}_r\cap K }\right \} = \left \{{\omega _1\cap M : M\in \mathcal {N}_q \cap N }\right \} , $

• • $\mathcal {N}_q\cap N \subseteq \mathcal {N}_r \cap K $ ,

• • $x\in K$ , and

• • $ r \Vdash _{\mathbb {P}_0}\!\!\!\text {"}\,\, \dot F_n \cap [ \nu ] \neq \emptyset \,\text {"} $ ,

and define

$$ \begin{align*} { \begin{array}{r@{}l} Y:=\Big\{g\in \left({2^{<\omega}}\right)^\omega :{\ } & \text{for any }k\in \omega \text{ and any }x\in \left[ \mathfrak{M}^P_0 \right]^{<\aleph_0}, \text{ there is } r\in \mathbb{P}_0\\ &\text{ which satisfies } \varphi_0(g\restriction k, x , r)\Big\}. \end{array} } \end{align*} $$

Since $\left \{{\dot F_n, \mathcal {N}_p\cap N}\right \} \in N \in \mathfrak {M}^P_0$ , we have $Y\in N$ .

We claim that Y is nowhere dense. To show this, let $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ . Take $\nu ' \in \left ({2^{<\omega }}\right )^{<\omega }$ and $s\leq _{ \mathbb {P}_0} q$ such that $\nu \subseteq \nu '$ , and

$$\begin{align*}s \Vdash_{\mathbb{P}_0}\ \text{"} \dot F_n \cap [\nu' ]= \emptyset \text{"}. \end{align*}$$

Let us show that $Y \cap [\nu '] =\emptyset $ . If not, there exists $g\in Y \cap [\nu ']$ . Let $k\in \omega $ be such that $\nu ' \subseteq g\restriction k$ . Then there exists $r \in \mathbb {P}_0$ which satisfies $\varphi _0(g\restriction k, \mathcal {N}_s, r)$ . Let us fix $K\in \mathcal {N}_r$ witness to $\varphi _0(g\restriction k, \mathcal {N}_s, r)$ . Define $r'\in U_0$ such that

$$\begin{align*}\begin{array}{r@{\ }l} \mathcal{N}_{r'}:= & \mathcal{N}_s \cup \mathcal{N}_r \\[5pt] & \cup \Big\{ \left( {\Psi_{M'} }^{-1} \circ \Psi_{K} \right) (M) : M'\in \mathcal{N}_r \text{ with } \omega_1\cap M'=\omega_1\cap K, M\in \mathcal{N}_s \Big\}. \end{array} \end{align*}$$

Then $r'$ is a common extension of s and r in $\mathbb {P}_0$ , and hence

$$\begin{align*}r' \Vdash_{\mathbb{P}_0}\ \text{"} \dot F_n \cap [\nu'] =\emptyset \text{ and } \dot F_n \cap [g \restriction k] \neq \emptyset \text{"} , \end{align*}$$

which is a contradiction.

We claim that $f_{\omega _1\cap N}$ belongs to $ Y$ . This contradicts the fact that $f_{\omega _1\cap N}$ is Cohen over N. To show that $f_{\omega _1\cap N} \in Y$ , assume that $f_{\omega _1\cap N} \not \in Y$ . Then there are $k\in \omega $ and $x\in \left [ \mathfrak {M}^P_0 \right ]^{<\aleph _0}$ such that there are no $r \in \mathbb {P}_0$ which satisfies $\varphi _0(f_{\omega _1\cap N} \restriction k, x , r)$ . Since $f_{\omega _1\cap N} \restriction k\in N$ and N is an elementary substructure of the expanded relational structure by $\mathbb {P}_0$ , there exists $x' \in \left [ \mathfrak {M}^P_0 \right ]^{<\aleph _0} \cap N$ such that there are no $r \in \mathbb {P}_0$ which satisfies $\varphi _0(f_{\omega _1\cap N} \restriction k, x' , r)$ . However, q satisfies $\varphi _0(f_{\omega _1\cap N} \restriction k, x' , q)$ , which is a contradiction. $\dashv$

Proof of  $\mathsf {(p)}_{\alpha +1}$ .

Suppose that $\alpha \in \omega _2$ , $p\in \mathbb {P}_{\alpha +1}$ , $N\in \mathcal {N}_p \cap \mathfrak {M}^P_{\alpha +1}$ which satisfies that $\{N\}\times ((\alpha +1)\cap N)\subseteq R_p$ , $\mathcal {D}\in N$ is a predense subset of $\mathbb {P}_{\alpha +1}$ , and ${q\leq _{\mathbb {P}_{\alpha +1}} p}$ . By extending q if necessary, we may assume that q is an extension of some member of $\mathcal {D}$ . Since $N \in \mathfrak {M}^P_{\alpha +1}$ , by Proposition 3.9(3) and the fact that $\alpha + 1 \in N$ (hence $\alpha \in N$ ), $N \in \mathfrak {M}^P_\alpha $ . So by the induction hypothesis $\mathsf {(p)}_\alpha $ , $q\restriction \alpha $ is $(N,\mathbb {P}_\alpha )$ -generic. It suffices to find $u\in \mathbb {P}_{\alpha + 1} \cap N$ which is compatible with q in $\mathbb {P}_{\alpha +1}$ such that u is an extension of some member of $\mathcal {D} \cap N$ . When $\alpha \not \in \operatorname {dom}(A_q)$ , the argument is similar to the proof of $\mathsf {(p)}_0$ . So we suppose that $\alpha \in \operatorname {dom}(A_q)$ .

Define $\mathcal {E}$ to be the set of the conditions u of $\mathbb {P}_{\alpha +1} $ such that

• • u is an extension of some member of $\mathcal {D}$ in $\mathbb {P}_{\alpha +1}$ ,

• • $\mathcal {N}_u\cap M = \mathcal {N}_q\cap N$ and $A_u(\alpha )\cap M = A_q(\alpha )\cap N$ for some $M\in {R_u}^{-1}[\{\alpha \}]$ , and

• • for any $\sigma \in A_q(\alpha )\setminus N$ with $\delta _\sigma> \omega _1\cap N$ and $C_{\delta _\sigma }\cap N \neq \emptyset $ ,

  $$\begin{align*}\max\left( C_{\delta_\sigma}\cap N \right) < \min \left\{{\delta_\tau : \tau \in A_u(\alpha) \setminus \left( A_q(\alpha) \cap N \right)}\right\}. \end{align*}$$

Then $q\in \mathcal {E}$ , and $\mathcal {E}$ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters $\mathcal {D}$ , $\mathcal {N}_q \cap N$ , $A_q(\alpha )\cap N$ , $\{ C_{\delta _\sigma } \cap N : \sigma \in A_q(\alpha )\setminus N, \delta _\sigma> \omega _1\cap N \} $ , all of which are in N. Since $q\restriction \alpha $ is $(N,\mathbb {P}_\alpha )$ -generic, for any $\eta \in \omega _1\cap N$ ,

$$ \begin{align*} { \begin{array}{r@{\ }l} \star \ q\restriction\alpha \Vdash_{\mathbb{P}_\alpha}\!\!\!&\text{"} \text{there exists }u\in \mathcal{E}\cap N \text{ such that } u\restriction \alpha \in \dot G_{\mathbb{P}_\alpha} \text{ and } \\ & \eta < \min \left\{{\delta_\tau : \tau \in A_u(\alpha) \setminus \left( A_q(\alpha) \cap N \right)}\right\}\! \text{"}. \end{array} } \end{align*} $$

Define $\dot Z$ to be a $\mathbb {P}_\alpha $ -name such that

$$ \begin{align*} { \begin{array}{r@{}l} \Vdash_{\mathbb{P}_\alpha}\ \text{"} & \dot Z \text{ is the set of the functions }g \text{ from }\omega \text{ into }2^{<\omega} \text{ such that there exists}\\ &u\in \mathcal{E}\cap N \text{ such that } u\restriction \alpha \in \dot G_{\mathbb{P}_\alpha} \text{ and, for any } \tau \in A_u(\alpha) \setminus \left( A_q(\alpha)\cap N \right),\\ &g ( \left| C_{\omega_1\cap N } \cap \delta_\tau \right|) \subseteq \dot r_\alpha \text{ and } g ( \left| C_{\omega_1\cap N } \cap (\gamma_\tau +1) \right|) \subseteq \dot r_\alpha \text{"}. \end{array} } \end{align*} $$

By (g-ob-2), (g-cl), and $N\in {R_q}^{-1}[\{\alpha \}]$ , there exists the unique $\sigma _0 \in A_q(\alpha )$ such that $\delta _{\sigma _0} = \omega _1\cap N$ . By (g-m), we can take $M\in \mathcal {N}_q \cap \mathfrak {M}^P_\alpha $ such that $N \in M$ , $\omega _1\cap M = \gamma _{\sigma _0}$ , and $\{ M\}\times (\alpha \cap M)\subseteq R_q$ . By $\star $ above, $q \restriction \alpha \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}\dot Z$ is dense open in $\left ({2^{<\omega }}\right )^\omega $ ”. $\dot Z$ is defined from the set $\left \{{N, A_q(\alpha ) \cap N , C_{\omega _1\cap N}, \dot r_\alpha }\right \}$ (which is in M), $\mathcal {E}$ and $\dot G_{\mathbb {P}_\alpha }$ , and is forced to be an open subset of $\left ({2^{<\omega }}\right )^\omega $ . Since M is an elementary substructure of the expanded relational structure by $\mathbb {P}_{\alpha }$ , by Proposition 5.1, $\dot Z$ can be considered as an element of M. By the induction hypothesis $\mathsf {(C)}_\alpha $ , $q \restriction \alpha \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}\,\, f_{\omega _1\cap M}$ is Cohen over $M[\dot G_{\mathbb {P}_\alpha }]\,\text {"}$ . It follows that $q \restriction \alpha \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}\,\, f_{\omega _1\cap M} \in \dot Z \,\text {"}$ . Take $r\in \mathbb {P}_\alpha $ and $u\in \mathcal {E}\cap N$ such that $r\leq _{\mathbb {P}_\alpha } q\restriction \alpha $ and

$$\begin{align*}r\Vdash_{\mathbb{P}_\alpha}\ \text{"} u\text{ is a witness that } f_{\omega_1\cap M} \in \dot Z \text{"}. \end{align*}$$

Let $r'$ be a common extension of r and $u\restriction \alpha $ . Define $q' \in U_{\alpha +1}$ such that

• • $\mathcal {N}_{q'}:=\mathcal {N}_{r'}$ ,

• • $R_{q'}:=R_{r'}$

  $$\begin{align*}\begin{array}{r@{\ }l} \cup \Big\{ \left\langle{\left( {\Psi_{M'} }^{-1} \circ \Psi_{N} \right) (K), \alpha}\right\rangle : & M' \in {R_q}^{-1}[\{\alpha\}] \text{ with } \omega_1\cap M' = \omega_1\cap N, \\ & K \in {R_u}^{-1}[\{\alpha\}] \setminus {R_q}^{-1}[\{\alpha\}] \Big\} , \end{array} \end{align*}$$

• • $A_{q'}\restriction \alpha := A_{r'}$ , and

• • $A_{q'}(\alpha ) = A_u(\alpha ) \cup A_q(\alpha )$ .

We claim that $q'$ is a condition of $\mathbb {P}_{\alpha +1}$ . Since $q'\restriction \alpha = r'$ , $q'\restriction \alpha $ is a condition of $\mathbb {P}_\alpha $ . Since $q\in \mathbb {P}_{\alpha +1}$ , $u \in \mathbb {P}_{\alpha +1} \cap N$ , $N\in \mathfrak {M}^P_{\alpha +1}$ , and $\alpha \in \omega _2$ , by Propositions 3.10 and 3.11, ${R_{q'}}^{-1}[\{\alpha \}]$ satisfies (el), (ho), (up), and (down). Since q and u are conditions of $\mathbb {P}_{\alpha +1}$ , $A_{q'}(\alpha )$ satisfies (g-ob), (g-ob-2), and (g-cl). It follows from the choice of r and u that $A_{q'}(\alpha )$ satisfies (g-w). Moreover, by $\alpha \in \omega _2$ and Proposition 3.10, $A_{q'}(\alpha )$ satisfies (g-m). Therefore $q'$ is a condition of $\mathbb {P}_{\alpha +1}$ . So $q'$ is a common extension of q and u in $\mathbb {P}_{\alpha +1}$ . By elementarity of N and the fact that $u\in \mathcal {E} \cap N$ , u is an extension of some member of $\mathcal {D}\cap N$ in $\mathbb {P}_{\alpha +1}$ . $\dashv$

Proof of Proposition 5.3.

Define $\mathcal {D}'$ to be the set of the conditions u of $\mathbb {P}_{\alpha +1}$ such that either $u\in \mathcal {D}$ or u is incompatible with any element of $\mathcal {D}$ in $\mathbb {P}_{\alpha +1}$ . Then $\mathcal {D}'$ is a predense subset of $\mathbb {P}_{\alpha +1}$ . Since N is an elementary substructure of the expanded relational structure by $\mathbb {P}_{\alpha +1}$ , by Proposition 5.1, there exists a maximal antichain $\mathcal {A}$ in N that is a subset of $\mathcal {D}'$ . By $\mathsf {(p)}_{\alpha +1}$ , p is $(N,\mathbb {P}_{\alpha +1})$ -generic. So there exists $q\in \mathcal {A} \cap N$ such that q is compatible with p in $\mathbb {P}_{\alpha +1}$ . Since $p \in \mathcal {D}$ , q has to be in $\mathcal {D}$ . $\dashv$

Proof of  $\mathsf {(C)}_{\alpha +1}$ .

Suppose that $p\in \mathbb {P}_{\alpha +1}$ , $N\in \mathcal {N}_p \cap \mathfrak {M}^P_{\alpha +1}$ which satisfies that $\{N\}\times ((\alpha +1)\cap N)\subseteq R_p$ , and $\left \{{\dot F_n: n\in \omega }\right \}$ is a set of $\mathbb {P}_{\alpha +1}$ -names for nowhere dense subsets of $\left ({2^{<\omega }}\right )^\omega $ such that $\left \{{\dot F_n: n\in \omega }\right \} \in N$ . Let us show that $p \Vdash _{\mathbb {P}_{\alpha +1}} \!\!\!\text {"}\,\, f_{\omega _1\cap N} \not \in \bigcup _{n\in \omega } \dot F_n \,\text {"}$ .

Suppose not, and let $q\leq _{\mathbb {P}_{\alpha +1}} p$ and $n\in \omega $ be such that

$$\begin{align*}q \Vdash_{\mathbb{P}_{\alpha+1}}\ \text{"} f_{\omega_1\cap N} \in \dot F_n \text{"}. \end{align*}$$

Let $\sigma _N \in A_q(\alpha )$ be such that $\delta _{\sigma _N} = \omega _1\cap N$ .

Only in this paragraph, for each set x (in the ground model), we denote by $\check x$ the canonical $\mathbb {P}_\alpha $ -name which represents the set x in the forcing extension. For each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ and each $A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ , define $\dot \Sigma '(\nu , A)$ to be the $\mathbb {P}_\alpha $ -name which consists of all the pairs $\left \langle {u, \check B}\right \rangle $ such that

• • $u\in \mathbb {P}_\alpha $ and $B\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ ,

• • there exists $u'\in \mathbb {P}_{\alpha +1}$ such that

  1. – $u\leq _{\mathbb {P}_\alpha }u'\restriction \alpha $ ,

  2. – ${R_q}^{-1}[\{\alpha \}]\cap N \subseteq {R_{u'}}^{-1}[\{\alpha \}]$ ,

  3. – $A_{u'}(\alpha ) = A \cup B$ ,

  4. – for any $\sigma \in A$ and any $\tau \in B$ , $\delta _\sigma < \delta _\tau $ , and

  5. – $ u' \Vdash _{\mathbb {P}_{\alpha +1}}\!\!\!\text {"}\,\, \dot F_n \cap [\nu ] = \emptyset \,\text {"} $ (here we omit the check-notation for $\mathbb {P}_{\alpha +1}$ ).

For each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ and each $A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ , $\dot \Sigma '(\nu , A) $ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters in $H(\kappa )$ . Moreover, if $A\in N$ , then $\dot \Sigma '(\nu , A) $ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters in N. By Proposition 5.1, there exists $\Sigma \in H(\kappa )$ such that

• • $\Sigma \subseteq \left ({2^{<\omega }}\right )^{<\omega } \times [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0} \times \mathbb {P}_\alpha \times [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ ,

• • for each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ and each $A, B\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ , the set $\mathcal {A}(\nu ,A,B) := \left \{{u\in \mathbb {P}_\alpha : \left \langle {\nu , A, u, B}\right \rangle \in \Sigma }\right \}$ is a maximal antichain (of size $\aleph _1$ ), and, for each $u\in \mathcal {A}(\nu ,A,B)$ , either $\left \langle {u, \check B}\right \rangle \in \dot \Sigma '(\nu ,A)$ , or no extension v of u in $\mathbb {P}_\alpha $ satisfies $\left \langle {v, \check B}\right \rangle \in \dot \Sigma '(\nu ,A)$ .

Since N is an elementary substructure of the expanded relational structure by $\mathbb {P}_{\alpha +1}$ , we may assume that $\Sigma \in N$ . For each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ and each $A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ , define $\dot \Sigma (\nu ,A) $ to be the $\mathbb {P}_\alpha $ -name such that

$$\begin{align*}\dot \Sigma(\nu,A) := \bigcup_{B\in [\omega_1\times\omega_1\times\omega_1]^{<\aleph_0}} \left\{{\left\langle{u, \check B}\right\rangle : u \in \mathcal{A}(\nu,A, B) \text{ and } \left\langle{u, \check B}\right\rangle\in \dot \Sigma'(\nu,A)}\right\}. \end{align*}$$

Then, the sequence $\left \langle {\dot \Sigma (\nu ,A) : \nu \in \left ({2^{<\omega }}\right )^{<\omega }, A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0} }\right \rangle $ belongs to N, and, for each $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ and each $A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}$ , $\Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}\dot \Sigma (\nu ,A) = \dot \Sigma '(\nu ,A) \,\text {"}$ . From now on, we omit the check-notation in the forcing language.

Since $\{N\}\times ( (\alpha +1)\cap N) \subseteq R_q$ , by $\mathsf {(p)}_{\alpha +1}$ and Proposition 5.3, there exists $q'\in \mathbb {P}_{\alpha +1}\cap N$ such that

• • $q'$ is compatible with q in $\mathbb {P}_{\alpha +1}$ ,

• • ${R_q}^{-1}[\{\alpha \}]\cap N \subseteq {R_{q'}}^{-1}[\{\alpha \}]$ ,

• • there exists $N_0 \in {R_{q'}}^{-1}[\{\alpha \}]$ such that

  1. – $A_{q'}(\alpha ) \cap N_0 = A_q(\alpha ) \cap N$ ,

  2. – the set

     $$\begin{align*}\begin{array}{r@{\ }l} \big\{ \left\langle{ \varepsilon_\sigma, C_{\delta_\sigma} \cap N_0 , f_{\gamma_\sigma} \restriction \left| C_{\delta_\sigma} \cap N_0 \right| }\right\rangle : & \sigma \in A_{q'}(\alpha)\setminus N_0 \text{ with } \varepsilon_\sigma \in N_0 \\ & \&\ \delta_\sigma\neq \omega_1\cap N_0 \big\} \end{array}\end{align*}$$
     is equal to the set
     $$\begin{align*}\left\{{ \left\langle{ \varepsilon_\sigma, C_{\delta_\sigma} \cap N, f_{\gamma_\sigma} \restriction \left| C_{\delta_\sigma} \cap N \right| }\right\rangle : \sigma \in A_{q}(\alpha) \setminus (N \cup\left\{{\sigma_N}\right\}) \text{ with } \varepsilon_\sigma \in N}\right\} , \end{align*}$$
     and

  3. – $N_0$ contains the sets $\left \{{\varepsilon _\sigma : \sigma \in A_{q}(\alpha ) \setminus N }\right \}\cap N $ , $\{C_{\gamma _\sigma }\cap N : \sigma \in A_{q}(\alpha ) \setminus (N \cup \left \{{\sigma _N}\right \}) \}$ and $\Big \langle \dot \Sigma (\nu ,A) : \nu \in \left ({2^{<\omega }}\right )^{<\omega }, A\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0} \Big \rangle $ as members.

Let $q^+$ be a common extension of $q'$ and q in $\mathbb {P}_{\alpha +1}$ . We notice that $N_0$ contains the set $\big \{ C_{\omega _1\cap N } \cap N_0 , f_{\gamma _{\sigma _N}} \restriction \left (\left | C_{\omega _1\cap N } \cap N_0 \right | + 1 \right ) \big \}$ and

$$\begin{align*}q^+\restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap N_0 \right|) \subseteq \dot r_\alpha \text{"}. \end{align*}$$

In a similar way as in the definition of $\dot \Sigma '(\nu ,A)$ before, we have a $\mathbb {P}_\alpha $ -name $\dot Z$ for a subset of $\left ({2^{<\omega }}\right )^{\omega }$ such that $\dot Z$ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters in N and, for any $u\in \mathbb {P}_\alpha $ and any $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ , if $u\Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}[\nu ]\subseteq \dot Z \,\text {"}$ , then there exists $u'\in \mathbb {P}_{\alpha +1}$ such that

• • $u \leq _{\mathbb {P}_\alpha } u' \restriction \alpha $ ,

• • $u' \leq _{\mathbb {P}_{\alpha +1}} q'$ ,

• • for each $\tau \in A_{u'}(\alpha )\cap N_0$ , if $\varepsilon _{\sigma _N} < \delta _\tau $ , then $\gamma _{\tau }+1 < \omega _1\cap N_0$ and

  $$\begin{align*}u'\restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap \delta_\tau \right|) \subseteq \dot r_\alpha \text{ and } f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap (\gamma_\tau +1) \right|) \subseteq \dot r_\alpha \text{"} , \end{align*}$$

• • $A_{u'}(\alpha )$ has $\sigma $ such that $\delta _\sigma =\omega _1\cap N_0$ , and

• • $u' \Vdash _{\mathbb {P}_{\alpha +1}}\!\!\!\text {"}\,\, \dot F_n \cap [\nu ] = \emptyset \,\text {"}$ .

Now $\dot F_n$ is a $\mathbb {P}_{\alpha +1}$ -name for a nowhere dense subset of $\left ({2^{<\omega }}\right )^{\omega }$ , $\sigma _N\in A_{q^+}(\alpha )$ , ${A_{q^+}}(\alpha )$ has $\sigma $ such that $\delta _\sigma =\omega _1\cap N_0$ , and $q^+ \leq _{\mathbb {P}_{\alpha +1}} q'$ , so

$$\begin{align*}q^+ \restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} \dot Z \text{ is a dense open subset of } \left({2^{<\omega}}\right)^{\omega} \text{"}. \end{align*}$$

Since $\dot Z$ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters in N, by Proposition 5.1, $\dot Z$ can be considered as an element of N. So by $\mathsf {(C)}_\alpha $ ,

$$\begin{align*}q^+\restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} f_{\omega_1\cap N} \in \dot Z \text{"}. \end{align*}$$

Thus, there are $r\leq _{\mathbb {P}_\alpha } q^+\restriction \alpha $ and $k\in \omega $ such that

$$\begin{align*}r \Vdash_{\mathbb{P}_\alpha}\ \text{"} [f_{\omega_1\cap N}\restriction k ] \subseteq \dot Z \text{"}. \end{align*}$$

Let $u_0\in \mathbb {P}_{\alpha +1}$ be a witness to $r \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}[f_{\omega _1\cap N}\restriction k ] \subseteq \dot Z \,\text {"} $ . Then

$$\begin{align*}r\Vdash_{\mathbb{P}_\alpha}\ \text{"} A_{u_0}\setminus N_0 \in \dot \Sigma(f_{\omega_1\cap N}\restriction k, A_{u_0}(\alpha)\cap N_0) \text{"}. \end{align*}$$

Let $\zeta _0 \in \omega _1\cap N_0$ be such that

• • for any $\tau \in A_{u_0}(\alpha )\cap N_0$ , $\gamma _\tau < \zeta _0$ ,

• • for any $\sigma \in A_q(\alpha ) \setminus N$ ,

  1. – if $\varepsilon _\sigma \in N$ , then $\varepsilon _\sigma < \zeta _0$ , and

  2. – $C_{\delta _\sigma }\cap N_0 \subseteq \zeta _0$ .