ASPERÓ–MOTA ITERATION AND THE SIZE OF THE CONTINUUM

Abstract

In this paper we build an Asperó–Mota iteration of length $\omega _2$ that adds a family of $\aleph _2$ many club subsets of $\omega _1$ which cannot be diagonalized while preserving $\aleph _2$. This result discloses a technical limitation of some types of Asperó–Mota iterations.

1 Introduction

Shelah introduced countable support iterations of proper forcing notions, which enable us to obtain a large number of consistency results. But by technical limitations, the size of the continuum cannot be larger than $\aleph _2$ in such consistency results. Asperó and Mota introduced a new iteration technique for proper forcing notions that enables us to obtain some consistency results with the continuum larger than $\aleph _2$ . Asperó–Mota iterations are equipped with symmetric systems of models as side conditions, idea due to Todorčević (see, e.g., [12, Section 4]). The main ingredient is not only the use of symmetric systems of models but also symmetric systems of models with markers. Asperó–Mota iterations are used in the papers [2–5, 10, 13]. In [5, 10], the iterations require that markers of models in symmetric systems also have symmetry in a suitable sense (Definition 4.1 (el), (ho), (up), and (down)).

The Asperó–Mota iterations used in [10] have length $\omega _2$ and are proper. The Asperó–Mota iterations used in [5] have length beyond $\omega _2$ and are claimed to be proper. The proof, however, contains a flaw, which has been acknowledged in personal communication. The problems of the proof from [5] are generated by the fact that the iteration in the paper is greater than $\omega _2$ . ¹

The forcing iteration in this paper deals with Asperó–Mota iterations with symmetric markers, like in [5, 10]. We disclose a technical limitation of this type of iterations; in fact our results show that the length of Asperó–Mota iterations with symmetric markers in the style of [5, 10] must be at most $\omega _2$ in order to ensure their properness.

To achieve our goal, we consider a family of club subsets of $\omega _1$ . Specifically, it is proved that Asperó–Mota iteration can force a certain assertion, which is called (c) in this paper, concerning the existence of a family of club subsets of $\omega _1$ which cannot be diagonalized while preserving $\aleph _2$ . The assertion was introduced by Justin Tatch Moore in personal communication and was inspired by the results in [1, Section 2]. Moreover, the natural proof of properness does not work when the length of the iteration is greater than $\omega _2$ .

This paper is organized as follows. Section 2 is devoted to the basic facts of the assertion (c): In Section 2.1, the assertion (c) is introduced; in Section 2.2, we introduce forcing notions to force the assertion (c). Forcing notions stated in Section 2.2 consist of finite objects. This idea is applied to the Asperó–Mota iterations presented in this paper. The rest of the sections are devoted to our main goal, that is, to prove that the assertion (c) can be forced by Asperó–Mota iterations. In Section 3, we introduce relational structures with which we will equip our Asperó–Mota style iterations. This notion is necessary for symmetric systems with symmetric markers. We define our iteration in Section 4, and in Section 5 we prove that it forces (c). As part of this proof, we show that the iteration is proper whenever its length is at most $\omega _2$ . In the last section we explain why the proof of properness breaks down when the length of our the iteration is greater than $\omega _2$ .

2 The assertion (c) and forcing (c) by finite approximations

2.1 The assertion (c)

Galvin showed that, if the Continuum Hypothesis holds, then for any family of $\aleph _2$ many club subsets of $\omega _1$ , there exists a subfamily of size $\aleph _1$ whose intersection is a club [8, Section 3.2]. Abraham and Shelah showed that the assumption of CH in this theorem of Galvin is necessary. More precisely, they showed that it is consistent that there exists a family of $\aleph _2$ many club subsets of $\omega _1$ such that the intersection of any uncountable subfamily is finite [1, Section 2]. (Notice that such a family cannot be diagonalized without collapsing $\aleph _1$ .) They proved this consistency result by an involved forcing construction using countable objects and ccc forcing notions. Justin Tatch Moore introduced the assertion (c), which is inspired by this result of Abraham–Shelah. His assertion (c) can be forced by a countable support iteration of proper forcing notions.

All definitions, propositions, and remarks in the subsection are due to Moore. Throughout the article, we assume the following.

Assumptions throughout the paper 2.1.

• • $\mathcal {C} =\left \langle {C_\delta : \delta \in \omega _1 \cap \mathsf {Lim}}\right \rangle $ is a ladder system on $\omega _1$ , that is, each $C_\delta $ is a cofinal subset of $\delta $ of order type $\omega $ ; moreover, we suppose that each $C_\delta $ consists of successor ordinals (hence, for any limit ordinals $\delta $ and $\gamma $ in $\omega _1$ with $\delta < \gamma $ , $C_\delta \cap (\gamma +1) = C_\delta \cap \gamma $ ),

• • the set $2^{<\omega } $ is equipped with the discrete topology, and $\left ( 2^{<\omega } \right )^\omega $ is considered as the product space of copies of the discrete space $2^{<\omega } $ ,

• • for each $\nu \in \left ( 2^{<\omega } \right )^{<\omega }$ , we denote

  $$\begin{align*}[\nu] := \left\{{ g \in \left( 2^{<\omega} \right)^\omega : \nu \subseteq g}\right\} , \end{align*}$$
  which is a basic open subset of the space $\left ( 2^{<\omega } \right )^\omega $ ; here we recall that $\nu \subseteq g$ means that, for every $n \in \operatorname {dom}(\nu )$ , $\nu (n) = g(n)$ .

Definition 2.2. For a set X of injective functions from $\omega $ into $2^{<\omega }$ and $r\in 2^\omega $ , we say that a club subset E of $\omega _1$ captures r relative to ( $\mathcal {C}$ and) X if, for any limit point $\delta $ of E, there are $f\in X$ and $\varepsilon \in \delta $ such that, for any $\xi \in (E\cap \delta )\setminus \varepsilon $ , $f( \left | C_\delta \cap \xi \right |) \subseteq r$ .

Proposition 2.3. Suppose that R is a set of reals and X is a set of injective functions from $\omega $ into $2^{<\omega }$ of size $\aleph _1$ . If, for each $r\in R$ , there exists a club subset $E_r$ of $\omega _1$ which captures r relative to X and if the set $\left \{{ E_r : r\in R}\right \}$ can be diagonalized, then the size of R is not larger than $\aleph _1$ .

Proof. Suppose that a club subset E of $\omega _1$ diagonalizes all club sets $E_r$ , $r\in R$ , (i.e., for each $r\in R$ , $E\setminus E_r$ is bounded in $\omega _1$ ) and R is of size $\geq \aleph _2$ . Then there exists $\eta \in \omega _1$ such that the set

$$\begin{align*}R' : = \left\{{r\in R : E\setminus\eta \subseteq E_r}\right\} \end{align*}$$

is of size $\geq \aleph _2$ . Let $\delta $ be a limit point of the set $E\setminus \eta $ . Since X is of size $\aleph _1$ , there are an injective function f in X and $\varepsilon \in \delta $ such that $\varepsilon \geq \eta $ and the set

$$\begin{align*}R":=\left\{{ r\in R' : \forall \xi\in (E_r\cap \delta)\setminus \varepsilon , f(\left| C_\delta \cap \xi \right| ) \subseteq r}\right\} \end{align*}$$

is of size $\geq \aleph _2$ . But then, for any $r\in R"$ ,

$$\begin{align*}\forall \xi\in (E\cap \delta)\setminus \varepsilon , f(\left| C_\delta \cap \xi \right| ) \subseteq r. \end{align*}$$

This contradicts the fact that $R"$ has at least two different reals.⊣

Definition 2.4. Define the assertion (c) to be the statement that there are a set X of injective functions from $\omega $ into $2^{<\omega }$ of size $\aleph _1$ and a collection of $\aleph _2$ -many reals each one of which can be captured by a club of $\omega _1$ relative to X.

Remark 2.5. By Proposition 2.3, any collection of $\aleph _2$ many club subsets each of which captures a real r relative to X cannot be diagonalized in any outer model with the same $\aleph _2$ if all the r’s are distinct.

Remark 2.6. Moore pointed out that, if X is a non-meager subset of injective functions from $\omega $ into $2^{<\omega }$ and $r\in 2^\omega $ , the forcing notion of all countable approximations to a club subset of $\omega _1$ that captures r is proper and adds no new reals (however it may not be $\sigma $ -closed). Moreover, if $\operatorname {\mathsf {CH}}$ holds, then it satisfies the $\aleph _2$ -proper isomorphic condition ( $\aleph _2$ -pic). Therefore, the assertion (c) can be forced by a countable support iteration.

2.2 Forcing (c) by finite approximations

In this subsection, we deal with a forcing notion to force the assertion (c) different from the one referred to in Remark 2.6. Our forcing notion is equipped with models as side conditions [12, Section 4]. The proofs of the basic facts of this forcing notion should help the reader understand the machinery of the proofs dealing with our Asperó–Mota iteration in Section 5.

Suppose that X is a non-meager subset of injective functions from $\omega $ into $2^{<\omega }$ , and $r\in 2^\omega $ . Let $\kappa := \left ( 2^{\aleph _0} \right )^+$ . Define $\mathfrak {M}(X,r)$ to be the set of countable elementary submodels of $H_\kappa $ which contain the set $\left \{{\mathcal {C}, X, r}\right \}$ . Each member of $\mathfrak {M}(X,r)$ is considered as a substructure of the structure $ \left \langle { H_\kappa ,\in , {\omega _1}, \mathcal {C}, X, r}\right \rangle $ . For each $M\in \mathfrak {M}(X,r)$ , the transitive collapse of M is considered as the structure $ \left \langle { \mathrm {trcl}(M),\in , {\omega _1}\cap M, \mathcal {C} \restriction M, X \cap M, r}\right \rangle $ , which is denoted by $\overline M$ . $\Psi _M$ denotes the transitive collapsing map from M onto $\overline M$ . For each $M\in \mathfrak {M}(X,r)$ , since M is countable and ${\omega _1}$ is of uncountable cofinality, $\omega _1\cap M$ is a countable ordinal. And if M and $M'$ in $\mathfrak {M}(X,r)$ have the same transitive collapse, then $\omega _1\cap M=\omega _1\cap M'$ , and the composition ${\Psi _{M'}}^{-1}\circ \Psi _M$ is an isomorphism from the structure $ \left \langle { M,\in , {\omega _1}, \mathcal {C} , X , r}\right \rangle $ onto the structure $ \left \langle { M', \in , {\omega _1}, \mathcal {C} , X , r}\right \rangle\!. $

In this subsection, we suppose $2^{\aleph _0} = \aleph _1$ . If M and $M'$ in $\mathfrak {M}(X,r)$ satisfy $\omega _1\cap M = \omega _1 \cap M'$ , then $\mathbb {R} \cap M = \mathbb {R}\cap M'$ . So the set of Borel codes in M coincides with those in $M'$ . Therefore, for any $f\in \left ({2^{<\omega }}\right )^\omega $ , f is Cohen over M iff f is Cohen over $M'$ . In this subsection, we identify a non-meager set X (which is of size $\aleph _1$ ) with some fixed enumeration of X of length $\omega _1$ . Then, if M and $M'$ in $\mathfrak {M}(X,r)$ satisfies $\omega _1\cap M = \omega _1 \cap M'$ , then $X\cap M = X \cap M'$ . We notice that, for any $M \in \mathfrak {M}(X,r)$ and any $f\in \left ( 2^{<\omega }\right )^\omega $ which is Cohen over M, the set $\left \{{ n \in \omega : f(n) \subseteq r}\right \}$ is infinite.

In Section 4, we will define an Asperó–Mota iteration of forcing notions playing the same role as the following forcing notions.

Definition 2.7. Define the forcing notion $\mathbb {Q}(X,r)$ consisting of the triples $p=\left \langle {\mathcal {N}_p^0, \mathcal {N}_p^1, A_p}\right \rangle $ such that

• (sym) $\mathcal {N}_p^0 \cup \mathcal {N}_p^1$ is a finite subset of $\mathfrak {M}(X,r)$ such that

  • • for each $M,M'\in \mathcal {N}^0_p\cup \mathcal {N}^1_p$ , if $\omega _1\cap M=\omega _1\cap M'$ , then $\overline M=\overline {M'}$ ,

  • • for each $M, M'\in \mathcal {N}^0_p\cup \mathcal {N}^1_p$ , if $\omega _1\cap M'<\omega _1\cap M$ , then there exists $M"\in \mathcal {N}^0_p\cup \mathcal {N}^1_p$ such that $\overline {M"}=\overline M$ and $M'\in M"$ ,

• (ob) $A_p$ is a finite set of tuples of the form $\sigma = \left \langle {\varepsilon _\sigma , \delta _\sigma , \gamma _\sigma , f_\sigma }\right \rangle $ such that $\varepsilon _\sigma , \delta _\sigma , \gamma _\sigma \in \omega _1$ , $\varepsilon _\sigma < \delta _\sigma < \gamma _\sigma $ , and $f_\sigma \in X$ ,

• (ob-2)

  • • the set $\left \{{\delta _\sigma : \sigma \in A_p}\right \} $ includes the set $\left \{{\omega _1\cap N : N \in \mathcal {N}_p^0}\right \}$ , and the set $\left \{{\gamma _\sigma : \sigma \in A_p}\right \} $ includes the set $\left \{{\omega _1\cap M : M \in \mathcal {N}_p^1}\right \}$ ,

  • • for any $\sigma \in A_p$ and any $N\in \mathcal {N}_p^0$ , if $\omega _1\cap N = \delta _\sigma $ , then there exists $M\in \mathcal {N}_p^1$ such that $N \in M$ and $\omega _1\cap M = \gamma _\sigma $ ,

  • • for any $\sigma \in A_p$ and any $M\in \mathcal {N}_p^1$ , if $\omega _1\cap M = \gamma _\sigma $ , then $f_\sigma $ is Cohen over M,

  • • for any $\sigma \in A_p$ and $N \in \mathcal {N}_p^0$ , if $\delta _\sigma < \omega _1\cap N$ , then $\sigma \in N$ ,

• (cl) for any $\left \{{\sigma ,\tau }\right \}\in \left [A_p \right ]^2$ , either $\gamma _\sigma < \delta _\tau $ or $\gamma _\tau < \delta _\sigma $ , and

• (w) for any $\sigma \in A_p$ , if $\delta _\sigma $ is a limit ordinal, then the set $\left \{{ n \in \omega : f_\sigma (n) \subseteq r }\right \}$ is infinite, and for any $\tau \in A_p\setminus \left \{{\sigma }\right \}$ with $\varepsilon _{\sigma } < \delta _\tau < \delta _\sigma $ , $f_{\sigma } ( \left | C_{\delta _\sigma } \cap \delta _\tau \right |) \subseteq r$ and $f_{\sigma } ( \left | C_{\delta _\sigma } \cap (\gamma _\tau +1) \right |) \subseteq r$ ,

for each $p,q \in \mathbb {Q}(X,r)$ , $q \leq _{\mathbb {Q}(X,r)} p $ if $\mathcal {N}_q^0 \supseteq \mathcal {N}_p^0$ , $\mathcal {N}_q^1 \supseteq \mathcal {N}_p^1$ , $A_q \supseteq A_p$ .

We notice that, for any $p\in \mathbb {Q}(X,r)$ , any $N\in \mathcal {N}^0_p$ and any $M \in \mathcal {N}^1_p$ , $\omega _1\cap N \neq \omega _1\cap M$ .

Lemma 2.8. For any non-meager set X of injective functions from $\omega $ into $2^{<\omega }$ and any $r\in 2^\omega $ , $\mathbb {Q}(X,r)$ is proper.

Proof. Let $\theta $ be a large enough regular cardinal, $N^*$ a countable elementary submodel of $H_\theta $ which contains $\left \{{\mathcal {C}, X , r, H_{\kappa }}\right \}$ , and $p \in \mathbb {Q}(X,r) \cap N^*$ . Let $M^*$ be a countable elementary submodel of $H_\theta $ which has the set $\{N^*\}$ , and let $\varepsilon _* \in \omega _1\cap N^*$ such that $\delta _\sigma <\varepsilon _*$ for every $\sigma \in A_p$ . We denote $N_* := N^* \cap H_{\kappa } $ and $M_* := M^* \cap H_{\kappa } $ . Since X is non-meager and $M^*$ is countable, there exists a function $f_*$ in X which is Cohen over $M^*$ . Define $ \sigma _* := \left \langle {\varepsilon _* , \omega _1\cap N_* , \omega _1\cap M_* , f_* }\right \rangle $ and $ p^+ : = \left \langle {\mathcal {N}_p^0 \cup \left \{{N_*}\right \}, \mathcal {N}_p^1\cup \left \{{M_*}\right \}, A_p \cup \left \{{ \sigma _*}\right \} }\right \rangle. $ $p^+$ is a condition of $\mathbb {Q}(X,r)$ , and hence is an extension of p.

Let us show that $p^+$ is an $N^*$ -generic condition of $\mathbb {Q}(X,r)$ . Let $\mathcal {D}$ be a predense subset of $\mathbb {Q}(X,r)$ which belongs to $N^*$ , and q an extension of $p^+$ in $\mathbb {Q}(X,r)$ . By extending q if necessary, we may assume that q is an extension of some member of $\mathcal {D}$ . Moreover, by extending q if necessary again, we may assume that the set $\left ( \mathcal {N}_q^0 \cup \mathcal {N}_q^1 \right ) \cap N_*$ includes the set

$$ \begin{align*} { \begin{array}{r@{}l} \Big\{ \left( \left(\Psi_{N_*}\right)^{-1} \circ \Psi_{N} \right) (K) :{\ } & N\in\mathcal{N}_q^0 \text{ with }\omega_1\cap N = \omega_1\cap N_* , \\ & K \in \left(\mathcal{N}_q^0 \cup \mathcal{N}_q^1\right) \cap N \Big\}. \end{array} } \end{align*} $$

Define $\mathcal {E}$ to be the set of the conditions u of $\mathbb {Q}(X,r) $ such that

• • u is an extension of some member of $\mathcal {D}$ in $\mathbb {Q}(X,r)$ ,

• • $\mathcal {N}_u^0 \cap N = \mathcal {N}_q^0 \cap N_*$ , $\mathcal {N}_u^1 \cap M = \mathcal {N}_q^1 \cap M_*$ ( $= \mathcal {N}_q^1 \cap N_*$ ), and $N \in M$ for some $N \in \mathcal {N}_u^0$ and some $M \in \mathcal {N}_u^1$ ,

• • $A_u \supseteq A_q\cap N_*$ , and, for any $\sigma \in A_u\setminus (A_q\cap N_*)$ and any $\tau \in A_q\setminus \left ( N_* \cup \left \{{ \sigma _* }\right \} \right )$ ,

  $$\begin{align*}\max\left( C_{\delta_\tau}\cap N_* \right) < \delta_\sigma. \end{align*}$$

Then $p^+ \in \mathcal {E}$ . Since the set

$$\begin{align*}\left\{{ \mathbb{Q}(X,r), \mathcal{D}, \mathcal{N}^0_q \cap N_*, \mathcal{N}^1_q\cap M_*, A_q\cap N_*, \left\langle{C_{\delta_\tau} \cap N_*: \tau \in A_q\setminus \left( N_* \cup \left\{{ \sigma_* }\right\} \right)}\right\rangle }\right\} \end{align*}$$

belongs to $N^*$ , by elementarity of $N^*$ , $\mathcal {E}$ belongs to $N^*$ . We note that

1. ♠ if $u\in \mathcal {E} \cap N_*$ , then for any $\sigma \in A_u\setminus (A_q\cap N_*)$ and any $\tau \in A_q\setminus \left ( N_* \cup \left \{{ \sigma _* }\right \} \right )$ ,

   $$\begin{align*}C_{\delta_\tau}\cap \delta_\sigma = C_{\delta_\tau}\cap \gamma_\sigma = C_{\delta_\tau}\cap N_* = C_{\delta_\tau} \cap \delta_{\sigma_*}. \end{align*}$$

It follows from elementarity of $N^*$ that

1. ♦ for any $\eta \in \omega _1\cap N_*$ , there exists $u\in \mathcal {E}\cap N^*$ such that, for every $\tau \in A_u \setminus (A_q\cap N_*)$ , $\eta < \delta _\tau $ , hence, for any $n\in \omega $ , there exists $u\in \mathcal {E}\cap N^*$ such that, for every $\tau \in A_u \setminus (A_q\cap N_*)$ , $\left | C_{\omega _1\cap N_*}\cap \gamma _\tau \right | \geq \left | C_{\omega _1\cap N_*}\cap \delta _\tau \right | \geq n$ .

Define Z to be the set of the functions g from $\omega $ into $2^{<\omega }$ such that there exists $u\in \mathcal {E} \cap N^*$ which satisfies that for any $\tau \in A_u\setminus \left ( A_q\cap N_* \right )$ , $g ( \left | C_{\omega _1\cap N_* } \cap \delta _\tau \right |) \subseteq r$ and $g ( \left | C_{\omega _1\cap N_* } \cap (\gamma _\tau +1) \right |) \subseteq r$ . Since Z is defined from $\mathcal {E}$ and $C_{\omega _1\cap N_*}$ ( $= C_{\delta _{\sigma _*}}$ ), and the set $\left \{{\mathcal {E}, N^*, C_{\omega _1\cap N_*}}\right \}$ is in $M^*$ , Z belongs to $M^*$ . By the property $\blacklozenge $ of $\mathcal {E} \cap N^*$ , Z is a dense open subset of the space $\left ( 2^{<\omega } \right )^\omega $ . Since $f_*$ is Cohen over $M^*$ , $f_*$ belongs to Z. Let u be a witness that $f_*$ belongs to Z.

Define $u' =\left \langle {\mathcal {N}_{u'}^0, \mathcal {N}_{u'}^1, A_{u'}}\right \rangle $ such that

$$\begin{align*}\begin{array}{r@{\ }l} \mathcal{N}_{u'}^0:= & \mathcal{N}_q^0 \cup \mathcal{N}_u^0 \\[5pt] & \cup \Big\{ \left( {\Psi_{N} }^{-1} \circ \Psi_{N_*} \right) (N') : N\in \mathcal{N}_q^0 \text{ with } \omega_1\cap N=\omega_1\cap N_*, N'\in \mathcal{N}_u^0 \Big\}, \end{array} \end{align*}$$

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

$$\begin{align*}A_{u'} := A_q \cup A_u. \end{align*}$$

Let us show that $u'$ is a condition of $\mathbb {Q}(X,r)$ . Then it follows that $u'$ is a common extension of q and u, which completes the proof. Since u and q satisfies (ob) and (ob-2), so does $u'$ . Since $A_{u'}$ is an end-extension of $A_q\cap N_*$ and $u'\in N^*$ , $A_{u'}$ satisfies (cl). We will check two non-trivial cases of (w) for $u'$ . Suppose that $\sigma \in A_{u}\setminus (A_q\cap N_*)$ and $\tau \in A_q\setminus \left ( N_* \cup \left \{{ \sigma _* }\right \} \right )$ . Then $\delta _\sigma < \delta _\tau $ . If $\varepsilon _\tau < \delta _\sigma $ , then $\varepsilon _\tau < \omega _1\cap N_*$ , and so, by $\spadesuit $ ,

$$\begin{align*}f_\tau ( \left| C_{\delta_\tau } \cap \delta_\sigma \right|) = f_\tau ( \left| C_{\delta_\tau } \cap (\gamma_\sigma +1) \right|) = f_\tau ( \left| C_{\delta_\tau } \cap \delta_{\sigma_*} \right|) \subseteq r. \end{align*}$$

This takes care of one non-trivial case. Since u is a witness that $f_* \in Z$ ,

$$\begin{align*}f_{\sigma_*} ( \left| C_{\delta_{\sigma_*} } \cap \delta_\sigma \right|) = f_* ( \left| C_{\omega_1\cap N_* } \cap \delta_\sigma \right|) \subseteq r \end{align*}$$

and

$$\begin{align*}f_{\sigma_*} ( \left| C_{\delta_{\sigma_*} } \cap (\gamma_\sigma +1) \right|) = f_* ( \left| C_{\omega_1\cap N_* } \cap (\gamma_\sigma +1) \right|) \subseteq r. \end{align*}$$

This takes care of the other non-trivial case. Therefore, $A_{u'}$ satisfies (w).⊣

Proposition 2.9. For any non-meager set X of injective functions from $\omega $ into $2^{<\omega }$ and any $r\in 2^\omega $ ,

$$\begin{align*}\Vdash_{\mathbb{Q}(X,r)}\ {"}\bigcup\left\{{ \left[ \delta_\sigma, \gamma_\sigma\right] : p\in \dot G, \sigma \in A_p }\right\} \supseteq (\omega_1\setminus \delta)\cap \mathsf{Lim} \text{ for some } \delta\in \omega_1 {"}, \end{align*}$$

where $[\delta , \gamma ] := \left \{{\xi \in \gamma +1 : \delta \leq \xi }\right \}$ , and $\mathsf {Lim}$ denotes the class of the limit ordinals.

Proof. Let $p\in \mathbb {Q}(X,r)$ and $\xi \in \omega _1\cap \mathsf {Lim}$ . Suppose that $\xi $ is not in the set $\displaystyle \bigcup _{\sigma \in A_p} \left [ \delta _\sigma , \gamma _\sigma \right ]$ and that there exists $\sigma _0 \in A_p$ such that $\gamma _{\sigma _0} < \xi $ . Since $\xi $ is a limit ordinal, $\gamma _{\sigma _0} +1 < \xi $ . We may assume that $\sigma _0$ is a largest tuple of $A_p$ with this property. Take $f_0 \in X \cap \bigcap \mathcal {N}^0_p$ .

If there are no $\tau \in A_p$ such that $\xi < \delta _\tau $ , then let $\delta \in \xi $ be a successor ordinal such that $\gamma _\sigma < \delta < \xi $ for every $\sigma \in A_p$ , and define $q:= \big \langle \mathcal {N}^0_p, \mathcal {N}^1_p, A_p\cup \left \{{ \left \langle {\gamma _{\sigma _0}, \delta , \xi +1, f_0}\right \rangle }\right \} \big \rangle $ . Suppose that $\xi < \gamma _\tau $ for some $\tau \in A_p$ . Let $\sigma _1\in A_p$ be the smallest tuple with the property that $\xi < \gamma _{\sigma _1}$ . Then by our assumption, $\xi < \delta _{\sigma _1}$ . If $\delta _{\sigma _1}$ is a successor ordinal, then define $q := \big \langle \mathcal {N}^0_p, \mathcal {N}^1_p, A_p\cup \left \{{ \left \langle {\gamma _{\sigma _0}, \gamma _{\sigma _0}+1, \delta _{\sigma _1} - 1 , f_0}\right \rangle }\right \} \big \rangle $ . If $\delta _{\sigma _1}$ is a limit ordinal, then take $\gamma \in \delta _{\sigma _1}$ such that, for any $\tau \in A_p$ with $\delta _\tau> \delta _{\sigma _1}$ , if $\delta _\tau $ is a limit ordinal, then

$$\begin{align*}C_{\delta_\tau} \cap (\gamma +1) = C_{\delta_\tau}\cap \delta_{\sigma_1}. \end{align*}$$

It follows that, if $\varepsilon _\tau < \delta _{\sigma _1}$ , then $f_\tau ( \left | C_{\delta _\tau } \cap (\gamma +1) \right |) \subseteq r$ . By extending $\gamma $ if necessary, we may assume that

$$\begin{align*}f_{\gamma_{\sigma_1}} ( | C_{\delta_{\sigma_1}} \cap (\gamma +1) | ) \subseteq r. \end{align*}$$

This can be done because the set $ \{{n\in \omega : f_{\gamma _{\sigma _1}} (n) \subseteq r} \}$ is infinite. Define $q := \big \langle \mathcal {N}^0_p, \mathcal {N}^1_p, A_p\cup \left \{{ \left \langle {\gamma _{\sigma _0}, \gamma _{\sigma _0}+1, \gamma , f_0}\right \rangle }\right \} \big \rangle $ .

In each case, q is a condition of $\mathbb {Q}(X,r)$ , and hence it is an extension of p in $\mathbb {Q}(X,r)$ and

$$\begin{align*}q \Vdash_{\mathbb{Q}(X,r)}\ \text{"} \xi \in \bigcup\left\{{ \left[ \delta_\sigma, \gamma_\sigma\right] : u \in \dot G, \sigma \in A_{u} }\right\}\! \text{"}, \end{align*}$$

which finishes the proof.⊣

It follows from this proposition that

$$ \begin{align*} \begin{array}{r@{}l} \Vdash_{\mathbb{Q}(X,r)}\text{"} & \text{the set } \left\{{ \delta_\sigma : p \in \dot G , \sigma \in A_p}\right\} \text{ is a club subset of }\omega_1, \text{ and captures } r\\ & \text{ relative to }X\text{"}. \end{array} \end{align*} $$

The following lemma shows that $\mathbb {Q}(X,r)$ almost preserves $\sqsubseteq ^{\mathrm {Cohen}}$ in the sense of Goldstern [9, Section 6, Application 3] (see also [7, Section 6.3.C]). It follows that $\mathbb {Q}(X,r)$ preserves non-meager sets of reals (from [7, Lemmas 6.3.16 and 6.3.17]). This is necessary to guarantee that a countable support iteration of forcing notions of the form $\mathbb {Q}(X,r)$ is still proper because the non-meagerness of X is used to prove properness of $\mathbb {Q}(X,r)$ . The preservation of $\sqsubseteq ^{\mathrm {Cohen}}$ is closed under countable support iterations [7, Theorems 6.1.13 and 6.3.20]. Therefore, for any non-meager set X of injective functions from $\omega $ into $2^{<\omega }$ , a countable support iteration of forcing notion of the form $\mathbb {Q}(X,r)$ is still proper and forces X to be non-meager.

Lemma 2.10. Let X be a non-meager set of injective functions from $\omega $ into $2^{<\omega }$ , $r\in 2^\omega $ , $\theta $ a regular cardinal such that $H_{\kappa } \in H_\theta $ , and $\lambda $ a regular cardinal such that $H_\theta \in H_{\lambda }$ . Then, for any countable elementary submodel $N^*$ of $H_\lambda $ which contains the set $\left \{{\mathcal {C}, X , r, H_{\kappa }, H_\theta }\right \}$ , any $c \in 2^\omega $ which is Cohen over $N^*$ , and any $p \in \mathbb {Q}(X,r) \cap N^*$ , there exists an extension $p^+$ of p in $\mathbb {Q}(X,r)$ such that $p^+$ is an $(N^*, \mathbb {Q}(X,r))$ -generic condition and

$$\begin{align*}p^+ \Vdash_{\mathbb{Q}(X,r)}\ \text{"} c \text{ is Cohen over } N^*[\dot G_{\mathbb{Q}(X,r)}] \text{"}. \end{align*}$$

Proof. As in the previous proof, let $N^*$ be a countable elementary submodel of $H_{\lambda }$ which contains $\left \{{\mathcal {C}, X , r, H_{\kappa }, H_\theta , p}\right \}$ , $M^*$ a countable elementary submodel of $H_\lambda $ which has the set $\{N^*\}$ , $\varepsilon _* \in \omega _1\cap N^*$ such that $\delta ^p_\sigma <\varepsilon _*$ for every $\sigma \in A_p$ , and $f_*$ a member of X which is Cohen over $M^*$ . Define $ \sigma _* := \left \langle {\varepsilon _* , \omega _1\cap N^*, \omega _1\cap M^*, f_* }\right \rangle $ and

$$\begin{align*}p^+ : = \left\langle{\mathcal{N}_p^0 \cup \left\{{N^* \cap H_{\kappa}}\right\}, \mathcal{N}_p^1\cup\left\{{M^* \cap H_{\kappa}}\right\}, A_p \cup\left\{{ \sigma_*}\right\} }\right\rangle. \end{align*}$$

We have shown in Lemma 2.8 that $p^+$ is $(N^*, \mathbb {Q}(X,r))$ -generic. Let us show that

$$\begin{align*}p^+ \Vdash_{\mathbb{Q}(X,r)}\ \text{"} c \text{ is Cohen over } N^*[\dot G_{\mathbb{Q}(X,r)}] \text{"}. \end{align*}$$

Suppose not. Then there are a $\mathbb {Q}(X,r)$ -name $\dot F$ for a nowhere dense subset of $2^{\omega }$ and $q\leq _{ \mathbb {Q}(X,r)} p$ such that $\dot F \in N^*$ and

$$\begin{align*}q\Vdash_{\mathbb{Q}(X,r)}\ \text{"} c \in \dot F \text{"}. \end{align*}$$

Define $\mathcal {D}$ to be the set of the conditions u of $\mathbb {Q}(X,r)$ such that there are countable elementary submodels $N'$ and $ M'$ of $H_\theta $ such that

• • $\left ( \mathcal {N}_u^0 \cup \mathcal {N}_u^1 \cup A_u \right ) \cap N' = \left ( \mathcal {N}_q^0 \cup \mathcal {N}_q^1 \cup A_q \right ) \cap N^*$ ,

• • $ \Big \{\mathcal {C}, X , r, H_{\kappa }, \dot F, \left \{{C_{\delta _\sigma } \cap N^* : \sigma \in A_q\setminus (N^* \cup \{\sigma _*\})}\right \}, \left \{{\varepsilon _\sigma : \sigma \in A_q}\right \} \cap N^* \Big \} $ $\in N' \in M'$ ,

• • $N' \cap H_\kappa \in \mathcal {N}_u^0$ and $M' \cap H_\kappa \in \mathcal {N}_u^1$ , and

• • there is a $\sigma \in A_u$ such that $\omega _1\cap N' = \delta _\sigma $ and $\omega _1\cap M' = \gamma _\sigma $ .

Then $q\in \mathcal {D} \in N^*$ . Since q is $(N^*, \mathbb {Q}(X,r))$ -generic, there exists $u \in \mathcal {D}\cap N^*$ that is compatible with q in $\mathbb {Q}(X,r)$ . Let $q^+$ be a common extension of q and u in $\mathbb {Q}(X,r)$ such that there are countable elementary submodels $N_0^*$ and $M_0^*$ of $H_\theta $ such that

• • $\left ( \mathcal {N}_u^0 \cup \mathcal {N}_u^1 \cup A_u \right ) \cap N_0^* = \left ( \mathcal {N}_q^0 \cup \mathcal {N}_q^1 \cup A_q \right ) \cap N^*$ ,

• • $ \Big \{\mathcal {C}, X , r, H_{\kappa }, \dot F, \left \{{C_{\delta _\sigma } \cap N^* : \sigma \in A_q\setminus (N^* \cup \{\sigma _*\})}\right \}, \left \{{\varepsilon _\sigma : \sigma \in A_q}\right \} \cap N^* \Big \} $ $\in N_0^* \in M_0^* \in N^*$ ,

• • $N_0 := N_0^* \cap H_\kappa \in \mathcal {N}_{q^+}^0$ and $M_0 := M_0^* \cap H_\kappa \in \mathcal {N}_{q^+}^1$ , and

• • there is a unique $\sigma _0 \in A_{q^+}$ such that $\omega _1\cap N_0 = \delta _{\sigma _0}$ and $\omega _1\cap M_0 = \gamma _{\sigma _0}$ .

As seen in the previous lemma, $q^+$ is $(N_0^*, \mathbb {Q}(X,r))$ -generic. By extending $q^+$ if necessary, we may assume that the set $\left ( \mathcal {N}_{q^+}^0 \cup \mathcal {N}_{q^+}^1 \right ) \cap N^*$ contains the set

$$\begin{align*}\begin{array}{r@{\ }l} \Big\{ \left( \left(\Psi_{N^* \cap H_\kappa}\right)^{-1} \circ \Psi_{N} \right) (K) : & N\in\mathcal{N}_{q^+}^0 \text{ with }\omega_1\cap N = \omega_1\cap N^*, \\ & K \in \left(\mathcal{N}_{q^+}^0 \cup \mathcal{N}_{q^+}^1\right) \cap N \Big\}. \end{array} \end{align*}$$

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

• • for every $\sigma \in A_q\cap N_0$ ( $= A_{q} \cap N^* = A_{q^+} \cap N_0$ ), $\gamma _\sigma < \zeta _0$ , and

• • for every $\sigma \in A_q \setminus N_0$ (then $\delta _\sigma \geq \omega _1\cap N^*> \omega _1\cap N_0 = \delta _{\sigma _0}$ ),

  1. – if $C_{\delta _\sigma }\cap N_0 \neq \emptyset $ , then $ \max \left ( C_{\delta _\sigma }\cap N_0 \right ) < \zeta _0 $ , and

  2. – if $\varepsilon _\sigma < \omega _1\cap N_0$ , then $\varepsilon _\sigma < \zeta _0$ .

For each $\nu \in 2^{<\omega }$ , each $\eta \in \omega _1$ , and each $x\in [\mathfrak {M}(X,r)]^{<\aleph _0}$ , define $\mathcal {E}(\nu ,\eta , x)$ to be the set of the conditions u of $\mathbb {Q}(X,r)$ such that

• • $A_{u} \cap (\eta ^3\times X) = A_{q}\cap N_0 $ and, for any $\tau \in A_u \setminus (A_q\cap N_0)$ , $\delta _\tau \geq \eta $ ,

• ♦ the set

  $$\begin{align*}\left\{{\left\langle{\varepsilon_\sigma, C_{ \delta_\sigma}\cap \eta, f_{\sigma} \restriction \left| C_{\delta_\sigma} \cap \eta \right| }\right\rangle : \sigma \in A_{u} \setminus \big( A_q \cap N_0\big) \text{ with } \varepsilon_\sigma < \eta}\right\} \end{align*}$$
  is equal to the set
  $$\begin{align*}\left\{{\left\langle{\varepsilon_\sigma, C_{\delta_\sigma}\cap N_0, f_{\sigma} \restriction \left| C_{\delta_\sigma} \cap N_0 \right| }\right\rangle : \sigma \in A_q \setminus N_0 \text{ with } \varepsilon_\sigma < \zeta_0}\right\} , \end{align*}$$

• • there are $N\in \mathcal {N}_u^0$ and $M\in \mathcal {N}_u^1$ such that $x \in N $ , $\mathcal {N}_u^0 \cap N = \mathcal {N}_{q}^0 \cap N_0$ , $ \omega _1\cap N = \min \left \{{\delta _\sigma : \sigma \in A_{u} \setminus (A_q \cap N_0)}\right \} $ , $\mathcal {N}_u^1 \cap M = \mathcal {N}_{q}^1 \cap M_0$ ( $=\mathcal {N}_{q}^1 \cap N_0$ ), $ \omega _1\cap M = \min \left \{{\gamma _\sigma : \sigma \in A_{u} \setminus (A_q \cap N_0)}\right \} $ , and $N \in M$ ,

• • $u \Vdash _{\mathbb {Q}(X,r)}\!\!\!\text {"}\,\, \dot F \cap [\nu ] \neq \emptyset \,\text {"} .$

By the choice of $N_0^*$ , the set

$$\begin{align*}\left\{{\mathcal{E}(\nu,\eta, x) : \nu \in 2^{<\omega}, \eta\in \omega_1, x \in \left[ \mathfrak{M}(X,r) \right]^{<\aleph_0} }\right\} \end{align*}$$

belongs to $N_0$ . Define

$$ \begin{align*} { \begin{array}{r@{\ }l} Y := \Big\{ a \in 2^\omega : & \text{ for any }k \in \omega, \text{ any }\eta\in \omega_1\setminus \zeta_0, \text{ and } \text{ any }x\in [\mathfrak{M}(X,r)]^{<\aleph_0}, \\ & \mathcal{E}(a\restriction k,\eta, x) \neq\emptyset \Big\}. \end{array} } \end{align*} $$

Y also belongs to $N_0$ .

We claim that Y is nowhere dense in $2^{\omega }$ . Let $\nu $ be in $2^{<\omega }$ . Take an extension $q'$ of $q^+$ in $\mathbb {Q}(X,r)$ and an end extension $\nu '$ of $\nu $ in $2^{<\omega }$ such that

$$\begin{align*}q' \Vdash_{\mathbb{Q}(X,r)}\ \text{"} \dot F \cap [\nu'] = \emptyset \text{"}. \end{align*}$$

By extending $q'$ if necessary, we may assume that the set $\left ( \mathcal {N}_{q'}^0 \cup \mathcal {N}_{q'}^1 \right ) \cap N_0$ includes the set

$$\begin{align*}\begin{array}{r@{\ }l} \Big\{ \left( \left(\Psi_{N_0}\right)^{-1} \circ \Psi_{N} \right) (K) : & N\in\mathcal{N}_{q'}^0 \text{ with }\omega_1\cap N = \omega_1\cap N_0, \\ & K \in \left(\mathcal{N}_{q'}^0 \cup \mathcal{N}_{q'}^1\right) \cap N \Big\}. \end{array} \end{align*}$$

Let us show that $Y \cap [\nu '] =\emptyset $ . Suppose not. Then there exists $a \in Y \cap [\nu '] $ . Let $k\in \omega $ be such that $\nu ' \subseteq a\restriction k$ , and let $\zeta _1 \in \omega _1\cap N_0$ be such that

• • for every $\sigma \in A_{q'}\cap N_0$ , $\gamma _\sigma < \zeta _1$ , and

• • for every $\sigma \in A_{q'} \setminus \left ( N_0 \cup \left \{{\sigma _0}\right \} \right )$ , if $C_{\delta _\sigma }\cap N_0 \neq \emptyset $ , then $ \max \left ( C_{\delta _\sigma }\cap N_0 \right ) < \zeta _1 $ .

Define Z to be the set of the functions h from $\omega $ into $2^{<\omega }$ such that there exists $u\in \mathcal {E}(a \restriction k, \zeta _1, \left (\mathcal {N}_{q'}^0 \cup \mathcal {N}_{q'}^1 \right )\cap N_0) \cap N_0^*$ which satisfies that, for any $\sigma \in A_{u} \setminus (A_q\cap N_0)$ , $h ( \left | C_{\omega _1\cap N_0 } \cap \delta _\sigma \right |) \subseteq r$ and $h ( \left | C_{\omega _1\cap N_0 } \cap ( \gamma _\sigma +1) \right |) \subseteq r$ . Since $a\in Y $ , it follows from elementarity of $N_0^*$ again that Z is a dense open subset of $\left ( 2^{<\omega } \right )^\omega $ . We note that Z is in $M_0^*$ . Since $f_{\sigma _0}$ is Cohen over $M_0^*$ , $f_{\sigma _0} $ is in Z. Take $u\in \mathcal {E}(a \restriction k, \zeta _1, \left ( \mathcal {N}_{q'}^0 \cup \mathcal {N}_{q'}^1 \right ) \cap N_0) \cap N_0^*$ which witnesses $f_{\sigma _0} \in Z$ . So there are $N\in \mathcal {N}_u^0$ and $M\in \mathcal {N}_u^1$ such that $ \omega _1\cap N = \min \left \{{\delta _\sigma : \sigma \in A_{u} \setminus (A_q \cap N_0)}\right \} $ , $ \omega _1\cap M = \min \left \{{\gamma _\sigma : \sigma \in A_{u} \setminus (A_q \cap N_0)}\right \} $ , $\left ( \mathcal {N}_{q'}^0 \cup \mathcal {N}_{q'}^1 \right ) \cap N_0 \in N \in M$ , $\mathcal {N}_u^0 \cap N = \mathcal {N}_{q}^0 \cap N_0$ , and $\mathcal {N}_u^1 \cap M = \mathcal {N}_{q}^1 \cap M_0$ . Then

$$\begin{align*}\mathcal{N}_u^0 \cap N = \mathcal{N}_{q}^0 \cap N_0 \subseteq \mathcal{N}_{q'}^0 \cap N_0 \in N \in \mathcal{N}_u^0 \end{align*}$$

and

$$\begin{align*}\mathcal{N}_u^1 \cap M = \mathcal{N}_u^1 \cap N = \mathcal{N}_{q}^1 \cap N_0 \subseteq \mathcal{N}_{q'}^1 \cap N_0 \in N \in M \in \mathcal{N}_u^1. \end{align*}$$

Therefore, if $A_u\cup A_{q'}$ satisfies (w), as in the proof of properness of $\mathbb {Q}(X,r)$ , u and $q'$ are compatible in $\mathbb {Q}(X,r)$ . However, a common extension of u and $q'$ in $\mathbb {Q}(X,r)$ forces both $\dot F\cap [\nu '] =\emptyset $ and $\dot F \cap [ a\restriction k] \neq \emptyset $ , which contradicts $\nu ' \subseteq a\restriction k$ . Therefore, if $A_u\cup A_{q'}$ satisfies (w), then $Y \cap [\nu '] = \emptyset $ .

We will show that $A_u\cup A_{q'}$ satisfies (w). The non-trivial case is that $\sigma \in A_u \setminus \left (A_q\cap N_0\right )$ such that $\varepsilon _\sigma < \zeta _1$ , and $\tau \in \left ( A_{q'} \cap N_0 \right ) \setminus A_q$ such that $\varepsilon _\sigma < \delta _\tau $ . Then $\delta _\tau < \zeta _1 < \delta _\sigma $ . Moreover by $\blacklozenge $ , there exists $\sigma ' \in A_q \setminus N_0$ such that

$$\begin{align*}\left\langle{\varepsilon_\sigma, C_{ \delta_\sigma}\cap \zeta_1, f_{\sigma} \restriction \left| C_{\delta_\sigma} \cap \zeta_1 \right| }\right\rangle = \left\langle{\varepsilon_{\sigma'}, C_{\delta_{\sigma'}}\cap N_0, f_{\sigma'} \restriction \left| C_{\delta_{\sigma'}} \cap N_0 \right| }\right\rangle. \end{align*}$$

Then $\sigma ' \in A_q \subseteq A_{q^+} \subseteq A_{q'}$ . Since $\tau \in A_{q'}\cap N_0$ , $\delta _\tau < \gamma _\tau < \zeta _1 <\omega _1\cap N_0$ . Thus $C_{\delta _\sigma }\cap \delta _\tau = C_{\delta _{\sigma '}}\cap \delta _\tau $ and $C_{\delta _\sigma }\cap (\gamma _\tau +1) = C_{\delta _{\sigma '}}\cap (\gamma _\tau +1)$ . Thus, since $\left \{{\sigma ', \tau }\right \}\subseteq A_{q'}$ and $\varepsilon _{\sigma '} = \varepsilon _\sigma < \delta _\tau < \omega _1\cap N_0 < \delta _{\sigma '}$ ,

$$\begin{align*}f_\sigma( \left| C_{\delta_\sigma}\cap \delta_\tau\right| ) = f_{\sigma'}( \left| C_{\delta_{\sigma'}}\cap \delta_\tau \right| ) \subseteq r \end{align*}$$

and

$$\begin{align*}f_\sigma( \left| C_{\delta_\sigma}\cap (\gamma_\tau +1)\right| ) = f_{\sigma'}( \left| C_{\delta_{\sigma'}}\cap (\gamma_\tau +1) \right| ) \subseteq r. \end{align*}$$

Since c is Cohen over $N^*$ , $c $ is not in $ Y$ . Thus, there are $k \in \omega $ , $\eta \in \omega _1\setminus \zeta _0$ , and $x\in [\mathfrak {M}(X,r)]^{<\aleph _0}$ such that $\mathcal {E}(c\restriction k,\eta , x) $ is empty. Since $c\restriction k \in N_0^*$ , by elementarity of $N_0^*$ , we may assume that $\eta \in \left (\omega _1 \cap N_0^*\right ) \setminus \zeta _0$ , and $x\in [\mathfrak {M}(X,r)]^{<\aleph _0} \cap N_0^*$ . But then q belongs to $\mathcal {E}(c\restriction k,\eta , x) $ , which is a contradiction.⊣

As in the proof of Proposition 5.1 that we will see later on, we can show that $\mathbb {Q}(X,r)$ has the $\aleph _2$ -chain condition ( $\aleph _2$ -cc). Moreover, as in [12, Section 4], we can show that $\mathbb {Q}(X,r)$ has the $\aleph _2$ -pic, which is defined by Shelah [11, Chapter VIII, Section 2] (see also [12, Section 4]). The $\aleph _2$ -pic is a stronger condition than the $\aleph _2$ -cc, and is closed under countable support iterations. Therefore, the following theorem is a consequence of the lemmas and observations in the present subsections.

Theorem 2.11. Suppose that $2^{\aleph _0} = \aleph _1$ , X is a non-meager subset of injective functions from $\omega $ into $2^{<\omega }$ , and some diamond principle (which is used in the book-keeping argument of a countable support iteration) holds. Then a countable support iteration of forcing notions of the form $\mathbb {Q}(X,r)$ with some booking argument forces the assertion ( c ) .

3 Symmetric systems of relational structures

This section is similar to Section 4 of the paper [10]. The idea of this section is due to Tadatoshi Miyamoto. The notion in this section will be used in the definition of our forcing notion which forces the assertion (c).

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 $ .

If M and $M'$ are countable elementary submodels of $H_\kappa $ with the set $\left \{{\vec {\mathbb R}}\right \}$ such that $\omega _1\cap M = \omega _1 \cap M'$ , then $\mathbb {R} \cap M = \mathbb {R}\cap M'$ . So, as in Section 2.2, the set of Borel codes in M coincides with the one in $M'$ . Therefore, for any $f\in \left ({2^{<\omega }}\right )^\omega $ , f is Cohen over M iff f is Cohen over $M'$ .

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.

Definition 3.3. $\mathfrak {M}_0$ is the set of countable elementary submodels N of $H_\kappa $ such that $\left \{{\vec {\mathbb {R}}, \vec X }\right \} \in N$ and the structure $\left \langle {N, \in ,\Phi \cap N}\right \rangle $ is an elementary substructure of the structure $\left \langle {H_\kappa ,\in ,\Phi }\right \rangle $ .

As in Section 2.2, we always consider the members of $\mathfrak {M}_0$ as substructures of the structure $ \left \langle { H_\kappa ,\in , {\omega _1}, \vec {\mathbb {R}}, \vec X, \Phi }\right \rangle $ . For each $M \in \mathfrak {M}_0$ , the transitive collapse of M is considered as the structure $ \left \langle { \mathrm {trcl}(M),\in , {\omega _1}\cap M, \vec {\mathbb {R}}\cap M, \vec X \cap M, \overline {\Phi \cap M}}\right \rangle $ , which is denoted by $\overline M$ , where $\overline {\Phi \cap M}$ is considered as the image, under the collapsing function of M, of $\Phi \cap M$ . As in Section 2.2, $\Psi _M$ denotes the transitive collapsing map from M onto $\overline M$ . So when M and $M'$ in $\mathfrak {M}_0$ are isomorphic, the composition ${\Psi _{M'}}^{-1}\circ \Psi _M$ is an isomorphism from the structure $ \Big \langle M,\in , {\omega _1}, \vec {\mathbb {R}} , \vec X , \Phi \cap M \Big \rangle $ onto the structure $ \Big \langle M', \in , {\omega _1}, \vec {\mathbb {R}} , \vec X , \Phi \cap M' \Big \rangle. $

Definition 3.4. A finite subset $\mathcal {M}$ of $\mathfrak {M}_0$ is called a symmetric system if

• (ho) for each $M,M'\in \mathcal {M}$ , if $\omega _1\cap M=\omega _1\cap M'$ , then $\overline M=\overline {M'}$ ,

• (up) for each $M, M'\in \mathcal {M}$ , if $\omega _1\cap M'<\omega _1\cap M$ , then there exists $M"\in \mathcal {M}$ such that $\overline {M"}=\overline M$ and $M'\in M"$ ,

• (down) for each $M_0, M_1 \in \mathcal {M}$ and each $M'\in \mathcal {M}\cap M_0$ , if $\overline {M_0} =\overline {M_1}$ , then $\left ({\Psi _{M_1}}^{-1}\circ \Psi _{M_0}\right ) (M')$ belongs to $\mathcal {M}$ , and

• (id) for each $M,M'\in \mathcal {M}$ , if $\omega _1\cap M=\omega _1\cap M'$ , then the function

  $$\begin{align*}\Big({\Psi_{M'}}^{-1}\circ\Psi_M\Big) \restriction ( M\cap M') \end{align*}$$
  is the identity.

The requirement (id) comes from the Asperó–Mota iteration [3]. This was used to show properness whenever the length of the iteration has uncountable cofinality. In this paper, the requirement (id) will be used in other places, for example, Propositions 3.10 and 3.11.

We will deal with symmetric systems of relational structures. To introduce such systems, we define the following notions, and mention some necessary propositions.

Definition 3.5. Let $( \mathbb {P}, \leq _{\mathbb {P}})$ be a forcing notion with the $\kappa $ -chain condition such that $\mathbb {P}\subseteq H_\kappa $ . We define the expanded relational structure by $\mathbb {P}$ to the relational structure

$$\begin{align*}\left\langle{ H_\kappa, \in, \mathbb{P}, \leq_{\mathbb{P}}, H^{\mathbb{P}}_\kappa, R^{\mathbb{P}}_=, R^{\mathbb{P}}_\in, \vec{\mathbb{R}} , \vec X , \Phi}\right\rangle , \end{align*}$$

where

• • $H^{\mathbb {P}}_\kappa := V^{\mathbb {P}}\cap H_\kappa $ , where $V^{\mathbb {P}}$ denotes the class of all $\mathbb {P}$ -names,

• • $R^{\mathbb {P}}_= :=\left \{{(p,\tau ,\pi )\in \left (\mathbb {P}\times V^{\mathbb {P}}\times V^{\mathbb {P}}\right )\cap H_\kappa : p\Vdash _{\mathbb {P}}\text {"} \tau =\pi \,\text {"}}\right \}$ , and

• • $R^{\mathbb {P}}_\in :=\left \{{(p,\tau ,\pi )\in \left (\mathbb {P}\times V^{\mathbb {P}}\times V^{\mathbb {P}}\right )\cap H_\kappa : p\Vdash _{\mathbb {P}}\text {"} \tau \in \pi \,\text {"}}\right \}$ .

For each $M\in \mathfrak {M}_0$ , M is also considered as the substructure

$$\begin{align*}\Big\langle M, \in\cap M^2 , \mathbb{P}\cap M, \leq_{\mathbb{P}}\cap M^2, H^{\mathbb{P}}_\kappa \cap M, R^{\mathbb{P}}_= \cap M^3, R^{\mathbb{P}}_\in \cap M^3, \vec{\mathbb{R}} , \vec X , \Phi\cap M \Big\rangle \end{align*}$$

of the expanded relational structure by $\mathbb {P}$ , and we write $M\prec \mathbb {P}$ when the structure M is an elementary substructure of the expanded relational structure by $\mathbb {P}$ .

The forcing notions we will define in Section 4 are not members of $H_\kappa $ , but subsets of $H_\kappa $ . The following can be proved in a similar way as in [11, Chapter III, Section 2]. Here, $p\in \mathbb {P}$ is called $(M,\mathbb {P})$ -generic if, for any predense subset $\mathcal {D}\in M$ of $\mathbb {P}$ , $\mathcal {D}\cap M$ is predense below p in $\mathbb {P}$ . For the proof of the following proposition, see, e.g., [11, Chapter III, Theorem 2.11].

Proposition 3.6. Suppose that $\mathbb {P}$ is a forcing notion with the $\kappa $ -chain condition such that $\mathbb {P}\subseteq H_\kappa $ .

1. 1. If $\theta $ is a large enough regular cardinal for $\mathbb {P}$ and $M^*$ is a countable elementary submodel of $H_\theta $ which contains the set

   $$\begin{align*}\Big\{ H_\kappa,\in, \mathbb{P}, \leq_{\mathbb{P}}, \vec{\mathbb{R}}, \vec X, \Phi \Big\} \end{align*}$$
   as a member, then $M^*\cap H_\kappa \in \mathfrak {M}_0$ and $M^*\cap H_\kappa \prec \mathbb {P}$ .

2. 2. For any $M\in \mathfrak {M}_0$ with $M\prec \mathbb {P}$ , and any $p\in \mathbb {P}$ , the followings are equivalent:

   • • p is $(M,\mathbb {P})$ -generic,

   • • $p\Vdash _{\mathbb {P}} {"} M[\dot G]\cap {H_\kappa }^V = M{"}$ , where ${H_\kappa }^V $ denotes $H_\kappa $ in the ground model, and

   • • $p\Vdash _{\mathbb {P}}{"} M[\dot G]\cap \kappa =M\cap \kappa{"}$ .

3. 3. For any $M\in \mathfrak {M}_0$ with $M\prec \mathbb {P}$ ,

   $$ \begin{align*} \begin{array}{r@{}l} \Vdash_{\mathbb{P}}\ \ &{"} \text{the structure } \\ & \Big\langle M[\dot G], \in \cap M[\dot G]^2, H_\kappa^ V\cap M[\dot G], \mathbb{P}\cap M[\dot G], \leq_{\mathbb{P}}\cap M[\dot G]^2, \dot G\cap M[\dot G], \\ &\hfill H^{\mathbb{P}}_\kappa \cap M[\dot G], R^{\mathbb{P}}_{=} \cap M[\dot G]^3, R^{\mathbb{P}}_\in \cap M[\dot G]^3, \vec{\mathbb{R}} , \vec X , \Phi \cap M[\dot G] \Big\rangle \\ & \text{ is an elementary substructure of the structure } \\ & \Big\langle H_\kappa^{V[\dot G]}, \in, H_\kappa^V, \mathbb{P}, \leq_{\mathbb{P}}, \dot G, H^{\mathbb{P}}_\kappa , R^{\mathbb{P}}_= , R^{\mathbb{P}}_\in , \vec{\mathbb{R}} , \vec{X}, \Phi \Big\rangle{"}. \end{array} \end{align*} $$

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 $ .

Definition 3.8. Let $\alpha \in \kappa +1$ , and $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ a sequence of forcing notions such that $\mathbb {P}_\xi \subseteq H_\kappa $ and $\mathbb {P}_\xi $ has the $\kappa $ -chain condition for each $\xi \leq \alpha $ . We define the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ to the structure

$$\begin{align*}\Big\langle H_\kappa, \in, \mathbb{P}_\alpha, \leq_{\mathbb{P}_\alpha}, H^{\mathbb{P}_\alpha}_\kappa, R^{\mathbb{P}_\alpha}_=, R^{\mathbb{P}_\alpha}_\in, \langle\!\langle H^{\mathbb{P}_\xi}_\kappa, R^{\mathbb{P}_\xi}_=, R^{\mathbb{P}_\xi}_\in: \xi \in \alpha \rangle\!\rangle \Big\rangle. \end{align*}$$

For each $M\in \mathfrak {M}_0$ , M is also considered as the substructure

$$ \begin{align*} { \begin{array}{l} \Big\langle M, \in \cap M^2 , \mathbb{P}_\alpha\cap M, \leq_{\mathbb{P}_\alpha}\cap M^2, H^{\mathbb{P}_\alpha}_\kappa \cap M, R^{\mathbb{P}_\alpha}_= \cap M^3, R^{\mathbb{P}_\alpha}_\in \cap M^3, \\ \hfill \langle\!\langle H^{\mathbb{P}_\xi}_\kappa \cap M, R^{\mathbb{P}_\xi}_= \cap M^3, R^{\mathbb{P}_\xi}_\in \cap M^3 : \xi \in \alpha \cap M \rangle\!\rangle \Big\rangle \end{array} } \end{align*} $$

of the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ , and we write $M\prec \left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ when the structure M is an elementary substructure of the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ .

The following is a variation of Proposition 3.6 for iterated forcing.

Proposition 3.9. Suppose that $\alpha \in \kappa +1$ , and $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ is a sequence of forcing notions such that $\mathbb {P}_\xi \subseteq H_\kappa $ and $\mathbb {P}_\xi $ has the $\kappa $ -chain condition for each $\xi \leq \alpha $ .

1. 1. If $\theta $ is a large enough regular cardinal for the iteration $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ and $M^*$ is a countable elementary submodel of $H_\theta $ which contains the set

   $$\begin{align*}\Big\{ H_\kappa,\in, \left\langle{\mathbb{P}_\xi:\xi\leq\alpha}\right\rangle, \vec{\mathbb{R}} , \vec X , \Phi \Big\} \end{align*}$$
   as a member, then $M^*\cap H_\kappa \in \mathfrak {M}_0$ and $M^*\cap H_\kappa \prec \left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ .

2. 2. If $\alpha <\kappa $ , then for any $M\in \mathfrak {M}_0$ with $M\prec \left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ , $\alpha $ belongs to M.

3. 3. For any $M\in \mathfrak {M}_0$ with $M\prec \left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ and any $\beta \in \alpha $ , if $\beta \in M$ , then $M\prec \left \langle {\mathbb {P}_\xi :\xi \leq \beta }\right \rangle $ .

The following is necessary for our symmetric systems of relational structures. This is the reason why we introduce the relational structures equipped with forcing notions that are subsets of $H_\kappa $ .

Proposition 3.10. Suppose that $M, N_0,N_1\in \mathfrak {M}_0$ are elementary substructures of the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ , $N_0$ and $N_1$ are isomorphic as substructures of the expanded relational structure by the sequence $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ (then the map $\Psi ={\Psi _{N_1}}^{-1}\circ \Psi _{N_0}$ is the isomorphism from $N_0$ onto $N_1$ as substructures of the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ ), $\beta \leq \alpha $ is such that $\Psi (\beta )=\beta $ , and $M\in \mathfrak {M}_0\cap N_0$ . Then

• • if $M\prec \left \langle {\mathbb {P}_\xi : \xi \leq \alpha }\right \rangle $ , then the structure

  $$ \begin{align*} { \begin{array}{l} \Big\langle M, \in \cap M^2 , \mathbb{P}_\beta\cap M, \leq_{\mathbb{P}_\beta}\cap M^2, H^{\mathbb{P}_\beta}_\kappa \cap M, R^{\mathbb{P}_\beta}_= \cap M^3, R^{\mathbb{P}_\beta}_\in \cap M^3, \\ \hfill \langle\!\langle H^{\mathbb{P}_\xi}_\kappa \cap M, R^{\mathbb{P}_\xi}_= \cap M^3, R^{\mathbb{P}_\xi}_\in \cap M^3 : \xi \in \beta \cap M \rangle\!\rangle \Big\rangle \end{array} } \end{align*} $$
  is an elementary substructure of the structure
  $$ \begin{align*} { \begin{array}{l} \Big\langle {N_0}, \in \cap {N_0}^2 , \mathbb{P}_\beta\cap {N_0}, \leq_{\mathbb{P}_\beta}\cap {N_0}^2, H^{\mathbb{P}_\beta}_\kappa \cap {N_0}, R^{\mathbb{P}_\beta}_= \cap {N_0}^3, R^{\mathbb{P}_\beta}_\in \cap {N_0}^3, \\ \hfill \langle\!\langle H^{\mathbb{P}_\xi}_\kappa \cap {N_0}, R^{\mathbb{P}_\xi}_= \cap {N_0}^3, R^{\mathbb{P}_\xi}_\in \cap {N_0}^3 : \xi \in \beta \cap {N_0} \rangle\!\rangle \Big\rangle , \end{array} }\end{align*} $$
  and

• • $ \Psi (M) \prec \left \langle {\mathbb {P}_\xi :\xi \leq \beta }\right \rangle $ , and the structure

  $$ \begin{align*} { \begin{array}{l} \Big\langle \Psi(M), \in \cap \Psi(M)^2, \mathbb{P}_\beta\cap \Psi(M), \leq_{\mathbb{P}_\beta}\cap \Psi(M)^2, H^{\mathbb{P}_\beta}_\kappa \cap \Psi(M), R^{\mathbb{P}_\beta}_= \cap \Psi(M)^3, \\ R^{\mathbb{P}_\beta}_\in \cap \Psi(M)^3, \langle\!\langle H^{\mathbb{P}_\xi}_\kappa \cap \Psi(M), R^{\mathbb{P}_\xi}_= \cap \Psi(M)^3, R^{\mathbb{P}_\xi}_\in \cap \Psi(M)^3 : \xi \in \beta \cap \Psi(M) \rangle\!\rangle \Big\rangle \end{array} }\end{align*} $$
  is an elementary substructure of the structure
  $$ \begin{align*} { \begin{array}{l} \Big\langle {N_1}, \in \cap {N_1}^2 , \mathbb{P}_\beta\cap {N_1}, \leq_{\mathbb{P}_\beta}\cap {N_1}^2, H^{\mathbb{P}_\beta}_\kappa \cap {N_1}, R^{\mathbb{P}_\beta}_= \cap {N_1}^3, R^{\mathbb{P}_\beta}_\in \cap {N_1}^3, \\ \hfill \langle\!\langle H^{\mathbb{P}_\xi}_\kappa \cap {N_1}, R^{\mathbb{P}_\xi}_= \cap {N_1}^3, R^{\mathbb{P}_\xi}_\in \cap {N_1}^3 : \xi \in \beta \cap {N_1} \rangle\!\rangle \Big\rangle. \end{array} }\end{align*} $$

The following is a key point of the proof of properness of our forcing notions.

Proposition 3.11. Suppose that $\alpha \in \omega _2\leq \kappa $ , $N_0,N_1\in \mathfrak {M}_0$ , and $N_0$ and $N_1$ are isomorphic as the substructures of the expanded relational structure by $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ . Then $N_0\cap \alpha = N_1\cap \alpha $ .

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

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

⊣

4 Definition of Asperó–Mota iteration to force (c)

In this section, we define our forcing notion $\mathbb {P}_\kappa $ that forces the assertion ( c ) . $\mathbb {P}_\kappa $ is defined by an Asperó–Mota iteration of forcing notions playing the same role as $\mathbb {Q}(X,r)$ in Section 2.2.

We notice that, for each $M\in \mathfrak {M}_0$ and $\alpha \in \kappa +1$ , any initial segment of $\alpha \cap M$ is of the form $\beta \cap M$ for some $\beta \in \alpha +1$ (which is not necessary unique). For each $\alpha \in \kappa +1$ , we will define the forcing notion $\mathbb {P}_\alpha $ to be a subset of the set

$$\begin{align*}U_\alpha:= \begin{array}[t]{l} \left[ \mathfrak{M}_0 \right]^{<\aleph_0} \\[6pt] \times \displaystyle \left\{ \bigcup_{\left\langle{M,\beta}\right\rangle\in Z}\{M\}\times(\beta \cap M): Z\in [\mathfrak{M}_0\times (\alpha+1)]^{<\aleph_0} \right\} \\[-8pt] \\ \times \displaystyle \left( \bigcup_{D\in [\alpha]^{<\aleph_0}} \left( \left[ \omega_1\times \omega_1\times\omega_1 \right]^{<\omega} \right)^D \right). \end{array} \end{align*}$$

Since $\mathfrak {M}_0$ is a subset of $H_\kappa $ , for each $\alpha \in \kappa +1$ , the forcing notion $\mathbb {P}_\alpha $ is a subset of $H_\kappa $ .

To define $\mathbb {P}_\alpha $ , we introduce the following notation. For each $\alpha \in \kappa +1$ and $p=(\mathcal {N}_p, R_p, A_p)\in U_\alpha $ ,

• • $\operatorname {dom}(R_p) :=\left \{{M : \text { there is } \zeta \in \alpha \text { so that } \left \langle {M,\zeta }\right \rangle \in R_p }\right \}$ ,

• • $\operatorname {ran}(R_p) :=\left \{{\zeta : \text { there is } M \in \mathfrak {M}_0 \text { so that } \left \langle {M,\zeta }\right \rangle \in R_p }\right \}$ ,

• • for each $I\subseteq \alpha $ ,

  $$\begin{align*}{R_p}^{-1}[ I ] := \big\{M : \text{ there is }\ \zeta\in I\ \text{so that } \left\langle{M,\zeta}\right\rangle\in R_p \big\} , \end{align*}$$

• • for each $M\in \operatorname {dom}(R_p)$ ,

  $$\begin{align*}R_p(M):=\left\{{\zeta\in\operatorname{ran}(R_p): \left\langle{M,\zeta}\right\rangle\in R_p}\right\} \!, \end{align*}$$

• • for each $\beta \in \alpha $ , define $p\restriction \beta = (\mathcal {N}_{p\restriction \beta }, R_{p\restriction \beta },A_{p\restriction \beta })$ to be the member of $U_\beta $ such that

  1. – $\mathcal {N}_{p\restriction \beta } :=\mathcal {N}_p$ ,

  2. – $ R_{p\restriction \beta } := R_p\cap \left (\mathfrak {M}_0\times \beta \right ) \!, $ and

  3. – $A_{p\restriction \beta }:=A_p\restriction \beta $ , the restriction of the function $A_p$ to the set $\beta $ .

For $p\in U_\alpha $ and $M\in \mathcal {N}_p$ , members of the set $R_p(M)$ are called markers of M.

We define a forcing notion $\mathbb {P}_\alpha $ satisfying the $\aleph _2$ -cc (under $\operatorname {\mathsf {CH}}$ ), by recursion on $\alpha \in \kappa +1$ . When we have defined the sequence $\left \langle {\mathbb {P}_\xi :\xi \leq \alpha }\right \rangle $ of forcing notions, we will define the subset $\mathfrak {M}_{\alpha }^P$ of $\mathfrak {M}_{0}$ by

$$\begin{align*}\mathfrak{M}^P_{\alpha}:= \left\{{ M\in \mathfrak{M}_{0}: M\prec \left\langle{\mathbb{P}_\xi:\xi\leq\alpha}\right\rangle }\right\}\!. \end{align*}$$

As in Proposition 3.9(2), if $\alpha \in \kappa $ and $M\in \mathfrak {M}^P_\alpha $ , then $\alpha \in M$ . As seen below, for each $\alpha \in \kappa $ , $\mathbb {P}_{\alpha }$ will be defined from the set

$$\begin{align*}\left\{{ {\omega_1}, \vec{\mathbb{R}}, \vec X, H_\kappa , \Phi, \langle\!\langle \mathfrak{M}_\xi^P:\xi\in\alpha \rangle\!\rangle }\right\}. \end{align*}$$

$\mathbb {P}_\kappa $ can be considered as the direct limit of $\left \langle {\mathbb {P}_\alpha :\alpha \in \kappa }\right \rangle $ .

Definition 4.1. The forcing notion $\mathbb {P}_\alpha $ is defined by recursion on $\alpha \in \kappa +1$ . However, each $\mathbb {P}_\alpha $ is defined uniformly. $\mathbb {P}_\alpha $ consists of the members $p=(\mathcal {N}_p, R_p, A_p)$ of $U_\alpha $ satisfying the following conditions:

• (ob) $\bullet $ $\mathcal {N}_p$ is finite and forms a symmetric system, and

  • • $\operatorname {dom}(R_p) \subseteq \mathcal {N}_p$ , and, for each $M\in \operatorname {dom}(R_p)$ , $R_p(M)$ is an initial segment of the set $\alpha \cap M$ .

• (el) For each $\zeta \in \alpha $ , ${R_p}^{-1}[\{ \zeta \}] \subseteq \mathfrak {M}^P_\zeta $ .

• (ho) For each $\zeta \in \alpha $ and each $M_0,M_1\in {R_p}^{-1}[\{ \zeta \}]$ , if $\omega _1\cap M_0=\omega _1\cap M_1$ , then the structure $\left \langle {M_0,\in , \vec {\mathbb {R}} , \vec X , \Phi \restriction M_0 , \langle \!\langle \mathbb {P}_\xi :\xi \in (\zeta +1)\cap M_0 \rangle \!\rangle }\right \rangle $ is isomorphic to the structure $\left \langle {M_1,\in , \vec {\mathbb {R}} , \vec X , \Phi \restriction M_1 , \langle \!\langle \mathbb {P}_\xi :\xi \in (\zeta +1) \cap M_1 \rangle \!\rangle }\right \rangle $ .

• (up) For each $\zeta \in \alpha $ and each $M,N_0\in {R_p}^{-1}[\{ \zeta \}]$ , if $\omega _1\cap M <\omega _1\cap N_0$ , then there exists $N_1\in {R_p}^{-1}[\{\zeta \}]$ such that $M\in N_1$ and $\omega _1\cap N_1=\omega _1\cap N_0$ .

• (down) For each $\zeta \in \alpha $ and each $M,N_0,N_1\in {R_p}^{-1}[\{ \zeta \}]$ , if $M\in N_0$ and $\omega _1\cap N_0=\omega _1\cap N_1$ , then $\Big ({\Psi _{N_1}}^{-1}\circ \Psi _{N_0}\Big )(M) \in {R_p}^{-1}[\{\zeta \}]$ .

• (g) If $\xi \in \operatorname {dom}(A_p)$ and $p\restriction \xi $ belongs to $\mathbb {P}_\xi $ , then $\Phi (\xi )=\left \{{\dot r_\xi }\right \}$ such that $\dot r_\xi $ is a $\mathbb {P}_\xi $ -name for a function from $\omega $ into $2$ . Moreover,

  • (g-ob) $A_p(\xi )$ is a finite set of triples of the form $\sigma = \left \langle {\varepsilon _\sigma , \delta _\sigma , \gamma _\sigma }\right \rangle $ such that $\varepsilon _\sigma \in \delta _\sigma \in \gamma _\sigma \in \omega _1$ ,

  • (g-ob-2) the set $\left \{{\delta _\sigma : \sigma \in A_p}\right \} $ includes the set $\left \{{\omega _1\cap N : N \in {R_p}^{-1}[\{\xi \}]}\right \}$ ,

  • (g-cl) for each $\left \{{\sigma , \tau }\right \} \in \left [ A_p(\xi ) \right ]^2$ , either $\gamma _\sigma < \delta _{\tau }$ or $\gamma _{\tau } < \delta _\sigma $ ,

  • (g-w) for any $\sigma \in A_p(\xi )$ , if $\delta _\sigma $ is a limit ordinal, then

    $$ \begin{align*} \begin{array}{r@{}l} \kern-10pt p\restriction \xi \Vdash_{\mathbb{P}_\xi}\ \text{"} & \left\{{ n \in \omega : f_{\gamma_\sigma}(n) \subseteq \dot r_\xi}\right\} \text{ is infinite, and, for any }\tau \in A_p(\xi) \setminus \left\{{\sigma}\right\}\\ &\text{ with } \varepsilon_{\sigma} < \delta_\tau < \delta_\sigma, f_{\gamma_\sigma} ( \left| C_{\delta_\sigma} \cap \delta_{\tau} \right| ) \subseteq \dot r_\xi \text{ and }\\ & f_{\gamma_\sigma} ( \left| C_{\delta_\sigma} \cap (\gamma_{\tau} +1) \right| ) \subseteq \dot r_\xi \text{"} , \end{array} \end{align*} $$

  • (g-m) for each $\sigma \in A_p(\xi )$ and each $N\in {R_p}^{-1}[\{\xi \}]$ (that is, $\{N\}\times \big ( (\xi +1)\cap N\big ) \subseteq R_p$ ) with $\omega _1\cap N = \delta _\sigma $ , there exists $M\in \mathcal {N}_p \cap \mathfrak {M}^P_\xi $ such that

    • • $N\in M$ ,

    • • $\omega _1\cap M = \gamma _\sigma $ , and

    • • $\{M\}\times (\xi \cap M) \subseteq R_p$ (then, by (g-ob-2) and (g-cl), $M \not \in {R_p}^{-1}[\{\xi \}]$ ).

By definition, we can check that, for each $p\in \mathbb {P}_\alpha $ and $\zeta \in \alpha $ , $p\restriction \zeta $ is a condition of $\mathbb {P}_\zeta $ . The order of $\mathbb {P}_\alpha $ is defined as follows: For each $p,q\in \mathbb {P}_{\alpha }$ , $q\leq _{\mathbb {P}_{\alpha }} p$ iff

• • $\mathcal {N}_q\supseteq \mathcal {N}_p$ ,

• • $R_q\supseteq R_p$ ,

• • $\operatorname {dom}(A_q)\supseteq \operatorname {dom}(A_p)$ ,

• • for each $\xi \in \operatorname {dom}(A_p)$ , $A_q(\xi )\supseteq A_p(\xi )$ .

By definition, we can check that, for each $p,q\in \mathbb {P}_\alpha $ with $q \leq _{\mathbb {P}_\alpha }p$ , and each $\zeta \in \alpha $ , $q\restriction \zeta \leq _{\mathbb {P}_\zeta }p \restriction \zeta $ . By definition, $\mathbb {P}_\kappa $ is equivalent to the direct limit of $\left \langle {\mathbb {P}_\alpha :\alpha \in \kappa }\right \rangle $ .

Lemma 4.2. Suppose that $\alpha ,\beta \in \kappa +1$ with $\beta <\alpha $ . Then $\mathbb {P}_\beta $ can be completely embeddable into $\mathbb {P}_\alpha $ .

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.⊣

Definition 4.3. For each $\xi \in \kappa $ , define the $\mathbb {P}_{\xi +1}$ -name $\dot E_\xi $ by

$$\begin{align*}\Vdash_{\mathbb{P}_{\xi+1}}\ \text{"} \dot E_\xi := \left\{{\delta_\sigma : q\in \dot G_{\mathbb{P}_{\xi+1}}, \sigma \in A_p(\xi) }\right\} \text{"}. \end{align*}$$

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.

5 Forcing (c) by $\mathbb {P}_{\omega _2}$

Proposition 5.1. For every $\alpha \in \kappa +1$ , $\mathbb {P}_\alpha $ has the $\aleph _2$ -chain condition. In fact, every subset of $\mathbb {P}_\alpha $ of size $\aleph _2$ has a pairwise compatible subset of size $\aleph _2$ .

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) 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 $p_{\zeta '}$ .⊣

Lemma 5.2. Each $\alpha $ in $\omega _2+1$ satisfies the following assertions.

• $\mathsf {(p)}_\alpha $ : For any $p\in \mathbb {P}_{\alpha }$ and any $N\in \mathcal {N}_p \cap \mathfrak {M}^P_\alpha $ such that $\{N\}\times (\alpha \cap N) \subseteq R_p$ , p is $(N,\mathbb {P}_\alpha )$ -generic.

• $\mathsf {(C)}_\alpha $ : For any $p\in \mathbb {P}_{\alpha }$ and any $N \in \mathcal {N}_p \cap \mathfrak {M}^P_\alpha $ such that $\{N \}\times (\alpha \cap N) \subseteq R_p$ ,

  $$\begin{align*}p \Vdash_{\mathbb{P}_\alpha}\ {"} f_{\omega_1\cap N} \text{ is Cohen over } N[\dot G_{\mathbb{P}_\alpha}]{"}. \end{align*}$$

This is proved by induction on $\alpha \in \omega _2+1$ . A point is that, if $\alpha \in \omega _2+1$ , $p\in \mathbb {P}$ , and $N\in \mathcal {N}_p$ , then $R_p(N) \subseteq \omega _2$ . In the following proof, we will use Propositions 3.10 and 3.11 frequently.

Proof of  $\mathsf {(p)}_0$ .

This proof is a standard proof in the context of the side condition method (see, e.g., [12, Lemma 4]), and similar to the proof in [10]. 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 $\mathbb {P}_{\alpha +1}$ . $\dashv$

The following proposition will be used in the rest of the proof.

Proposition 5.3. Suppose that $\alpha \in \omega _2$ , $\mathsf {(p)}_{\alpha +1}$ holds, $p\in \mathbb {P}_{\alpha +1}$ , $N\in \mathcal {N}_p \cap \mathfrak {M}^P_{\alpha +1}$ such that $\{N\}\times ((\alpha +1)\cap N)\subseteq R_p$ , and $\mathcal {D}$ is a definable class in the expanded relational structure by $\mathbb {P}_{\alpha +1}$ with parameters in N such that $p\in \mathcal {D}$ . Then there exists $q\in \mathcal {D}\cap N$ which is compatible with p in $\mathbb {P}_{\alpha +1}$ .

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 $\mathcal {D}$ . $\dashv$

The following proof has similarities with the proof of Lemma 2.10 (although it is not identical to it).

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

Since $\{N_0\}\times (\alpha \cap N_0) \subseteq R_{q^+} \subseteq R_r$ , by $\mathsf {(p)}_\alpha $ , r is $(N_0, \mathbb {P}_\alpha )$ -generic. Since the set $\left \{{f_{\omega _1\cap N}\restriction k, A_{u_0}(\alpha )\cap N_0}\right \}$ is in $N_0$ , $\dot \Sigma (f_{\omega _1\cap N}\restriction k, A_{u_0}(\alpha )\cap N_0) $ is also in $N_0$ . Moreover, $A_{u_0}(\alpha )\setminus N_0 $ has $\sigma $ such that $\delta _\sigma =\omega _1\cap N_0$ , which is larger than $\zeta _0$ . Hence, by Proposition 5.3, there are $r'\leq _{\mathbb {P}_\alpha } r$ and $B\in [\omega _1\times \omega _1\times \omega _1]^{<\aleph _0}\cap N_0$ such that

• • for every $\tau \in B$ , $\zeta _0 < \delta _\tau $ , and

• • $r' \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}B \in \dot \Sigma (f_{\omega _1\cap N}\restriction k, A_{u_0}(\alpha )\cap N_0) \,\text {"} $ .

Since $\{N\}\times (\alpha \cap N) \subseteq R_{q^+} \subseteq R_{r'}$ , by $\mathsf {(p)}_\alpha $ , $r'$ is $(N, \mathbb {P}_\alpha )$ -generic. Since N is an elementary substructure of the expanded relational structure by $\mathbb {P}_{\alpha +1}$ , there exists $u_1 \in \mathbb {P}_{\alpha +1}\cap N$ which witnesses $r' \Vdash _{\mathbb {P}_\alpha }\!\!\!\text {"}B \in \dot \Sigma (f_{\omega _1\cap N}\restriction k, A_{u_0}(\alpha )\cap N_0) \,\text {"} $ . Then $r'$ is a common extension of $u_1$ and $q\restriction \alpha $ in $\mathbb {P}_\alpha $ . Define $s\in U_{\alpha +1}$ such that

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

• • $R_{s}:=R_{u_1}\cup R_{q} $ $\cup \Big \{$ $\begin {array}[t]{@{}l@{\,}c@{\,}l} \left \langle {\left ( {\Psi _{M'} }^{-1} \circ \Psi _{N} \right ) (K), \alpha }\right \rangle & : & M'\in {R_{q}}^{-1}[\{\alpha \}] \text { with } \omega _1\cap M' = \omega _1\cap N, \\ & & K\in {R_{u_1}}^{-1}[\{\alpha \}] \Big \} , \end {array}$

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

• • $A_{s}(\alpha ) := A_{u_1}(\alpha ) \cup A_{q}(\alpha )$ .

Now $s\restriction \alpha =r' \in \mathbb {P}_\alpha $ . Since $u_1\in N$ and

$$\begin{align*}{R_q}^{-1}[\{\alpha\}]\cap N \subseteq {R_{u_1}}^{-1}[\{\alpha\}] \in N \in {R_q}^{-1}[\{\alpha\}] , \end{align*}$$

${R_s}^{-1}[\{\alpha \}]$ satisfies (el), (ho), (up), and (down). $A_s(\alpha )$ satisfies (g-ob), (g-ob-2), (g-cl), and (g-m). We will check that $A_s(\alpha )$ satisfies (g-w). Since $u_1\leq _{\mathbb {P}_{\alpha +1}} q'$ ,

$$\begin{align*}A_q(\alpha) \cap N = A_{q'}(\alpha) \cap N_0 \subseteq A_{u_1}(\alpha)\cap N_0. \end{align*}$$

By the choice of $u_1$ ,

$$\begin{align*}A_{u_1}(\alpha) = \left( A_{u_0}(\alpha)\cap N_0 \right) \cup B \subseteq N_0 \subseteq N. \end{align*}$$

Let $\tau \in A_{u_1}(\alpha )$ and $\sigma \in A_q(\alpha )\setminus (N\cup \left \{{\sigma _N}\right \})$ . Then there exists $\sigma ' \in A_{q'}(\alpha )\setminus N_0 $ such that

$$\begin{align*}\left\langle{\varepsilon_\sigma, C_{\delta_\sigma} \cap N_0 , f_{\gamma_\sigma} \restriction \left| C_{\delta_\sigma} \cap N_0 \right| }\right\rangle = \left\langle{\varepsilon_{\sigma'}, C_{\delta_{\sigma'}} \cap N , f_{\gamma_{\sigma'}} \restriction \left| C_{\delta_{\sigma'}} \cap N \right| }\right\rangle. \end{align*}$$

So if $\varepsilon _\sigma < \delta _\tau $ , then, since $u_1\leq _{\mathbb {P}_{\alpha +1}} q'$ ,

$$\begin{align*}\begin{array}{r@{\ }l} u_1\restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} &\!\! f_{\gamma_{\sigma}} ( \left| C_{\delta_\sigma} \cap \delta_\tau \right|) = f_{\gamma_{\sigma'}} ( \left| C_{\delta_{\sigma'}} \cap \delta_\tau \right|) \subseteq \dot r_\alpha \text{ and } \\ & f_{\gamma_{\sigma}} ( \left| C_{\delta_\sigma} \cap (\gamma_\tau +1) \right|) = f_{\gamma_{\sigma'}} ( \left| C_{\delta_{\sigma'}} \cap (\gamma_\tau +1) \right|) \subseteq \dot r_\alpha \text{"}. \end{array} \end{align*}$$

Let $\tau '\in A_{u_1}(\alpha )\cap N_0$ . By the choice of $u_1$ , if $\varepsilon _{\sigma _N} < \delta _\tau $ , then

$$\begin{align*}u_1 \restriction \alpha \Vdash_{\mathbb{P}_\alpha}\ \text{"} f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap \delta_\tau \right|) \subseteq \dot r_\alpha \text{ and } f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap (\gamma_\tau +1) \right|) \subseteq \dot r_\alpha \text{"}. \end{align*}$$

Let $\tau "\in A_{u_1}(\alpha )\setminus N_0$ . Then $\zeta _0 < \delta _{\tau "}$ , and hence $C_{\omega _1\cap N } \cap \delta _{\tau "} = C_{\omega _1\cap N } \cap (\gamma _{\tau "}+1) = C_{\omega _1\cap N } \cap N_0$ . Since $r' \leq _{\mathbb {P}_\alpha } q^+\restriction \alpha $ ,

$$\begin{align*}\begin{array}{r@{\ }l} r' \Vdash_{\mathbb{P}_\alpha}\ \text{"} &\!\! f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap \delta_{\tau"} \right|) = f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap (\gamma_{\tau"}+1) \right|) \\[4pt] & = f_{\gamma_{\sigma_N}} ( \left| C_{\omega_1\cap N } \cap N_0 \right|) \subseteq \dot r_\alpha \text{"}. \end{array} \end{align*}$$

Thus $A_s(\alpha )$ satisfies (g-w). Therefore s is a condition of $\mathbb {P}_{\alpha +1}$ . It follows that s is a common extension of $u_1$ and q. However, then

$$\begin{align*}s\Vdash_{\mathbb{P}_{\alpha+1}}\ \text{"} \dot F_n \cap [f_{\omega_1\cap N}\restriction k] = \emptyset \text{ and } f_{\omega_1\cap N} \in \dot F_n \text{"} , \end{align*}$$

which is a contradiction, and finishes the proof of $\mathsf {(C)}_{\alpha +1}$ . $\dashv$

We will use Proposition 5.5 in the proof of $\mathsf {(p)}_{\alpha }$ and $\mathsf {(C)}_{\alpha }$ for a limit ordinal $\alpha $ .

Proposition 5.4. Suppose that $\alpha \in \omega _2+1$ , $\mathsf {(p)}_{\beta }$ and $\mathsf {(C)}_\beta $ hold for every $\beta <\alpha $ , $p\in \mathbb {P}_{\alpha }$ , $\xi \in \operatorname {dom}(A_p)$ , $M\in {R_p}^{-1}[\{\xi \}] \cap \mathfrak {M}^P_{\xi +1}$ , and $N\in \mathfrak {M}^P_{\xi } \cap M$ contains the set $ \left \{{ \mathcal {N}_p\cap M , \left \langle {A_p(\zeta )\cap M : \zeta \in \operatorname {dom}(A_p) \cap (\xi +1) \cap M}\right \rangle }\right \} $ and is such that, for every $\zeta \in \operatorname {dom}(A_p)\cap (\xi +1) \cap N$ and every $\sigma \in A_p(\zeta )$ with $\varepsilon _\sigma < \omega _1 \cap N < \delta _\sigma $ ,

$$\begin{align*}p\restriction \zeta \Vdash_{\mathbb{P}_{\zeta}} \ {"} f_{\gamma_\sigma} ( \left| C_{\delta_\sigma} \cap \omega_1\cap N \right|) \subseteq \dot r_\zeta {"}. \end{align*}$$

Then there is some $q\leq _{\mathbb {P}_{\xi +1}} p\restriction (\xi +1)$ such that $\left \langle {N, \xi }\right \rangle \in R_q$ and $A_q = A_p \cup \{\sigma \}$ for some $\sigma $ with $\delta _\sigma = \omega _1\cap N$ .

Proof of Proposition 5.4.

We will prove this by induction on $\xi $ . Let $\xi _0 : = \max \big ( \operatorname {dom}(A_p) \cap \xi \cap M \big )$ . Note that $\xi _0 \in N$ , and hence $N \in \mathfrak {M}^P_{\xi _0}\cap M$ . By the induction hypothesis, there exists $p_0 \leq _{\mathbb {P}_{\xi _0+1}} p\restriction (\xi _0+1)$ such that $N \in {R_{p_0}}^{-1}[\{\xi _0 \}]$ . Let $p_0' = \left \langle { \mathcal {N}_{p_0}, R_{p_0} \cup R_p, A_{p_0} \cup \left ( A_p \restriction [\xi _0+1, \xi ) \right )}\right \rangle $ , which is a canonical common extension of $p_0$ and p in $\mathbb {P}_\xi $ . Since $p_0'$ is $(M, \mathbb {P}_{\xi +1})$ -generic, by Proposition 5.3, we can take an extension $p_1 \leq _{\mathbb {P}_{\xi +1}} p_0'$ and $N_1 \in {R_{p_1}}^{-1}[\{\xi \}] \cap M$ which contains N as a member. Then, by (g-ob-2) and (g-w), for every $\zeta \in \operatorname {dom}(A_{p_1}) \cap N$ and every $\sigma \in A_{p_1}(\zeta )$ with $\varepsilon _\sigma < \omega _1 \cap N_1 < \delta _\sigma $ ,

$$\begin{align*}p_1 \restriction \zeta \Vdash_{\mathbb{P}_{\zeta}} \ \text{"} f_{\gamma_\sigma} ( \left| C_{\delta_\sigma} \cap \omega_1\cap N_1 \right|) \subseteq \dot r_\zeta \text{"}. \end{align*}$$

Recall that, for each limit ordinal $\delta $ , $C_\delta \cap \omega _1\cap N_1 = C_\delta \cap (\omega _1\cap N_1 +1)$ . Take $\varepsilon \in \omega _1\cap N$ such that

$$\begin{align*}\max\left\{{\gamma_\sigma : \sigma \in A_{p}(\xi)\cap M}\right\} < \varepsilon. \end{align*}$$

Define $q \in U_{\xi + 1}$ such that $\mathcal {N}_q := \mathcal {N}_{p_1}$ ,

$$\begin{align*}\begin{array}{r@{\ }l} R_q := R_{p_1\restriction \xi} \cup \Big\{ \left\langle{\left( {\Psi_{K} }^{-1} \circ \Psi_{M} \right) (N), \xi}\right\rangle : & K\in {R_{p}}^{-1}[\{\xi\}] \\ & \text{ with } \omega_1\cap K = \omega_1\cap M \Big\} , \end{array}\end{align*}$$

$A_q \restriction \xi := A_{p_1}\restriction \xi $ , and

$$\begin{align*}A_q(\xi) := A_p(\xi) \cup \left\{{\left\langle{\varepsilon, \omega_1\cap N, \omega_1 \cap N_1}\right\rangle}\right\}. \end{align*}$$

By Propositions 3.10 and 3.11, $R_{q}$ satisfies (el), (ho), (up), and (down) in Definition 4.1. By $\mathsf {(C)}_\xi $ and the fact that $\{N_1\}\times (\xi \cap N_1) \subseteq R_{p_1\restriction \xi } \subseteq R_q$ ,

$$ \begin{align*} {\begin{array}{r@{\ }l} q\restriction\xi \Vdash_{\mathbb{P}_\xi}\!\!\!&\text{"} f_{\omega_1\cap N_1} \text{ is Cohen over } N_1[\dot G_{\mathbb{P}_\xi}] \text{ and }\dot r_\xi \in N_1, \text{ hence }\\ &\left\{{n\in \omega : f_{\omega_1\cap N_1}(n) \subseteq \dot r_\xi}\right\} \text{ is infinite"}. \end{array}} \end{align*} $$

Moreover, by the roles of $\varepsilon $ and $N_1$ , $A_{q}(\xi )$ satisfies (g-ob), (g-ob-2) (g-cl), and (g-w). By the role of the set $N_1 $ , $A_{q}(\xi )$ satisfies (g-m) for $R_{q}$ . Therefore, q is a condition of $\mathbb {P}_{\xi +1}$ . This q is what we want. (We notice that $q\restriction \xi $ is an extension of $p_1\restriction \xi $ in $\mathbb {P}_\xi $ ; however, q may not be an extension of $p_1$ in $\mathbb {P}_{\xi +1}$ .) $\dashv$

Proposition 5.5. Suppose that $\alpha \in \omega _2+1$ , $\mathsf {(p)}_{\beta }$ and $\mathsf {(C)}_\beta $ hold for every $\beta <\alpha $ , $q,r\in \mathbb {P}_{\alpha }$ , and $\xi < \alpha $ is such that

• • $q\restriction \xi $ and $r\restriction \xi $ are compatible in $\mathbb {P}_{\xi }$ ,

• • $\xi \not \in \operatorname {dom}(A_q)$ and $\xi \in \operatorname {dom}(A_r)$ ,

• • there exists $N \in {R_q}^{-1}[\{\xi \}]$ such that $r\in N$ and, for any $M\in {R_q}^{-1}[\{\xi \}]$ , $\omega _1\cap M \geq \omega _1\cap N$ and $M \in \mathfrak {M}^P_{\xi +1}$ .

Then there exists a common extension $q' \in \mathbb {P}_{\xi +1}$ of $q\restriction (\xi +1)$ and $r\restriction (\xi +1)$ in $\mathbb {P}_{\xi +1}$ such that

$$\begin{align*}\left\{{\delta_\sigma : \sigma \in A_{q'}(\xi)\setminus A_r(\xi) }\right\} = \left\{{\omega_1\cap M : M \in {R_q}^{-1}[\{\xi\}]}\right\}. \end{align*}$$

Proof of Proposition 5.5.

Let $p_{-1} \in \mathbb {P}_{\xi +1}$ be such that

• • $p_{-1}\restriction \xi $ is a common extension of $q\restriction \xi $ and $r\restriction \xi $ in $\mathbb {P}_\xi $ ,

• • $p_{-1}$ is an extension of r in $\mathbb {P}_{\xi +1}$ ,

• • ${R_{p_{-1}}}^{-1}[\{\xi \}] = {R_{r}}^{-1}[\{\xi \}]$ , and

• • $A_{p_{-1}}(\xi ) = A_r(\xi )$ .

Take a maximal $\in $ -chain $\left \{{M_i : i \leq n}\right \}$ of ${R_q}^{-1}[\{\xi \}]$ such that $M_0 = N$ . For each $i\leq n$ , since $\{M_i\}\times (\xi \cap M_i) \subseteq R_{p_{-1}}$ , for every $\zeta \in \operatorname {dom}(A_{p_{-1}})\cap \xi \cap M_i$ and every $\sigma \in A_{p_{-1}}(\zeta )$ with $\varepsilon _\sigma < \omega _1 \cap M_i < \delta _\sigma $ ,

$$\begin{align*}p\restriction \zeta \Vdash_{\mathbb{P}_{\zeta}} \ \text{"} f_{\gamma_\sigma} ( \left| C_{\delta_\sigma} \cap \omega_1\cap M_i \right|) \subseteq \dot r_\zeta \text{"}. \end{align*}$$

By using Proposition 5.4 $(n+1)$ times repeatedly, for each $i\leq n$ , we can construct an extension $p_i \in \mathbb {P}_{\xi +1}$ such that

• • $p_i\restriction \xi $ is an extension of $p_{i-1}$ ,

• • $\left \langle {M_i, \xi }\right \rangle \in R_{p_{i}}$ , and

• • $A_{p_i}(\xi ) = A_{p_{i-1}}(\xi ) \cup \left \{{\sigma _i}\right \}$ for some $\sigma _i$ such that $\delta _{\sigma _i} = \omega _1\cap M_i$ .

Then $p_{n}$ is what we want. $\dashv$

Proof of  $\mathsf {(p)}_{\alpha }$  for a limit ordinal  $\alpha $ .

Suppose that $\alpha \in \omega _2+1$ is a limit ordinal, $p\in \mathbb {P}_{\alpha }$ , $N\in \mathcal {N}_p \cap \mathfrak {M}^P_\alpha $ satisfies that $\{N\}\times (\alpha \cap N) \subseteq R_p$ , $\mathcal {D} \in N$ is a predense subset of $\mathbb {P}_{\alpha }$ , and $q\leq _{\mathbb {P}_{\alpha }} p$ is an extension of some member of $\mathcal {D}$ . By extending q if necessary, we may assume that there exists $u\in \mathcal {D}$ such that $q\leq _{\mathbb {P}_\alpha }u$ . It suffices to find $u'\in \mathcal {D}\cap N$ which is compatible with q in $\mathbb {P}_{\alpha }$ . We have the case when $\alpha $ has uncountable cofinality and the case when it has countable cofinality. In the latter case, $\alpha \cap N$ is cofinal in $\alpha $ and hence we can take $\beta \in \alpha \cap N$ such that $\max (\operatorname {dom}(A_q))<\beta $ . But we may not be able to take such a $\beta $ in the former case, that is, it may happen that $\operatorname {dom}(A_q)$ is not bounded by $\sup (\alpha \cap N)$ . So we need more argument for the former case than for the latter case.

Suppose that $\alpha $ is of uncountable cofinality and $\operatorname {dom}(A_q) \not \subseteq \sup (\alpha \cap N)$ . (If $\operatorname {dom}(A_q) \subseteq \sup (\alpha \cap N)$ , then the proof will be simpler, same to the proof of the case that $\alpha $ is of countable cofinality.) Since $\mathcal {N}_q$ forms a symmetric system, for each $M'\in \mathcal {N}_q$ with $\omega _1\cap M'<\omega _1\cap N$ , there exists $M\in \mathcal {N}_q$ such that $\omega _1\cap M=\omega _1\cap N$ and $M'\in M$ , and then, by the requirement (id) in Definition 3.4,

$$\begin{align*}\begin{array}{r@{\ }c@{\ }l} \sup(M'\cap N\cap \alpha) & = & \sup(\left({\Psi_{{N\cap H_\kappa}}}^{-1}\circ \Psi_M\right)(M')\cap M\cap\alpha) \\[0.5em] & \leq & \sup(\left({\Psi_{{N\cap H_\kappa}}}^{-1}\circ \Psi_M\right)(M')\cap\alpha). \end{array} \end{align*}$$

Since N thinks that the set $\left ({\Psi _{{N\cap H_\kappa }}}^{-1}\circ \Psi _M\right )(M')$ is countable and $\alpha $ is of uncountable cofinality,

$$\begin{align*}\sup(\left({\Psi_{{N\cap H_\kappa}}}^{-1}\circ \Psi_M\right)(M')\cap\alpha) \in N\cap\alpha. \end{align*}$$

So there are large enough $\beta \in \alpha \cap N$ and $\zeta \in \omega _1\cap N$ , which means that

• • $\max (\operatorname {dom}(A_q)\cap \sup (\alpha \cap N))<\beta $ ,

• • $\max (\big \{\sup (R_q(M)) : M\in \operatorname {dom}(R_q)\big \}\cap N)<\beta $ ,

• • for every $M'\in \mathcal {N}_q$ with $\omega _1\cap M'<\omega _1\cap N$ ,

  $$\begin{align*}\sup(M'\cap N\cap \alpha)<\beta , \end{align*}$$

• • $\left \{{\omega _1\cap M : M \in \mathcal {N}_q \cap N }\right \} \subseteq \zeta $ ,

• • for any $\xi \in \operatorname {dom}(A_q) \cap N$ and any $\sigma \in A_q(\xi ) \cap N$ , $\gamma _\sigma < \zeta $ .

By the second and the third requirements on $\beta $ , we observe that

1. 〈1〉 for every $\xi \in [\beta , \alpha )\cap N$ and every $K\in \mathcal {N}_q$ with $\omega _1\cap K<\omega _1\cap N$ , $ K\not \in \mathfrak {M}_\xi ^P $ (because if K was in $\mathfrak {M}_\xi ^P$ , $\xi $ would be in K).

Define

$$ \begin{align*} { \begin{array}{r@{}l} \mathcal{E}:=\Big\{r\restriction\beta:{\ } & r\in \mathbb{P}_\alpha \text{ such that }\\ &\bullet\ \text{ there exists }u\in\mathcal{D} \text{ so that }r\leq_{\mathbb{P}_\alpha} u, \text{ and }\\ &\bullet\ \left\{{\omega_1\cap K:K\in\mathcal{N}_r}\right\}\cap \zeta =\left\{{\omega_1\cap K:K\in\mathcal{N}_q}\right\}\cap N \Big\}. \end{array} } \end{align*} $$

We notice that $q\restriction \beta \in \mathcal {E}$ and $\mathcal {E}$ is a definable class in the expanded relational structure by $\mathbb {P}_\alpha $ with parameters in N. By Proposition 3.9, since $\beta \in N\in \mathfrak {M}^P_\alpha $ , $N\in \mathfrak {M}^P_\beta $ . Moreover, it follows that

$$\begin{align*}\{N\}\times (\beta\cap N) \subseteq {R_{p\restriction \beta}} \subseteq {R_{q\restriction\beta}}. \end{align*}$$

So, by the induction hypothesis $\mathsf {(p)}_\beta $ , $q\restriction \beta $ is $(N,\mathbb {P}_\beta )$ -generic. Hence there exists $p_1$ in the set $\mathcal {E}\cap N$ which is compatible with the condition $q\restriction \beta $ in $\mathbb {P}_\beta $ . Let $r\in \mathbb {P}_\alpha \cap N$ and $u\in \mathcal {D}\cap N$ witness that $p_1\in \mathcal {E}$ .

Let us show that q and r are compatible in $\mathbb {P}_\alpha $ , which finishes the proof of this case. Let $p_2\in \mathbb {P}_\beta $ be a common extension of $q\restriction \beta $ and $p_1$ ( $=r\restriction \beta $ ). We note that $\operatorname {dom}(A_{p_2})\subseteq \beta $ and

1. 〈2〉 $ \operatorname {dom}(A_q)\cap \operatorname {dom}(A_r)\cap [\beta ,\alpha ) =\emptyset $ , more precisely,

$$\begin{align*}\beta < \min\left( \operatorname{dom}(A_r)\setminus \beta\right) \leq \max\left( \operatorname{dom}(A_r)\right) \hspace{7em} \end{align*}$$

$$\begin{align*}\hspace{7em} < \sup(\alpha\cap N) < \min\left( \operatorname{dom}(A_q)\setminus \beta\right) , \end{align*}$$

because $\operatorname {dom}(A_r)\subseteq N$ and $\max (\operatorname {dom}(A_q)\cap \sup (\alpha \cap N))<\beta $ . Since $r\in N$ ,

1. 〈3〉 $\mathcal {N}_r\subseteq N$ ,

2. 〈4〉 for each $\xi \in \left [ \beta , \alpha \right )\cap N$ ,

   • – if $\xi \in \operatorname {dom}(A_r)$ , then $A_r(\xi )\subseteq N$ ,

   • – $ \left \{{K\in {R_{q}}^{-1}\big [\,[\xi ,\alpha )\,\big ] \cap \mathfrak {M}_\xi ^P: \omega _1\cap K<\omega _1\cap N}\right \} =\emptyset $ (which follows from $\langle $ 1 $\rangle $ ), and

   • – if $M\in {R_q}^{-1}[\{\xi \}]$ , then $\omega _1\cap M \geq \omega _1\cap N$ and $\sup (R_q(M)) \geq \sup (\alpha \cap N)$ (by the role of $\beta $ ), and hence $M \in {R_q}^{-1}[\{\xi +1\}] \subseteq \mathfrak {M}^P_{\xi +1}$ ,

   and

3. 〈5〉 for each $\xi \in \left [ \beta , \alpha \right )\setminus N$ , $ \mathcal {N}_r\cap \mathfrak {M}_\xi ^P = \emptyset $ , in fact, no element of $\mathcal {N}_r$ contains $\xi $ as a member.

Let $\xi _0 := \min \big ( \operatorname {dom}(A_r) \setminus \beta \big )$ . Define $p_2' \in U_{\xi _0}$ such that

• • $R_{p_2'} := \begin {array}[t]{l@{\,}l} R_{p_2} \cup R_{q\restriction \xi _0} \\ \cup \Big \{ \left \langle {\left ( {\Psi _{M} }^{-1} \circ \Psi _{N} \right ) (K) , \xi }\right \rangle : & \left \langle {K, \xi }\right \rangle \in R_{r\restriction \xi _0}, \ M\in {R_{q}}^{-1}[\{\xi \}] \\ & \text {with } \omega _1\cap M = \omega _1\cap N \Big \} , \end {array} $

• • $\mathcal {N}_{p_2'} := \operatorname {dom}(R_{p_2'}) $ , and

• • $A_{p_2'} := A_{p_2}$ .

Now $\operatorname {dom}(A_r) \cap \xi _0 = \operatorname {dom}(A_r) \cap \beta $ and $\operatorname {dom}(A_q) \cap \xi _0 = \operatorname {dom}(A_q) \cap \beta $ . By Propositions 3.10 and 3.11 and the fact that $\alpha \in \omega _2$ , $R_{p_2'}$ satisfies (el), (ho), (up), and (down) in Definition 4.1, and so $p_2'$ is a condition of $\mathbb {P}_{\xi _0}$ . Hence $p_2'$ is a common extension of $r\restriction \xi _0$ and $q\restriction \xi _0$ .

By $\langle $ 4 $\rangle $ , we can apply Proposition 5.5 to find a common extension $q_{\xi _0}' $ of $p_2'$ , $r\restriction (\xi _0+1)$ , and $q \restriction (\xi _0 +1)$ in $\mathbb {P}_{\xi _0+1}$ . Let $\left \{{\xi _i : i \leq m}\right \}$ be the increasing enumeration of the set $\operatorname {dom}(A_r)\setminus \beta $ . By $\langle $ 4 $\rangle $ again, for each $i \leq m$ with $i\neq 0$ , we can apply Proposition 5.5 to find a common extension $q_{\xi _i}'$ of $q_{\xi _{i-1}}'$ , $r\restriction (\xi _i+1)$ , and $q \restriction (\xi _i +1)$ in $\mathbb {P}_{\xi _i+1}$ .

Define $q^{\prime }_\alpha \in U_\alpha $ such that

• • $ R_{q^{\prime }_\alpha }:= R_{q^{\prime }_{\xi _m}}\cup R_q$

  $$\begin{align*}\begin{array}{r} \cup \bigcup \Big\{ \left\{{\left( {\Psi_{M} }^{-1} \circ \Psi_{N} \right) (K)}\right\} \times \left( (\zeta+1)\cap \left( {\Psi_{M} }^{-1} \circ \Psi_{N} \right) (K) \right) : \hspace{3em} \ \\ \hfill \zeta \geq \beta, \left\langle{K,\zeta}\right\rangle \in R_r, \left\langle{M,\zeta}\right\rangle\in R_q \text{ with } \omega_1\cap M =\omega_1\cap N \Big\} , \end{array} \end{align*}$$

• • $ \mathcal {N}_{q_\alpha '}:= \operatorname {dom}(R_{q_\alpha '}), $ and

• • $A_{q_\alpha '} := A_{q^{\prime }_{\xi _m}}\cup \left ( A_q \restriction [\beta , \alpha ) \right ) $ .

By Propositions 3.10 and 3.11 and the fact that $\alpha \in \omega _2$ again, $R_{q_\alpha '}$ satisfies (el), (ho), (up), and (down) in Definition 4.1. By $\langle $ 5 $\rangle $ , for each $\xi \in \operatorname {dom}(A_q) \cap [\beta ,\alpha )$ , $A_{p_\alpha '}(\xi )$ satisfies (g) for $R_{q_\alpha '}$ . Therefore, $q_\alpha '$ is a condition of $\mathbb {P}_\alpha $ , and so is a common extension of r and q in $\mathbb {P}_\alpha $ . This finishes the proof of this case.

Suppose that $\alpha $ is of countable cofinality. Take a large enough ordinal $\beta \in \alpha \cap N$ , which means that

• • $\max (\operatorname {dom}(A_q)) <\beta $ , and

• • for each $M\in \operatorname {dom}({R_q})$ , either $R_q(M)\subseteq \beta $ or $R_q(M)$ is cofinal in $\alpha $ ,

and define

$$ \begin{align*} { \begin{array}{r@{}l} \mathcal{E}:=\Big\{r\restriction\beta:{\ } & r\in \mathbb{P}_\alpha \text{ such that}\\ &\bullet\ \text{ there exists }u'\in\mathcal{D} \text{ so that }r\leq_{\mathbb{P}_\alpha} u',\\ &\bullet\ \operatorname{dom}(A_r)\subseteq \beta, \text{ and }\\ &\bullet\ \text{ for each }M\in {\mathcal{N}_r}, \text{ either }R_r(M)\subseteq\beta \text{ or }R_r(M) \text{ is } \text{ cofinal in }\alpha \Big\}. \end{array} } \end{align*} $$

We note that $q\in \mathcal {E}$ and $\mathcal {E}$ is a definable class in the expanded relational structure by $\mathbb {P}_\alpha $ with parameters in N. By the induction hypothesis $\mathsf {(p)}_\beta $ and the fact that $\beta \in N\in \mathfrak {M}_\alpha ^P$ , $q\restriction \beta $ is $(N,\mathbb {P}_\beta )$ -generic. So there exists $p_1\in \mathcal {E}\cap N$ which is compatible with the condition $q\restriction \beta $ in $\mathbb {P}_\beta $ . Let $r\in \mathbb {P}_\alpha \cap N$ and $u' \in \mathcal {D} \cap N$ witness that $p_1\in \mathcal {E}$ , and let $p_2\in \mathbb {P}_\beta $ be a common extension of $q\restriction \beta $ and $p_1$ ( $=r\restriction \beta $ ).

Define $q' \in U_\alpha $ such that

• • $ R_{q'}:= R_{p_2}\cup R_r \cup R_{q} $

  $ \kern20pt \cup \bigcup \Big \{ \left \{{\left ( {\Psi _{M} }^{-1} \circ \Psi _{N} \right ) (K)}\right \} \times \left ( \alpha \cap \left ( {\Psi _{M} }^{-1} \circ \Psi _{N} \right ) (K) \right ) : $

  $ \kern20pt \xi \geq \beta , \left \langle {K,\xi }\right \rangle \in R_r, \left \langle {M,\xi }\right \rangle \in R_q \text { with } \omega _1\cap M =\omega _1\cap N \Big \}, $

• • $\mathcal {N}_{q'}:=\operatorname {dom}(R_{q'})$ , and

• • $A_{q'} := A_{p_2}$ .

Then $q'$ is a condition of $\mathbb {P}_\alpha $ , and is a common extension of q, r, and $u'$ in $\mathbb {P}_\alpha $ . $\dashv$

The following proof is similar to one of Lemma 2.10.

Proof of  $\mathsf {(C)}_{\alpha }$  for a limit ordinal  $\alpha $ .

Suppose that $\alpha \in \omega _2+1$ is a limit ordinal, $p\in \mathbb {P}_{\alpha }$ , $N\in \mathcal {N}_p \cap \mathfrak {M}^P_\alpha $ satisfies that $\{N\}\times (\alpha \cap N)\subseteq R_p$ , and $\left \{{\dot F_n: n\in \omega }\right \}$ is a set of $\mathbb {P}_\alpha $ -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 $p \Vdash _{\mathbb {P}_{\alpha }} \!\!\!\text {"} f_{\omega _1\cap N} \not \in \bigcup _{n\in \omega } \dot F_n \,\text {"}$ .

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

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

As in the proof of $\mathsf {(p)}_\alpha $ when $\alpha $ is a limit ordinal, we need to separate two cases.

Suppose that $\alpha $ is of uncountable cofinality and $\operatorname {dom}(A_q) \not \subseteq \sup (\alpha \cap N)$ . Let $\beta \in \alpha \cap N$ and $\zeta \in \omega _1\cap N$ be large enough ordinals for the condition q as in the proof of $\mathsf {(p)}_\alpha $ . By the induction hypothesis $\mathsf {(p)_\beta }$ and the fact that $\beta \in N \in \mathfrak {M}^P_\alpha $ , $q\restriction \beta $ is $(N,\mathbb {P}_\beta )$ -generic. For each $\nu \in \left ( 2^{<\omega } \right )^{<\omega }$ , each $\beta ' \in \alpha \setminus \beta $ , each $\eta \in \omega _1$ and each $u\in \mathbb {P}_\alpha $ , define $\varphi _\alpha (\nu , \beta ', \eta , u)$ to be the assertion that

• • $\operatorname {dom}(A_u) \cap \beta ' = \operatorname {dom}(A_p) \cap \beta $ ,

• • $ \left \{{\omega _1\cap K:K\in \mathcal {N}_u }\right \} \cap \eta =\left \{{\omega _1\cap K:K\in \mathcal {N}_q }\right \} \cap N $ , and

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

Define a $\mathbb {P}_\beta $ -name $\dot Y$ such that

$$ \begin{align*} { \begin{array}{r@{\ }l} \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot Y = \Big\{ g \in \left({2^{<\omega}}\right)^\omega : & \text{ for any }k\in \omega, \text{ any }\beta'\in \alpha \setminus \beta, \text{ and } \text{ any }\eta \in \omega_1\setminus \zeta,\\ &\text{ there exists }u\in \mathbb{P}_{\alpha} \text{ such that } u\restriction \beta \in \dot G_{\mathbb{P}_{\beta}} \text{ and}\\ &\varphi_\alpha( g \restriction k, \beta', \eta, u) \Big\} \text{"}. \end{array} }\end{align*} $$

Then $\dot Y$ is a definable class in the expanded relational structure by $\mathbb {P}_\alpha $ with parameters in N. $\dot Y$ is forced to be a closed subset of $\left ({2^{<\omega }}\right )^\omega $ . So by Proposition 5.1, $\dot Y$ can be considered as an element of N.

We claim that

$$\begin{align*}q\restriction \beta \Vdash_{\mathbb{P}_\beta} \ \text{"} \dot Y \text{ is nowhere dense} \text{"}. \end{align*}$$

Let $r \leq _{\mathbb {P}_\beta } q \restriction \beta $ and $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ , and let $q'$ be a common extension of both r and q in $\mathbb {P}_{\alpha }$ . Then there are $q"\leq _{\mathbb {P}_\alpha } q'$ and an end-extension $\nu '$ of $\nu $ in $\left ({2^{<\omega }}\right )^{<\omega }$ such that

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

$q"\restriction \beta $ is an extension of r in $\mathbb {P}_\beta $ . Let us show that

$$\begin{align*}q" \restriction \beta \not \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot Y \cap [\nu'] \neq\emptyset \text{"}. \end{align*}$$

Assume not. Let $\dot g$ a $\mathbb {P}_\beta $ -name such that

$$\begin{align*}q" \restriction \beta \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot g \in \dot Y \cap [\nu'] \text{"}. \end{align*}$$

Take $s\leq _{\mathbb {P}_{\beta }} q"\restriction \beta $ , $k\in \omega $ , and $\nu " \in \left ({2^{<\omega }}\right )^{<\omega }$ such that

$$\begin{align*}s \Vdash_{\mathbb{P}_{\beta}}\ \text{"} \nu' \subseteq \dot g \restriction k = \nu" \text{"}. \end{align*}$$

Let $\beta '\in (\alpha \cap N)\setminus \beta $ and let $\eta \in (\omega _1\cap N) \setminus \zeta $ be large enough ordinals for $q"$ as in the proof of $\mathsf {(p)}_\alpha $ . By the definition of $\dot Y$ , $\left \{{\dot Y, \beta ', \eta , \nu "}\right \} \in N$ , and the fact that $q"\restriction \beta $ is also $(N, \mathbb {P}_{\beta })$ -generic and that N is an elementary substructure of the expanded relational structure by $\mathbb {P}_\alpha $ , there exists an extension $s'$ of s in $\mathbb {P}_{\beta }$ and $u\in \mathbb {P}_\alpha \cap N$ such that $s' \leq _{\mathbb {P}_{\beta }} u\restriction {\beta } $ and $\varphi _\alpha (\nu ", \beta ', \eta , u)$ . As in the proof of $\mathsf {(p)}_\alpha $ , by the roles of $\beta '$ and $\eta $ , we can build a common extension t of $s'$ , $q"$ , and u in $\mathbb {P}_{\alpha }$ . (To build t, for each coordinate $\xi $ in $\left (\operatorname {dom}(A_{q"}) \cup \operatorname {dom}(A_u) \right ) \cap [\beta , \sup (\alpha \cap N))$ , we construct a preparatory condition $t_\xi \in \mathbb {P}_{\xi +1}$ like $q_{\xi _i}'$ as in the proof of $\mathsf {(p)}_\alpha $ .) Then

$$\begin{align*}t \Vdash_{\mathbb{P}_{\alpha}}\ \text{"} \dot F_n \cap [\nu'] = \emptyset \text{ and } \dot F_n \cap [\nu"] \neq \emptyset \text{"} , \end{align*}$$

which is a contradiction.

We claim that

$$\begin{align*}q\restriction \beta \Vdash_{\mathbb{P}_\beta}\ \text{"} f_{\omega_1\cap N} \in \dot Y \text{"}. \end{align*}$$

This contradicts the induction hypothesis $\mathsf {(C)}_\beta $ , and finishes the proof of this case.

Assume not. Then there exists $r\leq _{\mathbb {P}_\beta } q \restriction \beta $ such that

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta}\text{"} f_{\omega_1\cap N} \not\in \dot Y \text{"}. \end{align*}$$

By extending r if necessary, we may assume that there are $k\in \omega $ , $\beta ' \in \alpha \setminus \beta $ , and $\eta \in \omega _1\setminus \zeta $ such that

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta}\text{"} \text{there are no } u \in \mathbb{P}_\alpha \text{ such that } u\restriction {\beta} \in \dot G_{\mathbb{P}_{\beta}} \text{ and } \varphi_\alpha( f_{\omega_1\cap N} \restriction k, \beta', \eta, u) \text{"}. \end{align*}$$

By the induction hypothesis $\mathsf {(p)}_\beta $ , r is $(N,\mathbb {P}_\beta )$ -generic. So by extending r again if necessary, we may assume that $\beta '\in (\alpha \cap N)\setminus \beta $ and $\eta \in (\omega _1\cap N) \setminus \zeta $ . However, then

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta}\ \text{"} q\restriction \beta \in \dot G_{\mathbb{P}_{\beta}} \text{ and } \varphi_\alpha( f_{\omega_1\cap N} \restriction k, \beta', \eta, q) \text{"}, \end{align*}$$

which is a contradiction.

Suppose that $\alpha $ is of countable cofinality. The proof below is similar to the one in the case of uncountable cofinality. The differences are the necessary property of $\beta $ and the definition of $\dot Y$ . Let $\beta \in \alpha \cap N$ be a large enough ordinal as in the proof of $\mathsf {(p)}_\alpha $ . Let $\zeta \in \omega _1\cap N$ be such that

$$\begin{align*}\left\{{\omega_1\cap K:K\in\mathcal{N}_q}\right\} \cap \zeta =\left\{{\omega_1\cap K:K\in\mathcal{N}_q}\right\} \cap N. \end{align*}$$

For each $\nu \in \left ( 2^{<\omega } \right )^{<\omega }$ , each $\eta \in \omega _1$ , and each $u\in \mathbb {P}_\alpha $ , define $\varphi _\alpha (\nu , \eta , u)$ to be the assertion that

• • $\operatorname {dom}(A_u) \subseteq \beta $ ,

• • for each $M\in {\mathcal {N}_u}$ , either $R_u(M)\subseteq \beta $ or $R_u(M)$ is cofinal in $\alpha $ ,

• • $ \left \{{\omega _1\cap K:K\in \mathcal {N}_u }\right \} \cap \eta =\left \{{\omega _1\cap K:K\in \mathcal {N}_q }\right \} \cap N $ , and

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

and define $\dot Y$ to be a $\mathbb {P}_\beta $ -name such that

$$ \begin{align*} { \begin{array}{r@{\ }l} \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot Y = \Big\{ g \in \left({2^{<\omega}}\right)^\omega : & \text{ for any }k\in\omega \text{ and } \text{ any }\eta\in \omega_1\setminus \zeta, \text{ there exists }u\in \mathbb{P}_{\alpha}\\ &\text{ such that } u\restriction \beta \in \dot G_{\mathbb{P}_\beta} \text{ and } \varphi_\alpha(g \restriction k, \eta, u) \Big\}\text{"}. \end{array} }\end{align*} $$

Then $\dot Y$ is a definable class in the expanded relational structure by $\mathbb {P}_\alpha $ with parameters in N, and is forced to be a closed subset of $\left ({2^{<\omega }}\right )^\omega $ . So by Proposition 5.1, $\dot Y$ can be considered as an element of N.

We claim that

$$\begin{align*}q\restriction \beta \Vdash_{\mathbb{P}_\beta} \ \text{"} \dot Y \text{ is nowhere dense} \text{"}. \end{align*}$$

Let $r \leq _{\mathbb {P}_\beta } q \restriction \beta $ and $\nu \in \left ({2^{<\omega }}\right )^{<\omega }$ , and let $q'$ be a common extension of both r and q in $\mathbb {P}_{\alpha }$ . Then there are $q"\leq _{\mathbb {P}_\alpha } q'$ and an end-extension $\nu '$ of $\nu $ in $\left ({2^{<\omega }}\right )^{<\omega }$ such that

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

$q"\restriction \beta $ is an extension of r in $\mathbb {P}_\beta $ . Let us show that

$$\begin{align*}q" \restriction \beta \not \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot Y \cap [\nu'] \neq\emptyset \text{"}. \end{align*}$$

Assume not, and let $\dot g$ be a $\mathbb {P}_\beta $ -name such that

$$\begin{align*}q" \restriction \beta \Vdash_{\mathbb{P}_\beta}\ \text{"} \dot g \in \dot Y \cap [\nu'] \text{"}. \end{align*}$$

Take $s\leq _{\mathbb {P}_{\beta }} q"\restriction \beta $ , $k\in \omega $ and $\nu " \in \left ({2^{<\omega }}\right )^{<\omega }$ such that

$$\begin{align*}s \Vdash_{\mathbb{P}_{\beta}}\ \text{"} \nu' \subseteq \dot g \restriction k = \nu" \text{"}. \end{align*}$$

Take $\eta \in \omega _1\cap N$ such that $\eta \geq \zeta $ and, for any $\xi \in \operatorname {dom}(A_{q"})\cap [\beta , \alpha )\cap N$ (then $N\in R_{q"}[\{\xi \}]$ ) and any $\sigma \in A_{q"}(\xi ) \cap N$ , $\gamma _\sigma < \eta $ . As in the previous case, we take an extension s of $q" \restriction \beta $ in $\mathbb {P}_\beta $ and $u\in \mathbb {P}_\alpha \cap N$ such that $s \leq _{\mathbb {P}_\beta } u\restriction \beta $ and $\varphi _\alpha (\nu ", \eta , u)$ . For any $\xi \in \operatorname {dom}(A_{q"}) \cap [\beta ,\alpha )$ ,

• • if $\xi \in N$ , then

  1. – $N\in {R_{q"}}^{-1}[\{\xi \}] \cap \mathfrak {M}^P_{\xi +1}$ ,

  2. – ${R_u}^{-1}[\{\xi \}] = {R_u}^{-1}[\{\xi +1\}] \subseteq \mathfrak {M}^P_{\xi +1}$ (because u satisfies $\varphi _\alpha (\nu ", \eta , u)$ ),

  3. – $\left \{{\omega _1\cap M : M \in {R_u}^{-1}[\{\xi \}]}\right \} \cap \eta \subseteq \left \{{\delta _\sigma : \sigma \in A_{q"}(\xi )}\right \}$ ,

  4. – for any $\sigma \in A_q(\xi )\cap N$ , $\gamma _\sigma < \eta $ , and

  5. – $\max \left \{{\omega _1\cap M : M \in {R_u}^{-1}[\{\xi \}]}\right \} < \omega _1\cap N$ ,

• • if $\xi \not \in N$ , by Proposition 3.11 and the fact that $u\in N$ , ${R_u}^{-1}[\{\xi \}] = \emptyset $ .

Hence, as the construction of $q^{\prime }_{\xi _m}$ in the proof of $\mathsf {(p)}_\alpha $ before, we can find a common extension t of s, u, and $q"$ . But then t forces a contradiction.

We claim that

$$\begin{align*}q\restriction \beta \Vdash_{\mathbb{P}_\beta}\ \text{"} f_{\omega_1\cap N} \in \dot Y \text{"}. \end{align*}$$

This contradicts the induction hypothesis $\mathsf {(C)}_\beta $ , and finishes the proof of this case.

Assume not, then there exists $r\leq _{\mathbb {P}_\beta } q \restriction \beta $ such that

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta} \ \text{"} f_{\omega_1\cap N} \not\in \dot Y \text{"}. \end{align*}$$

By extending r if necessary, we may assume that there are $k\in \omega $ and $\eta \in \omega _1\setminus \zeta $ such that

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta}\ \text{"} \text{there are no } u \in \mathbb{P}_\alpha \text{ such that } u\restriction \beta \in \dot G_{\mathbb{P}_\beta} \text{ and } \varphi_\alpha(f_{\omega_1\cap N} \restriction k, \eta, u) \text{"}. \end{align*}$$

By the induction hypothesis $\mathsf {(p)}_\beta $ , r is $(N,\mathbb {P}_\beta )$ -generic. So by extending r again if necessary, we may assume that $\eta \in \left ( \omega _1 \setminus \zeta \right ) \cap N$ . However then

$$\begin{align*}r \Vdash_{\mathbb{P}_\beta}\ \text{"} q\restriction \beta \in \dot G_{\mathbb{P}_\beta} \text{ and } \varphi_\alpha(f_{\omega_1\cap N} \restriction k, \eta, q) \text{"} , \end{align*}$$

which is a contradiction. $\dashv$

Lemma 5.6. For any $\mathbb {P}_{\omega _2}$ -name $\dot r$ for a member of $2^\omega $ ,

$$\begin{align*}\Vdash_{\mathbb{P}_{\omega_2}}\ {"} \text{there is } \alpha \in \omega \text{ such that } \dot E_\alpha \text{ captures } \dot r \text{ relative to the set } X{"}. \end{align*}$$

Proof. $\dot E_\xi $ is defined in Definition 4.3. By Proposition 5.1, we may assume that $\dot r$ belongs to $H(\kappa )$ . Let $p\in \mathbb {P}_{\omega _2}$ . Take $\alpha \in \omega _2$ such that $\Phi (\alpha ) = \left \{{ \dot r_\alpha }\right \} = \left \{{\dot r}\right \}$ , and $\operatorname {ran}(R_p) \subseteq \alpha $ . Then $\operatorname {dom}(A_p) \subseteq \alpha $ .

Let $\theta $ be a large enough regular cardinal for the forcing notion $\mathbb {P}_{\alpha +1}$ . Take any $\varepsilon \in \omega _1$ , and take countable elementary submodels $N_0^*$ and $N^*_1$ of $H_\theta $ such that $\left \{{\vec {\mathbb {R}}, \vec X, H_\kappa , \mathbb {P}_\alpha , p, \alpha , \varepsilon }\right \} \in N^*_0 \in N^*_1$ . Then both $N^*_0 \cap H_\kappa $ and $N^*_1 \cap H_\kappa $ are in $\mathfrak {M}^P_\alpha \cap \mathfrak {M}^P_{\alpha + 1} $ . By Proposition 5.4, there exists an extension q of p in $\mathbb {P}_{\alpha +1}$ such that ${R_q}^{-1}[\{\alpha \}] = \left \{{ N_0^* \cap H_\kappa }\right \}$ and $A_q(\alpha ) = \left \{{\left \langle {\varepsilon , \omega _1\cap N^*_0, \omega _1\cap N^*_1 }\right \rangle }\right \}$ . Then, by Lemma 5.2, $\mathbb {P}_{\alpha +1}$ is proper and q is $(N_0^*\cap H_\kappa , \mathbb {P}_{\alpha +1})$ -generic. So as seen in Observation 4.4,

$$\begin{align*}q \Vdash_{\mathbb{P}_{\omega_2}}\ \text{"} \dot E_\alpha \text{ is a club subset of } \omega_1 \text{ and captures } \dot r\ \text{(}\!= \dot r_\alpha\text{)} \text{ relative to } X\text{"}.\\[-38pt] \end{align*}$$

⊣

Since $\mathbb {P}_{\omega _2}$ forces that $2^{\aleph _0}=\aleph _2$ , we conclude the following.

Theorem 5.7. $\mathbb {P}_{\omega _2}$ forces the assertion ( c ) .

6 Properness and the length of the iteration

For each $\xi \in \kappa $ , define the $\mathbb {P}_{\xi +1}$ -name $\dot S_\xi $ by

$$\begin{align*}\Vdash_{\mathbb{P}_{\xi+1}}\ \text{"} \dot S_\xi := \left\{{ \omega_1\cap N : p \in \dot G_{\mathbb{P}_{\xi+1}}, N \in {R_p}^{-1}[\{\xi\}]}\right\} \text{"}. \end{align*}$$

Let $p\in \mathbb {P}_{\omega _2}$ and $\xi \in \operatorname {dom}(A_p)$ (then $\xi \in \omega _2$ ). If M belongs to ${R_p}^{-1}[\{\xi \}]$ , that is, $\{ M \} \times \big ( (\xi + 1) \cap M \big ) \subseteq R_p$ , then

$$\begin{align*}p \restriction (\xi+1) \Vdash_{\mathbb{P}_{\xi +1}}\ \text{"} \omega_1\cap M \in \dot S_\xi \text{"}. \end{align*}$$

Moreover, if M also belongs to $\mathfrak {M}^P_{\xi +1}$ , then p is $(M,\mathbb {P}_{\xi +1})$ -generic by Lemma 5.2, and therefore

$$\begin{align*}p \restriction (\xi+1) \Vdash_{\mathbb{P}_{\xi +1}} \ \text{"} \omega_1\cap M \text{ is a limit point of } \dot S_\xi \text{"}. \end{align*}$$

We notice that

$$\begin{align*}p \restriction (\xi+1) \Vdash_{\mathbb{P}_{\xi +1}} \ \text{"} \dot S_\xi \text{ is a stationary subset of } \omega_1 \text{"}. \end{align*}$$

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

$$ \begin{align*} { \begin{array}{r@{}l} \Vdash_{\mathbb{P}_{\xi+1}}\ \text{"} & \text{ the set } \dot{S}_\xi \text{ is a stationary subset of }\omega_1, \text{ and captures }r \text{ relative to }X,\\ &\text{ that is, for any limit point }\delta \text{ of }\dot S_\xi \text{ (which means that } \delta \in \dot{S}_\xi \text{ and }\\ &\dot{S}_\xi \cap \delta \text{ is cofinal in }\delta), \text{ there exists }f\in X \text{ and }\varepsilon \in \delta \text{ such that, for any}\\ & \xi \in (\dot{S}_\xi \cap \delta) \setminus \varepsilon, f ( \left| C_{\delta } \cap \xi \right|) \subseteq r \text{"}. \end{array} }\end{align*} $$

Suppose that $p\in \mathbb {P}_{\omega _2}$ , $\xi , \zeta \in \operatorname {dom}(A_p)$ with $\xi <\zeta $ , and $M\in {R_p}^{-1}[\{\zeta \}]$ (then p forces $\omega _1\cap M$ to be in $\dot S_\zeta $ ). If $\xi \in M$ , then

$$\begin{align*}p\Vdash_{\mathbb{P}_\zeta}\ \text{"} \omega_1\cap M \in \dot S_\xi \text{"}. \end{align*}$$

Therefore, by Proposition 3.10 and the requirements (ho) and (up) in the definition of $\mathbb {P}_\alpha $ ,

$$\begin{align*}p\Vdash_{\mathbb{P}_\zeta}\ \text{"} \dot S_\zeta\setminus M \subseteq \dot S_\xi \text{"}. \end{align*}$$

Therefore, $\mathbb {P}_{\omega _2}$ forces that there are $R\in \left [ 2^\omega \right ]^{\aleph _2}$ and a sequence $\left \langle {S_\xi : \xi \in \omega _2}\right \rangle $ of stationary subsets of $\omega _1$ such that

• • for each $r\in R$ , there exists $\xi \in \omega _2$ such that $S_\xi $ captures r relative to X.

• • for each $\xi , \zeta \in \omega _2$ , if $\xi < \zeta $ , then $S_\zeta \setminus S_\xi $ is bounded in $\omega _1$ .

As in the proof of Proposition 2.3, the set $\left \{{S_\xi : \xi \in \omega _2}\right \}$ cannot be diagonalized by any stationary subset of $\omega _1$ without collapsing $\aleph _2$ . This observation leads to the following conclusion.

Lemma 6.1. If $\kappa>\omega _2$ , then $\mathsf {(p)}_{\omega _2+1}$ in Lemma 5.2 fails.

This suggests that $\mathbb {P}_{\omega _2 +1}$ should not be proper.

Proof. Suppose that $\mathsf {(p)}_{\omega _2+1}$ in Lemma 5.2 holds. Then, as we have already proved, $\dot S_{\omega _2}$ is a $\mathbb {P}_{\omega _2+1}$ -name for a stationary subset of $\omega _1$ . For each $\eta \in \omega _2$ , the set of conditions p of $\mathbb {P}_{\omega _2+1}$ such that there are $M\in \mathcal {N}_p$ and $\xi \in \omega _2\setminus \eta $ such that $\{\xi , \omega _2\} \in M$ , $\{M\}\times \big ( (\omega _2+1) \cap M\big ) \subseteq R_p$ , and $\left \{{\xi , \omega _2}\right \} \subseteq \operatorname {dom}(A_p)$ is dense in $\mathbb {P}_{\omega _2+1}$ . Therefore

$$ \begin{align*} { \begin{array}{r@{}l} \Vdash_{\mathbb{P}_{{\omega_2}+1}}\ \text{"} & \left\{{\xi : \exists p\in \dot G_{\mathbb{P}_{{\omega_2}+1}} \left( \xi \in \operatorname{dom}(A_p)\cap {\omega_2} \right)}\right\} \text{ is of size }\aleph_2, \\[10pt] & \dot S_{\omega_2} \text{ is a stationary subset of }\omega_1 \text{ and diagonalizes the set } \\[5pt] & \left\{{\dot S_\xi : \exists p\in \dot G_{\mathbb{P}_{{\omega_2}+1}} \left( \xi \in \operatorname{dom}(A_p)\cap {\omega_2} \right)}\right\} \text{"}. \end{array} }\end{align*} $$

However, this is a contradiction.⊣

Remark 6.2. There are reasons to suspect that $\mathbb {P}_\alpha $ , for $\alpha> \omega _2$ , may fail to be proper, even disregarding the working parts of the forcing. Suppose that the length of the iteration is $\omega _2+1$ , $q\in \mathbb {P}_{\omega _2+1}$ , $\{M,N\}\subseteq {R_q}^{-1}[\{\omega _2\}]$ , $r\in \mathbb {P}_{\omega _2+1}\cap N$ which is a nice copy of q inside N as in the proof of $\mathsf {(p)}_\alpha $ for a limit ordinal $\alpha $ , $\left \{{M_0,M_1,M_2,M_3}\right \}\subseteq {R_r}^{-1}[\{\omega _2\}]\setminus M$ such that $M_2 \in M_0$ , $M_3\in M_1$ , $M_0\cap \omega _1 =M_1\cap \omega _1$ , $M_2\cap \omega _1 =M_3\cap \omega _1$ , and $M_i':=\left ({\Psi _M}^{-1}\circ \Psi _N\right )(M_i) \in \mathcal {N}_{q'}\setminus \left (\mathcal {N}_q\cup \mathcal {N}_r\right )$ for each $i\in \{0,1,2,3\}$ , as in the following figure.

Then $\left \{{M_0',M_1',M_2',M_3'}\right \}\subseteq {R_{q'}}^{-1}[\{\omega _2\}]$ , and ${R_{q'}}^{-1}[\{\omega _2\}]$ forms a symmetric system. Let $\zeta \in \alpha $ . It should be satisfied that ${R_{q'}}^{-1}[\{\zeta \}] \subseteq \mathfrak {M}^P_\zeta $ and ${R_{q'}}^{-1}[\{\zeta \}]$ forms a symmetric system. Now we have no guarantees that the assertion ${R_{q'}}^{-1}[\{\zeta \}] \subseteq \mathfrak {M}^P_\zeta $ is true. And even if this is true, since Proposition 3.11 may fail for $\alpha =\omega _2$ , ${R_{q'}}^{-1}[\{\zeta \}] $ may fail to form a symmetric system. For example, it may happen that $\zeta \in M_1'$ , $\zeta \in M_2'$ (hence $\zeta \in M_0'$ ), and $\zeta \not \in M_3'$ . Then $ {R_{q'}}^{-1}[\{\zeta \}]$ does not satisfy (down). Therefore, in this case, q and r may fail to be compatible in $\mathbb {P}_{\omega _2+1}$ .

7 Acknowledgments

In the workshop “Set theory of the Reals” at Casa Matemática Oaxaca, I asked Justin Tatch Moore about assertions which imply that the size of the continuum is $\aleph _2$ . The aim of my question was to find some types of Asperó–Mota iterations whose length is greater than $\omega _2$ and not proper. He gave me the idea presented in Section 2.1. He encouraged me to complete this paper and assisted me on the draft of the introduction. I am very thankful to him.

I would also like to thank Diego A. Mejía and Tadatoshi Miyamoto for many useful comments on an earlier draft of the paper. To conclude, I am grateful to the referee for many worthwhile comments.