Countable-support fusion at a limit

Because $\mathrm{cf}(\eta )=\omega$ and $\eta \in M$, choose in $M$ an increasing cofinal sequence $0={\eta }_0<{\eta }_1<\cdots <\eta ,$ and enumerate the dense subsets of $P_{\eta }$ which belong to $M$ as $⟨D_n:n<\omega ⟩$, repeating a dense set if necessary. Put $q_0=1_{P_0}$ and let ${\dot{p}}_0=\check{p}$. Then $q_0$ is the trivial $(M,P_0)$-master and forces $p_0\upharpoonright {\eta }_0\in G_0$.

Recursively suppose that $q_n$ is an $(M,P_{{\eta }_n})$-master and forces that $p_n\in P_{\eta }\cap M$ with $p_n\upharpoonright {\eta }_n\in G_{{\eta }_n}$. Work in a $P_{{\eta }_n}$-generic extension containing $q_n$ and resolve $p_n$. This value is a ground-model condition in $M$, so the ground set $E_n=\{u\in P_{{\eta }_n}:u\perp p_n\upharpoonright {\eta }_n\text{ or }(\mathrm{\exists }r\le p_n)[r\in D_n\mathrm{\&}u\le r\upharpoonright {\eta }_n]\}\mathrm{.}$ This set belongs to $M$ and is dense: below a condition compatible with $p_n\upharpoonright {\eta }_n$, take a common extension, paste it to the tail of $p_n$, and strengthen the resulting $P_{\eta }$-condition into $D_n$. By F2 the generic below $q_n$ meets $E_n\cap M$. Its member cannot take the incompatible alternative because $p_n\upharpoonright {\eta }_n$ is in the same generic. Elementarity therefore gives a name ${\dot{p}}_{n+1}$ forced to satisfy $p_{n+1}\in D_n\cap M,p_{n+1}\le p_n,p_{n+1}\upharpoonright {\eta }_n\in G_{{\eta }_n}\mathrm{.}$ Apply F1 from ${\eta }_n$ to ${\eta }_{n+1}$ to obtain an $(M,P_{{\eta }_{n+1}})$-master $q_{n+1}$ such that $q_{n+1}\upharpoonright {\eta }_n=q_n$ and $q_{n+1}$ forces $p_{n+1}\upharpoonright {\eta }_{n+1}\in G_{{\eta }_{n+1}}$.

Define $q={\bigcup }_{n<\omega }{q}_n\mathrm{.}$ The displayed coherence makes this a function whose restriction to every ${\eta }_n$ is exactly $q_n$. Its support is contained in the countable union of the countable supports of the $q_n$, hence is countable; cofinality of the ${\eta }_n$ leaves no unfilled coordinate below $\eta$. Thus F3 gives $q\in P_{\eta }$. This is the fusion step. It takes no lower bound of the sequence $⟨q_n(\xi ):n<\omega ⟩$ inside a single iterand: after coordinate $\xi$ first appears, later conditions preserve the already constructed initial segment containing it.

The conclusion that $q$ forces each $p_{n+1}$ into the full generic is the limit conclusion of F1 applied to exactly the recursion in steps 1.1--3.1. It does not follow merely from compatibility of all bounded restrictions, and no such inverse-limit compactness is asserted here. F1 therefore gives $q⊩p_{n+1}\in D_n\cap M\cap {\dot{G}}_{\eta }$ for every $n$. It follows that every $D_n\cap M$ is predense below $q$, so $q$ is an $(M,P_{\eta })$-master. Also F1 gives $q⊩\check{p}\in {\dot{G}}_{\eta }$, so $q$ is compatible with $p$. Choose a common extension $q^{\mathrm{\prime }}\le q,p$; mastery and all displayed forced conclusions persist below $q^{\mathrm{\prime }}$. Hence $q^{\mathrm{\prime }}$ is the promised master literally below $p$. The index $n=0$, the empty initial stage, one-coordinate supports, and a finite list of dense sets are all covered by the same recursion.