composition preserves chain condition

Let $\kappa$ be a regular cardinal. Let $P$ be a forcing  notion satisfying the $\kappa$ chain condition. Let $\hat{Q}$ be a $P$-name such that $\Vdash_{P}\hat{Q}$ is a forcing notion satisfying the $\kappa$ chain condition. Then $P*Q$ satisfies the $\kappa$ chain condition.

Outline

We prove that there is some $p$ such that any generic subset of $P$ including $p$ also includes $\kappa$ of the $p_{i}$. Then, since $Q[G]$ satisfies the $\kappa$ chain condition, two of the corresponding $\hat{q}_{i}$ must be compatible. Then, since $G$ is directed, there is some $p$ stronger than any of these which forces this to be true, and therefore makes two elements of $S$ compatible.

Let $S=\langle p_{i},\hat{q}_{i}\rangle_{i<\kappa}\subseteq P*Q$.

Claim: There is some $p\in P$ such that $p\Vdash|\{i\mid p_{i}\in\hat{G}\}|=\kappa$

(Note: $\hat{G}=\{\langle p,p\rangle\mid p\in P\}$, hence $\hat{G}[G]=G$)

If no $p$ forces this then every $p$ forces that it is not true, and therefore $\Vdash_{P}|\{i\mid p_{i}\in G\}|\leq\kappa$. Since $\kappa$ is regular, this means that for any generic $G\subseteq P$, $\{i\mid p_{i}\in G\}$ is bounded. For each $G$, let $f(G)$ be the least $\alpha$ such that $\beta<\alpha$ implies that there is some $\gamma>\beta$ such that $p_{\gamma}\in G$. Define $B=\{\alpha\mid\alpha=f(G)\}$ for some $G$.

Claim: $|B|<\kappa$

If $\alpha\in B$ then there is some $p_{\alpha}\in P$ such that $p\Vdash f(\hat{G})=\alpha$, and if $\alpha,\beta\in B$ then $p_{\alpha}$ must be incompatible with $p_{\beta}$. Since $P$ satisfies the $\kappa$ chain condition, it follows that $|B|<\kappa$.

Since $\kappa$ is regular, $\alpha=\operatorname{sub}(B)<\kappa$. But obviously $p_{\alpha+1}\Vdash p_{\alpha+1}\in\hat{G}$. This is a contradiction   , so we conclude that there must be some $p$ such that $p\Vdash|\{i\mid p_{i}\in\hat{G}\}|=\kappa$.

If $G\subseteq P$ is any generic subset containing $p$ then $A=\{\hat{q}_{i}[G]\mid p_{i}\in G\}$ must have cardinality $\kappa$. Since $Q[G]$ satisfies the $\kappa$ chain condition, there exist $i,j<\kappa$ such that $p_{i},p_{j}\in G$ and there is some $\hat{q}[G]\in Q[G]$ such that $\hat{q}[G]\leq\hat{q}_{i}[G],\hat{q}_{j}[G]$. Then since $G$ is directed, there is some $p^{\prime}\in G$ such that $p^{\prime}\leq p_{i},p_{j},p$ and $p^{\prime}\Vdash\hat{q}[G]\leq\hat{q}_{1}[G],\hat{q}_{2}[G]$. So $\langle p^{\prime},\hat{q}\rangle\leq\langle p_{i},\hat{q}_{i}\rangle,\langle p% _{j},\hat{q}_{j}\rangle$.

Title composition preserves chain condition CompositionPreservesChainCondition 2013-03-22 12:54:40 2013-03-22 12:54:40 Henry (455) Henry (455) 5 Henry (455) Result msc 03E40 msc 03E35