composition preserves chain condition

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

Proof:

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 ${\widehat{q}}_i$ 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 .

Claim: There is some $p\in P$ such that $p⊩|\{i\mid p_i\in \widehat{G}\}|=\kappa$

(Note: $\widehat{G}=\{⟨p,p⟩\mid p\in P\}$, hence $\widehat{G}[G]=G$)

If no $p$ forces this then every $p$ forces that it is not true, and therefore ${⊩}_P|\{i\mid p_i\in G\}|\le \kappa$. 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 implies that there is some $\gamma >\beta$ such that $p_{\gamma }\in G$. Define $B=\{\alpha \mid \alpha =f(G)\}$ for some $G$.

Claim:

If $\alpha \in B$ then there is some $p_{\alpha }\in P$ such that $p⊩f(\widehat{G})=\alpha$, 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 .

Since $\kappa$ is regular, . But obviously $p_{\alpha +1}⊩p_{\alpha +1}\in \widehat{G}$. This is a contradiction, so we conclude that there must be some $p$ such that $p⊩|\{i\mid p_i\in \widehat{G}\}|=\kappa$.

If $G\subseteq P$ is any generic subset containing $p$ then $A=\{{\widehat{q}}_i[G]\mid p_i\in G\}$ must have cardinality $\kappa$. Since $Q[G]$ satisfies the $\kappa$ chain condition, there exist such that $p_i,p_j\in G$ and there is some $\widehat{q}[G]\in Q[G]$ such that $\widehat{q}[G]\le {\widehat{q}}_i[G],{\widehat{q}}_j[G]$. Then since $G$ is directed, there is some $p^{\prime }\in G$ such that $p^{\prime }\le p_i,p_j,p$ and $p^{\prime }⊩\widehat{q}[G]\le {\widehat{q}}_1[G],{\widehat{q}}_2[G]$. So $⟨p^{\prime },\widehat{q}⟩\le ⟨p_i,{\widehat{q}}_i⟩,⟨p_j,{\widehat{q}}_j⟩$.

Title	composition preserves chain condition
Canonical name	CompositionPreservesChainCondition
Date of creation	2013-03-22 12:54:40
Last modified on	2013-03-22 12:54:40
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	5
Author	Henry (455)
Entry type	Result
Classification	msc 03E40
Classification	msc 03E35