# FS iterated forcing preserves chain condition

Let $\kappa$ be a regular cardinal and let $\langle\hat{Q}_{\beta}\rangle_{\beta<\alpha}$ be a finite support iterated forcing where for every $\beta<\alpha$, $\Vdash_{P_{\beta}}\hat{Q}_{\beta}$ has the $\kappa$ chain condition.

By induction:

$P_{0}$ is the empty set.

If $P_{\alpha}$ satisfies the $\kappa$ chain condition then so does $P_{\alpha+1}$, since $P_{\alpha+1}$ is equivalent to $P_{\alpha}*Q_{\alpha}$ and composition preserves the $\kappa$ chain condition for regular $\kappa$.

Suppose $\alpha$ is a limit ordinal and $P_{\beta}$ satisfies the $\kappa$ chain condition for all $\beta<\alpha$. Let $S=\langle p_{i}\rangle_{i<\kappa}$ be a subset of $P_{\alpha}$ of size $\kappa$. The domains of the elements of $p_{i}$ form $\kappa$ finite subsets of $\alpha$, so if $\operatorname{cf}(\alpha)>\kappa$ then these are bounded, and by the inductive hypothesis, two of them are compatible.

Otherwise, if $\operatorname{cf}(\alpha)<\kappa$, let $\langle\alpha_{j}\rangle_{j<\operatorname{cf}(\alpha)}$ be an increasing sequence of ordinals cofinal in $\alpha$. Then for any $i<\kappa$ there is some $n(i)<\operatorname{cf}(\alpha)$ such that $\operatorname{dom}(p_{i})\subseteq\alpha_{n(i)}$. Since $\kappa$ is regular and this is a partition of $\kappa$ into fewer than $\kappa$ pieces, one piece must have size $\kappa$, that is, there is some $j$ such that $j=n(i)$ for $\kappa$ values of $i$, and so $\{p_{i}\mid n(i)=j\}$ is a set of conditions of size $\kappa$ contained in $P_{\alpha_{j}}$, and therefore contains compatible members by the induction hypothesis.

Finally, if $\operatorname{cf}(\alpha)=\kappa$, let $C=\langle\alpha_{j}\rangle_{j<\kappa}$ be a strictly increasing, continuous sequence cofinal in $\alpha$. Then for every $i<\kappa$ there is some $n(i)<\kappa$ such that $\operatorname{dom}(p_{i})\subseteq\alpha_{n(i)}$. When $n(i)$ is a limit ordinal, since $C$ is continuous, there is also (since $\operatorname{dom}(p_{i})$ is finite) some $f(i) such that $\operatorname{dom}(p_{i})\cap[\alpha_{f(i)},\alpha_{i})=\emptyset$. Consider the set $E$ of elements $i$ such that $i$ is a limit ordinal and for any $j, $n(j). This is a club, so by Fodor’s lemma there is some $j$ such that $\{i\mid f(i)=j\}$ is stationary.

For each $p_{i}$ such that $f(i)=j$, consider $p^{\prime}_{i}=p_{i}\upharpoonright j$. There are $\kappa$ of these, all members of $P_{j}$, so two of them must be compatible, and hence those two are also compatible in $P$.

Title FS iterated forcing preserves chain condition FSIteratedForcingPreservesChainCondition 2013-03-22 12:57:14 2013-03-22 12:57:14 Henry (455) Henry (455) 4 Henry (455) Result msc 03E35 msc 03E40