# FS iterated forcing preserves chain condition

Let $\kappa $ be a regular cardinal and let $$ be a finite support iterated forcing where for every $$, ${\u22a9}_{{P}_{\beta}}{\widehat{Q}}_{\beta}$ has the $\mathrm{\kappa}$ 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 $$. Let $$ 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 $\mathrm{cf}(\alpha )>\kappa $ then these are bounded^{}, and by the inductive hypothesis, two of them are compatible.

Otherwise, if $$, let $$ be an increasing sequence of ordinals^{} cofinal in $\alpha $. Then for any $$ there is some $$ such that $\mathrm{dom}({p}_{i})\subseteq {\alpha}_{n(i)}$. 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 $\mathrm{cf}(\alpha )=\kappa $, let $$ be a strictly increasing, continuous^{} sequence cofinal in $\alpha $. Then for every $$ there is some $$ such that $\mathrm{dom}({p}_{i})\subseteq {\alpha}_{n(i)}$. When $n(i)$ is a limit ordinal, since $C$ is continuous, there is also (since $\mathrm{dom}({p}_{i})$ is finite) some $$ such that $\mathrm{dom}({p}_{i})\cap [{\alpha}_{f(i)},{\alpha}_{i})=\mathrm{\varnothing}$. Consider the set $E$ of elements $i$ such that $i$ is a limit ordinal and for any $$, $$. 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 ${p}_{i}^{\prime}={p}_{i}\upharpoonright j$. 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 |
---|---|

Canonical name | FSIteratedForcingPreservesChainCondition |

Date of creation | 2013-03-22 12:57:14 |

Last modified on | 2013-03-22 12:57:14 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 4 |

Author | Henry (455) |

Entry type | Result |

Classification | msc 03E35 |

Classification | msc 03E40 |