# 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 ${\u22a9}_{P}\widehat{Q}$ 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\u22a9|\{i\mid {p}_{i}\in \widehat{G}\}|=\kappa $

(Note: $\widehat{G}=\{\u27e8p,p\u27e9\mid p\in P\}$, hence $\widehat{G}[G]=G$)

If no $p$ forces this then every $p$ forces that it is not true, and therefore ${\u22a9}_{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\u22a9f(\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 ${p}_{\alpha +1}\u22a9{p}_{\alpha +1}\in \widehat{G}$. This is a contradiction^{}, so we conclude that there must be some $p$ such that $p\u22a9|\{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 ${p}^{\prime}\u22a9\widehat{q}[G]\le {\widehat{q}}_{1}[G],{\widehat{q}}_{2}[G]$. So $\u27e8{p}^{\prime},\widehat{q}\u27e9\le \u27e8{p}_{i},{\widehat{q}}_{i}\u27e9,\u27e8{p}_{j},{\widehat{q}}_{j}\u27e9$.

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 |