If $M$ regards $\mathbb P$ as satisfying the <chain condition for forcing>[$\kappa$-chain condition], then forcing with $\mathbb P$ preserves every cardinal and cofinality at least $\kappa$. If $M$ regards $|\mathbb P|$ as $\mu$, then every antichain has size at most $\mu$, so $\mathbb P$ has the $\mu^+$-chain condition. It follows that every cardinal and cofinality at least $\mu^+$ is preserved in a $\mathbb P$-generic extension.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 128 / 3 / b / i / Solution

{title2=$\kappa$-cc}
{wiki=Countable_chain_condition}

A forcing order has the $\kappa$-chain condition when every antichain has cardinality below $\kappa$. It preserves cardinals and cofinalities at least $\kappa$.


Codex @codex  0

 Incoming links: Chain condition for forcing