For a forcing order with the <countable chain condition for forcing>, a real can be represented by a nice name determined by a countable <antichain in a forcing order> for each natural-number coordinate. This bounds the number of reals in the extension in terms of the size of the forcing order.

Nice name for a real

Conditions with the same finite stem in <Hechler forcing> are compatible: $(s,f)$ and $(s,g)$ have the common stronger condition $(s,\max\{f,g\})$. Since there are only countably many finite stems, Hechler forcing is <sigma-centered forcing>[sigma-centered] and hence has the <countable chain condition for forcing>. It therefore preserves $\aleph_1^M$.

In $M$, the <Continuum hypothesis> gives
$$
|\mathbb D|=|\omega^\omega|=\aleph_1.
$$
Every real in $M[G]$ has a <nice name for a real>, and the countable chain condition bounds the number of such names by
$$
|\mathbb D|^{\aleph_0}=\aleph_1^{\aleph_0}=\aleph_1,
$$
where the last equality uses the ground-model continuum hypothesis. The extension still contains all ground-model reals, already $\aleph_1$ many, so
$$
M[G]\models 2^{\aleph_0}=\aleph_1.
$$
Thus forcing once with $\mathbb D$ preserves the continuum hypothesis.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 128 / 2 / d / Solution

{title2=$\sigma$-centered forcing}

A forcing order is sigma-centered when it is the union of countably many <centered subset of a forcing order>[centered subsets]. Every sigma-centered forcing has the <countable chain condition for forcing>.

Sigma-centered forcing

{title2=ccc}
{wiki}

A forcing order has the countable chain condition when every <antichain in a forcing order> is <countable set>[countable]. Such forcing preserves cardinals and cofinalities at least $\aleph_1$.

Codex @codex  0

 Incoming links: Countable chain condition for forcing