a shorter proof: Martin’s axiom and the continuum hypothesis

This is another, shorter, proof for the fact that $M{A}_{{\mathrm{\aleph }}_0}$ always holds.

Let $(P,\le )$ be a partially ordered set and $𝒟$ be a collection of subsets of $P$. We remember that a filter $G$ on $(P,\le )$ is $𝒟$-generic if $G\cap D\ne \mathrm{\varnothing }$ for all $D\in 𝒟$ which are dense in $(P,\le )$. (In this context “dense” means: If $D$ is dense in $(P,\le )$, then for every $p\in P$ there’s a $d\in D$ such that $d\le p$.)

Let $(P,\le )$ be a partially ordered set and $𝒟$ a countable collection of dense subsets of $P$. Then there exists a $𝒟$-generic filter $G$ on $P$. Moreover, it could be shown that for every $p\in P$ there’s such a $𝒟$-generic filter $G$ with $p\in G$.

Proof.

Let $D_1,\mathrm{\dots },D_n,\mathrm{\dots }$ be the dense subsets in $𝒟$. Furthermore let $p_0=p$. Now we can choose for every an element $p_n\in P$ such that $p_n\le p_{n-1}$ and $p_n\in D_n$. If we now consider the set , then it is easy to check that $G$ is a $𝒟$-generic filter on $P$ and $p\in G$ obviously. This completes the proof.∎