SomethingRelatedToMartinsAxiomAndTheContinuumHypothesis 2003-08-24 20:32:54 2004-03-15 17:17:13 Proof Martin's axiom and the continuum hypothesis 0 $\mathcal D$-generic generic dense Martins Axiom MA_{\aleph_0} countable collections of dense subsets internal forcing theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \PMlinkescapeword{completes} This is another, shorter, proof for the fact that $MA_{\aleph_0}$ always holds. Let $(P,\leq)$ be a partially ordered set and $\mathcal D$ be a collection of subsets of $P$. We remember that a filter $G$ on $(P,\leq)$ is $\mathcal D$-generic if $G \cap D \neq \varnothing$ for all $D \in \mathcal D$ which are dense in $(P,\leq)$. (In this context ``dense'' means: If $D$ is dense in $(P,\leq)$, then for every $p \in P$ there's a $d \in D$ such that $d \leq p$.) Let $(P,\leq)$ be a partially ordered set and $\mathcal D$ a countable collection of dense subsets of $P$. Then there exists a $\mathcal D$-generic filter $G$ on $P$. Moreover, it could be shown that for every $p \in P$ there's such a $\mathcal D$-generic filter $G$ with $p \in G$. \begin{proof} Let $D_1,\dots, D_n, \dots$ be the dense subsets in $\mathcal D$. Furthermore let $p_0 = p$. Now we can choose for every $1 \leq n < \omega$ an element $p_n \in P$ such that $p_n \leq p_{n-1}$ and $p_n \in D_n$. If we now consider the set $G:=\{ q \in P \mid \exists \; n < \omega \text{ s.t. } p_n \leq q \}$, then it is easy to check that $G$ is a $\mathcal D$-generic filter on $P$ and $p \in G$ obviously. This completes the proof. \end{proof}