Martin's axiom

Remarks

Example: $\mathrm{MA}_{\aleph_0}$ is provable in ZFC. Check: list the dense sets as $D_0, D_1, \ldots$, choose $p_0 \ge p_1 \ge \cdots$ with $p_n \in D_n$, and take the elements above some $p_n$. So the continuum hypothesis implies MA. Non-example: $\mathrm{MA}_{2^{\aleph_0}}$ is false. Take the ccc order of finite partial functions from $\omega$ to $2$, ordered by reverse inclusion. Consider the dense sets "$n$ is in the domain" and "the function differs from $x$", for all $n$ and all $x \in 2^\omega$. A filter that meets them all would give a total function different from every $x$. Solovay and Tennenbaum showed that MA together with $2^{\aleph_0} > \aleph_1$ is consistent, if ZFC is consistent. MA with $2^{\aleph_0} > \aleph_1$ implies Suslin's hypothesis, and MA implies $2^\kappa = 2^{\aleph_0}$ for each infinite $\kappa < 2^{\aleph_0}$.