Martin's Axiom

A partial order $P$ has the countable chain condition (ccc) when every family of pairwise incompatible elements of $P$ is countable. For a cardinal $\kappa$, $\mathrm{M}\mathrm{A}(\kappa )$ asserts:

for every ccc partial order $P$ and every family $\mathcal{D}$ of at most $\kappa$ dense subsets of $P$, there is a filter on $P$ meeting every member of $\mathcal{D}$.

Martin's Axiom (MA) is the assertion that $\mathrm{M}\mathrm{A}(\kappa )$ holds for every $\kappa <2^{{\mathrm{\aleph }}_0}$.

Three facts fix its status.

(a) $\mathrm{M}\mathrm{A}({\mathrm{\aleph }}_0)$ is a theorem of ZFC (the Rasiowa-Sikorski lemma), so MA is not vacuous but its content is entirely in the uncountable cases.

(b) CH implies MA, trivially, since under CH there is no $\kappa$ with ${\mathrm{\aleph }}_0<\kappa <2^{{\mathrm{\aleph }}_0}$. So MA alone decides nothing that CH does not.

(c) If ZFC is consistent, then so is ZFC + MA + (not CH). This is Solovay and Tennenbaum (1971), and it is where finite-support iterated ccc forcing was invented: one iterates ccc posets ${\mathrm{\aleph }}_2$ times, catching every ccc poset of size less than the continuum along the way, and the iteration is itself ccc so no cardinal is collapsed.

What MA + (not CH) buys. Every set of reals of cardinality less than $2^{{\mathrm{\aleph }}_0}$ is Lebesgue null and meagre; the union of fewer than $2^{{\mathrm{\aleph }}_0}$ meagre sets is meagre; $2^{\kappa }=2^{{\mathrm{\aleph }}_0}$ for every infinite $\kappa <2^{{\mathrm{\aleph }}_0}$; and the product of two ccc spaces is ccc, so ccc is productive.

• Not proved in this library. Neither the consistency result nor any of the consequences is proved here.

• What would prove it. Iterated forcing with finite support, the ccc preservation theorem for such iterations, and a bookkeeping argument for the ${\mathrm{\aleph }}_2$-length iteration. This is the forcing track named in Cohen 1963: ZF does not prove the Axiom of Choice ‡, one level beyond a single-step extension.

• Why it matters here. MA is the standard hypothesis under which the continuum behaves as if it were ${\mathrm{\aleph }}_1$ for the purposes of category and measure while nevertheless being large. It is the axiom that makes the two sides of The Suslin hypothesis is independent of ZFC, in both directions ‡ and of The normal Moore space conjecture is independent, and its consistency needs a large cardinal ‡ possible, and it is the reason a topology page cannot state "ccc is productive" as a theorem: that statement is a consequence of MA + (not CH) and is refuted by a Suslin line. Notice also what MA is not: it is not a choice principle and not a size axiom about Cardinal (initial ordinal) and cardinality ↗; it is a genericity assumption, and its interaction with CH is recorded in The continuum hypothesis, and what this page does not prove ↗.

• Conditional discipline. Clause (c) is relative to the consistency of ZFC. Clauses (a) and (b) are ZFC theorems. Nothing here asserts MA.

• On the citation. The Solovay-Tennenbaum paper is behind a paywall that refuses automated access, so the reference url points at the survey account of the result; the full bibliographic details are in the reference title.