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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]], and read $q \\le p$ as \"$q$ extends $p$\".\nTwo elements $p, q$ are compatible if some $r \\in P$ has $r \\le p$ and $r \\le q$.\n\n1. An antichain of $P$ is a subset whose elements are pairwise incompatible, and $P$ has the countable chain condition, ccc, if each antichain is countable.\n2. A set $D \\subseteq P$ is dense if for each $p \\in P$ there is $q \\in D$ with $q \\le p$.\n3. A filter on $P$ is a nonempty $G \\subseteq P$ that is upward closed and in which any two elements have a common extension in $G$.\n\nLet $\\kappa$ be a [[cardinal|0QKPNZKNEA]].\nThe statement $\\mathrm{MA}_\\kappa$ concerns each ccc partial order $P$ and each family $\\mathcal{D}$ of at most $\\kappa$ dense subsets of $P$.\nIt says that some filter on $P$ meets every member of $\\mathcal{D}$.\nMartin's axiom, MA, is the statement that $\\mathrm{MA}_\\kappa$ holds for every $\\kappa \u003c 2^{\\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].\n\n## Remarks\n\nExample: $\\mathrm{MA}_{\\aleph_0}$ is provable in ZFC.\nCheck: 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 $p_n$.\nSo the continuum hypothesis implies MA.\nNon-example: $\\mathrm{MA}_{2^{\\aleph_0}}$ is false.\nTake the ccc order of finite partial functions from $\\omega$ to $2$, ordered by reverse inclusion.\nConsider the dense sets \"$n$ is in the domain\" and \"the function differs from $x$\", for all $n$ and all $x \\in 2^\\omega$.\nA filter that meets them all would give a total function different from every $x$.\nSolovay and Tennenbaum showed that MA together with $2^{\\aleph_0} \u003e \\aleph_1$ is consistent, if ZFC is consistent.\nMA with $2^{\\aleph_0} \u003e \\aleph_1$ implies Suslin's hypothesis, and MA implies $2^\\kappa = 2^{\\aleph_0}$ for each infinite $\\kappa \u003c 2^{\\aleph_0}$.\n",
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