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  "body": "\n## Definition\n\nPartial orders, dense sets and filters are as in [[Martin's axiom|Z8XMCCZFA4]].\nFor an uncountable set $\\lambda$, let $[\\lambda]^{\\omega}$ be the set of countable subsets of $\\lambda$.\nA set $C \\subseteq [\\lambda]^\\omega$ is club if both conditions hold.\nIt contains the union of each increasing sequence $x_0 \\subseteq x_1 \\subseteq \\cdots$ of its members.\nEach countable subset of $\\lambda$ is contained in a member of $C$.\nA set $S \\subseteq [\\lambda]^\\omega$ is stationary if it meets each club subset of $[\\lambda]^\\omega$.\nA partial order $P$ is proper if [[forcing|F2DTWW9HP7]] with $P$ preserves stationarity: for each uncountable $\\lambda$, each stationary $S \\subseteq [\\lambda]^\\omega$ stays stationary in the generic extension.\nThe proper forcing axiom, PFA, is the following statement.\nLet $P$ be a proper partial order, and let $\\mathcal{D}$ be a family of at most $\\aleph_1$ dense subsets of $P$.\nThen some filter on $P$ meets every member of $\\mathcal{D}$.\n\n## Remarks\n\nExample: each ccc partial order is proper, so PFA implies $\\mathrm{MA}_{\\aleph_1}$.\nEach countably closed partial order is proper too.\nNon-example: the partial order that collapses $\\omega_1$ to be countable, by finite partial functions from $\\omega$ to $\\omega_1$, is not proper.\nProper forcings preserve $\\omega_1$, while this one makes $\\omega_1$ countable.\nBaumgartner proved that PFA is consistent relative to a supercompact cardinal.\nPFA implies $2^{\\aleph_0} = \\aleph_2$, and it implies that every two $\\aleph_1$-dense sets of reals are order isomorphic.\nThese notions extend the [[stationary sets|6FDYJVG1CK]] of a cardinal to sets of countable subsets.\n",
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