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  "body": "\n## Summary\n\nThe proper forcing axiom, PFA, is a strong extension of [[Martin's axiom|V8KGT8MRFE]] at $\\aleph_1$.\nIt replaces the countable chain condition by the much weaker condition of properness.\nIt says that for each proper forcing notion one can meet $\\aleph_1$ dense sets with a single filter; the [[precise definition|KJT0AXWQXP]] is given separately.\nIts consistency is known only from a large cardinal hypothesis, a supercompact cardinal.\nIt decides many questions of combinatorial set theory, and it implies that $2^{\\aleph_0} = \\aleph_2$.\n\n## Explanations\n\n### L1 human\n\nA forcing axiom says that certain constructions of length $\\omega_1$ can always be completed.\nMartin's axiom allows this for a narrow class of construction methods, the ccc ones.\nThe proper forcing axiom allows it for a much wider class, the proper ones, which include all ccc methods and all countably closed ones.\nWith this freedom, many open-ended questions about sets of size $\\aleph_1$ get definite answers.\n",
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