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  "body": "\n## Summary\n\nDescriptive set theory studies the definable subsets of [[Polish spaces|584GXXY73M]] such as the real line, the Cantor space and the [[Baire space|P2WS16HKD7]].\nIt sorts sets by how complicated their definitions are, in the [[Borel hierarchy|4P4Z4ZGXCY]] and the [[projective hierarchy|8DZTNVTGY2]].\nIts classical results show that simply defined sets, such as Borel and [[analytic sets|3RWWZYPAFV]], behave well.\nFor example, they are Lebesgue measurable, and they are either countable or contain a perfect set.\nFor more complex sets these questions depend on axioms beyond ZFC, such as [[determinacy|QXECDPAYTN]] and [[large cardinals|X1FG7PSCTY]].\n\n## Explanations\n\n### L1 human\n\nSome sets of real numbers are easy to describe, like intervals or the set of rational numbers.\nOthers are only given by long chains of operations, or not by any explicit rule at all.\nDescriptive set theory measures how hard a set is to describe, and shows that easy sets behave well.\nFor example, an uncountable closed set of reals always contains a copy of the Cantor set, so it has the size of the continuum.\n",
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