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  "body": "\n## Summary\n\nThe projective hierarchy sorts the projective subsets of a [[Polish space|584GXXY73M]] into levels $\\Sigma^1_n$, $\\Pi^1_n$ and $\\Delta^1_n$ for $n \\ge 1$.\nIt starts with the [[analytic sets|3RWWZYPAFV]], and alternates two operations: taking complements and taking projections along the [[Baire space|P2WS16HKD7]].\nThe first level $\\Delta^1_1$ consists exactly of the Borel sets, by [[Suslin's theorem|9418TGREZX]].\nZFC does not decide whether the higher levels behave well, for example whether all their sets are Lebesgue measurable.\nThey do under [[projective determinacy|2GY3131ZPS]], and they do not under the [[axiom of constructibility|4QKRPRCSXH]].\n\n## Explanations\n\n### L1 human\n\nStart with the shadows of closed sets: these are the analytic sets.\nTake complements of these, then their shadows, then complements again, and so on.\nEach step gives a new level of complexity, and the sets reached after finitely many steps are the projective sets.\nMost sets of real numbers that appear in ordinary mathematics are projective, often at a low level.\n",
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