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Many questions in descriptive set theory ultimately depend upon [[set theory|set-theoretic]] considerations and the properties of [[ordinal number|ordinal]] and [[cardinal number]]s. This phenomenon is particularly apparent in the '''projective sets'''. These are defined via the [[projective hierarchy]] on a Polish space ''X'':
* A set is declared to be <math>\mathbf{\Sigma}^1_1</math> if it is analytic.
* A set
* A set ''A'' is <math>\mathbf{\Sigma}^1_{n+1}</math> if there is a <math>\mathbf{\Pi}^1_n</math> subset ''B'' of <math>X \times X</math> such that ''A'' is the projection of ''B'' to the first coordinate.
* A set ''A'' is <math>\mathbf{\Pi}^1_{n+1}</math> if there is a <math>\mathbf{\Sigma}^1_n</math> subset ''B'' of <math>X \times X</math> such that ''A'' is the projection of ''B'' to the first coordinate.
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