Editing Analytical hierarchy (section)

== Extensions ==
As is the case with the [[arithmetical hierarchy]], a relativized version of the analytical hierarchy can be defined. The language is extended to add a constant set symbol ''A''. A formula in the extended language is inductively defined to be <math>\Sigma^{1,A}_n</math> or <math>\Pi^{1,A}_n</math> using the same inductive definition as above. Given a set <math>Y</math>, a set is defined to be <math>\Sigma^{1,Y}_n</math> if it is definable by a <math>\Sigma^{1,A}_n</math> formula in which the symbol <math>A</math> is interpreted as <math>Y</math>; similar definitions for <math>\Pi^{1,Y}_n</math> and <math>\Delta^{1,Y}_n</math> apply. The sets that are <math>\Sigma^{1,Y}_n</math> or <math>\Pi^{1,Y}_n</math>, for any parameter ''Y'', are classified in the [[projective hierarchy]], and often denoted by boldface Greek letters to indicate the use of parameters.<ref>P. D. Welch, [https://people.maths.bris.ac.uk/~mapdw/det17.pdf "Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions"] (2010 draft ver., p. 3). Accessed 31 July 2022.</ref>