The Analytical Hierarchy of Formulas
The notation indicates the class of formulas in the language of second-order arithmetic with no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarchy for details.
A formula in the language of second-order arithmetic is defined to be if it is logically equivalent to a formula of the form where is . A formula is defined to be if it is logically equivalent to a formula of the form where is . This inductive definition defines the classes and for every natural number .
Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is or for some . Because meaningless quantifiers can be added to any formula, once a formula is given the classification or for some it will be given the classifications and for all greater than .
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