Meaning of The Notation
The following meanings can be attached to the notation for the arithmetical hierarchy on formulas.
The subscript in the symbols and indicates the number of alternations of blocks of universal and existential number quantifiers that are used in a formula. Moreover, the outermost block is existential in formulas and universal in formulas.
The superscript in the symbols, and indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type are functions that map the set of objects of type to the natural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers, is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers over numbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects), the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, and so on.
Read more about this topic: Arithmetical Hierarchy
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