Bounded Quantifiers in Set Theory
Suppose that L is the language of the Zermelo–Fraenkel set theory, where the ellipsis may be replaced by term-forming operations such as a symbol for the powerset operation. There are two bounded quantifiers: and . These quantifiers bind the set variable x and contain a term t which may not mention x but which may have other free variables.
The semantics of these quantifiers is determined by the following rules:
A ZF formula which contains only bounded quantifiers is called, and . This forms the basis of the Levy hierarchy, which is defined analogously with the arithmetical hierarchy.
Bounded quantifiers are important in Kripke-Platek set theory and constructive set theory, where only Δ0 separation is included. That is, it includes separation for formulas with only bounded quantifiers, but not separation for other formulas. In KP the motivation is the fact that whether a set x satisfies a bounded quantifier formula only depends on the collection of sets that are close in rank to x (as the powerset operation can only be applied finitely many times to form a term). In constructive set theory, it is motivated on predicative grounds.
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