Axiom schema of predicative separation

In axiomatic set theory, the axiom schema of predicative separation, or of restricted, or Δ₀ separation, is a schema of axioms which is a restriction of the usual axiom schema of separation in Zermelo–Fraenkel set theory. It only asserts the existence of a subset of a set if that subset can be defined without reference to the entire universe of sets. The axiom appears in the systems of constructive set theory CST and CZF, as well as in the system of Kripke–Platek set theory. The name Δ₀ comes from the Levy hierarchy (in analogy with the arithmetic hierarchy).

The formal statement of this is the same as full separation schema, but with a restriction on the formulas that may be used. For any formula φ:

\forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge \phi (z))

provided, as usual, that the variable y is not free in φ; but also provided that φ contains only bounded quantifiers. That is, all quantifiers in φ (if there are any) must appear in the form $\exists x\in y\;\psi (x)$ or $\forall x\in y\;\psi (x)$ for some sub-formula ψ.

The meaning of this is that, given any set x, and any predicate φ there is a set y whose elements are the elements of x which satisfy φ,

y=\{z\mid \phi (z)\}\subset x

provided φ only quantifies over existing sets, and never quantifies over all sets. This restriction is necessary from a predicative point of view, since the universe of all sets contains the set being defined. If it were referenced in the definition of the set, the definition would be circular.

Although the schema contains one axiom for each restricted formula φ, it is possible in CZF to replace this schema with a finite number of axioms.

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