Axiom schema of predicative separation

Schema of axioms in set theory

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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. This name Δ0 stems from the Lévy hierarchy, in analogy with the arithmetic hierarchy.

## Statement[edit]

The axiom asserts only the existence of a subset of a set if that subset can be defined without reference to the entire universe of sets. 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,

∀ x ∃ y ∀ z ( z ∈ y ↔ z ∈ x ∧ φ ( z ) ) {\displaystyle \forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge \varphi (z))}

provided that φ contains only bounded quantifiers and, as usual, that the variable y is not free in it. So all quantifiers in φ, if any, must appear in the forms

∃ u ∈ v ψ ( u ) {\displaystyle \exists u\in v\;\psi (u)}
∀ u ∈ v ψ ( u ) {\displaystyle \forall u\in v\;\psi (u)}

for some sub-formula ψ and, of course, the definition of  v {\displaystyle v} is bound to those rules as well.

### Motivation[edit]

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.

## Theories[edit]

The axiom appears in the systems of constructive set theory CST and CZF, as well as in the system of Kripke–Platek set theory.

### Finite axiomatizability[edit]

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.[citation needed]

## See also[edit]

* Constructive set theory
* Axiom schema of separation

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