# Axiom schema of predicative separation

__: Schema of axioms in set theory__

**Short description**In axiomatic set theory, the **axiom schema of predicative separation**, or of **restricted**, or **Δ _{0}** separation, is a schema of axioms which is a restriction of the usual axiom schema of separation in Zermelo–Fraenkel set theory.
This name Δ

_{0}stems from the Lévy hierarchy, in analogy with the arithmetic hierarchy.

## Statement

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,

- [math]\displaystyle{ \forall x \; \exists y \; \forall z \; (z \in y \leftrightarrow z \in x \wedge \phi(z)) }[/math]

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

- [math]\displaystyle{ \exists u \in v \; \psi(u) }[/math]
- [math]\displaystyle{ \forall u \in v \; \psi(u) }[/math]

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

### Motivation

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

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

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.

## See also

- Constructive set theory
- Axiom schema of separation

Original source: https://en.wikipedia.org/wiki/Axiom schema of predicative separation.
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