axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

, the axiom schema of predicative separation, or of restricted, or Δ₀ separation, is a

schema The word schema comes from the Greek word ('), which means ''shape'', or more generally, ''plan''. The plural is ('). In English, both ''schemas'' and ''schemata'' are used as plural forms. Schema may refer to: Science and technology * SCHEMA ...

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s 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 set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This i ...

, in analogy with the

arithmetic hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...

Statement

The axiom asserts only the existence of a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

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, :

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

provided that φ contains only

bounded quantifiers In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and " ...

and, as usual, that the variable ''y'' is not free in it. So all quantifiers in φ, if any, must appear in the forms :

\exists u \in v \; \psi(u)

\forall u \in v \; \psi(u)

for some sub-formula ψ and, of course, the definition of

v

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.

Statement

Motivation

Theories

Finite axiomatizability

See also