In axiomatic set theory, 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 ...

of axioms which is a restriction of the usual

axiom schema of separation In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...

. 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

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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 The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the 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 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 Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "=" and "\in" of classical set theory is usually used, so this is not to be confused with a con ...

CST and CZF, as well as in the system of

Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its fo ...

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