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 mathema ...

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

schema Schema may refer to: Science and technology * SCHEMA (bioinformatics), an algorithm used in protein engineering * Schema (genetic algorithms), a set of programs or bit strings that have some genotypic similarity * Schema.org, a web markup vocab ...

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 ...

s 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 (''Aussonderungsaxiom''), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is ...

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 suc ...

. 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 is ...

, in analogy with the arithmetic hierarchy.

Statement

The axiom asserts only the existence of a

subset In mathematics, a 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 a ...

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. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...

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 \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 :

\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 Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in the Soviet Union in ...

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 Zermelo–Fraenkel set theory (ZFC) and is considerably weak ...

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.

References

* Constructivism (mathematics) Axioms of set theory {{settheory-stub

Statement

Motivation

Theories

Finite axiomatizability

See also

References