Axiom schema of predicative separation

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

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,

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

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 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

References

• Tennant, Neil (2021). "Does Choice Really Imply Excluded Middle? Part II: Historical, Philosophical, and Foundational Reflections on the Goodman–Myhill Result†". Philosophia Mathematica. 29: 28–63. doi:10.1093/philmat/nkaa010.