Single Idea 13031

[catalogued under 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension]

Full Idea

Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ.

Gist of Idea

Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ)

Source

Kenneth Kunen (Set Theory [1980], §1.5)

Book Reference

Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.11


A Reaction

Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential.