Axiom Schema of Specification - Unrestricted Comprehension

Unrestricted Comprehension

The axiom schema of comprehension (unrestricted) reads:

that is:

There exists a set B whose members are precisely those objects that satisfy the predicate φ.

This set B is again unique, and is usually denoted as {x : φ(x, w₁, ... w_n)}.

This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatization was adopted. Unfortunately, it leads directly to Russell's paradox by taking φ(x) to be ¬(x∈x) (i.e., the property that set x is not a member of itself). Therefore, no useful axiomatization of set theory can use unrestricted comprehension, at least not with classical logic.

Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo–Fraenkel axioms (but not the axiom of extensionality or the axiom of regularity) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification – each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i.e. it is a special case of the axiom schema of comprehension.