Axiom of Subsets

The axiom of Zermelo-Fraenkel set theory which asserts the existence for any set  and a formula  of a set  consisting of all elements of  satisfying ,

where  denotes exists,  means for all,  denotes "is an element of,"  means equivalent, and  denotes logical AND.

This axiom is called the subset axiom by Enderton (1977), while Kunen (1980) calls it the comprehension axiom. Itô (1986) terms it the axiom of separation, but this name appears to not be used widely in the literature and to have the additional drawback that it is potentially confusing with the separation axioms of Hausdorff arising in topology.

This axiom was introduced by Zermelo.

REFERENCES:

Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.

Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146-148, 1986.

Kunen, K. Set Theory: An Introduction to Independence Proofs. Dordrecht, Netherlands: Elsevier, 1980.