Zermelo-Fraenkel Axioms

It would be very convenient indeed for set theorists if any collection of objects with a given property describable by the language of set theory could be called a set. Unfortunately, as shown by paradoxes such as Russells Paradox, we must put some restrictions on which collections to call sets. The Zermelo Fraenkel axiom system, developed by Ernst Zermelo and Abraham Fraenkel, does precisely this.

The Null Set Axiom

This axiom ensures that there is at least one set. Statement: There exists a set which contains no elements.

The Axiom of Subset Selection

This axiom declares subsets of a given set as sets themselves. Statement: Given a set , and a formula with one free variable, there exists a set whose elements are precisely those elements of which satisfy .

The Power Set Axiom

This axiom allows us to construct a bigger set from a given set. Statement: Given a set , there is a set containing all the subsets of A and no other element.