The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1), stands for exists, for for all, for "is an element of," for the empty set, for implies, for AND, for OR, and for "is equivalent to."

1. Axiom of Extensionality: If and have the same elements, then .

2. Axiom of the Unordered Pair: For any and there exists a set that contains exactly and . (also called Axiom of Pairing)

3. Axiom of Subsets: If is a property (with parameter ), then for any and there exists a set that contains all those that have the property . (also called Axiom of Separation or Axiom of Comprehension)

4. Axiom of the Sum Set: For any there exists a set , the union of all elements of . (also called Axiom of Union)

5. Axiom of the Power Set: For any there exists a set , the set of all subsets of .

6. Axiom of Infinity: There exists an infinite set.

7. Axiom of Replacement: If is a function, then for any there exists a set .

8. Axiom of Foundation: Every nonempty set has an -minimal element. (also called Axiom of Regularity)

9. Axiom of Choice: Every family of nonempty sets has a choice function.

The system of axioms 1-8 is called Zermelo-Fraenkel set theory, denoted "ZF." The system of axioms 1-8 minus the axiom of replacement (i.e., axioms 1-6 plus is called Zermelo set theory, denoted "Z." The set of axioms 1-9 with the axiom of choice is usually denoted "ZFC."

However, note that there seems to be some disagreement in the literature about just what axioms constitute "Zermelo set theory." Mendelson (1997) does not include the axioms of choice or foundation in Zermelo set theory, but does include the axiom of replacement. Enderton (1977) includes the axioms of choice and foundation, but does not include the axiom of replacement. Itô includes an Axiom of the empty set, which can be gotten from (6) and (3), via and .

Abian (1969) proved consistency and independence of four of the Zermelo-Fraenkel axioms.