Definition:Zermelo-Fraenkel Axioms

The Zermelo-Fraenkel axioms are the most well-known basis for axiomatic set theory.

There is no standard numbering for them, and their exact formulation varies. Certain axioms can in fact be derived from other axioms, so their status as "axioms" can be questioned.

The axioms are as follows:

The Axiom of Extension: Two sets are equal if and only if they have the same contents.

Otherwise known as the Axiom of Extensionality or Axiom of Extent.

The Axiom of Existence: There exists a set that has no elements.

This can be deduced from the Axiom of Infinity and Axiom of Subsets and some treatments exclude it from the list.

It is sometimes referred to as the Axiom of the Empty Set.

The Axiom of Pairing: For any two sets, there exists a set to which only those two sets belong.

This can be deduced from the Axiom of Infinity and the Axiom of Replacement.

It is sometimes referred to as the Axiom of the Unordered Pair.

The Axiom of Subsets: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.

This can be deduced from the Axiom of Replacement.

It is sometimes called:

The Axiom of Specification;

The Axiom of Comprehension;

The Axiom of Selection;

The Axiom of Separation.

The Axiom of Unions: For every collection of sets, there exists a set that contains all the elements that belong to at least one of the sets in the collection.

It is otherwise known as the Axiom of the Sum Set.

The Axiom of Powers: For each set, there exists a collection of sets that contains amongst its elements all the subsets of the given set.

It is otherwise known as the Axiom of the Power Set.

The Axiom of Infinity: There exists a set containing a set with no elements and the successor of each of its elements.

The Axiom of Replacement: Let $$P \left({y, z}\right)$$ be a propositional function, which determines a function.

For any set $$S$$, there exists a set $$x$$ such that, for any element $$y$$ of $$S$$, if there exists an element $$z$$ satisfying the condition $$P \left({y, z}\right)$$, then such $$z$$ appear in $$x$$.

The Axiom of Foundation: For all non-null sets, there is an element of the set that shares no member with the set.

Otherwise known as the Axiom of Regularity.

In addition to the above axioms, there is a further axiom which is still hotly debated as to its validity:

The Axiom of Choice: For every set, we can provide a mechanism for choosing one element of any non-empty subset of the set.