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 Russell's 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 Axiom of Extensionability

This axiom establishes the most basic property of sets - a set is completely characterized by its elements alone.
Statement: If two sets have the same elements, they are identical

The Null Set Axiom

This axiom ensures that there is at least one set.
Statement: There exists a set called the null 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 $A$ , and a formula $\phi(a)$ with one free variable, there exists a set whose elements are precisely those elements of $A$ which satisfy $\phi$ .

The Power Set Axiom

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

The Axiom of Replacement

This axiom allows us, given a set, to construct other sets of the same size.
Statement: Given a set $A$ and a bijective binary relation describable in the language of set theory, there is a set which consists of exactly those elements related to elements in $A$ .

The Axiom of Union

This axiom allows us to take unions of two or more sets.
Statement: Given a set $A$ , there exists a set with exactly those elements which belong to some element of $A$ .

The Axiom of Infinity

This gives us at least one infinite set.
Statement: There exists a set $A$ containing the null set, such that for all $a$ in $A$ , $\{a\}$ is also in $A$ .

The Axiom of Foundation

This makes sure no set contains itself, thus avoiding certain paradoxical situations.
Statement: The relation belongs to is well-founded.

The Axiom of Choice

This allows to find a choice set for any arbitrary collection of sets.
Statement: Given any collection of sets, there exists a set (called the choice set) containing precisely one element of each set in the collection.

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