Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory (abbreviated ZF) is a system of axioms used to describe set theory. When the axiom of choice is added to ZF, the system is called ZFC. It is the system of axioms used in set theory by most mathematicians today.
After Russell's paradox was found in the 1901, mathematicians wanted to find a way to describe set theory that did not have contradictions. Ernst Zermelo proposed a theory of set theory in 1908. In 1922, Abraham Fraenkel proposed a new version based on Zermelo's work.
Axioms.
An axiom is a statement which is accepted without question, and which has no proof. ZF contains eight axioms.
Axiom of choice.
The axiom of choice says that it is possible to take one object out of each of the elements of a set and make a new set. For example, given the set formula_14, the axiom of choice would show that a set such as formula_28 exists. For finite sets, this axiom can be proved from the other axioms, but not for infinite sets.