What is Zermelo-Fraenkel set theory?

Zermelo-Fraenkel set theory is an axiomatic system proposed in the early twentieth century by Ernst Zermelo and Abraham Fraenkel to formulate a theory of sets free from paradoxes like Russell's paradox. When combined with the axiom of choice, it becomes ZFC, which serves as the most common foundation for modern mathematics. The theory formalizes pure sets and prevents models from containing urelements or a universal set.

What we know

• Zermelo-Fraenkel set theory was proposed in the early twentieth century by mathematicians Ernst Zermelo and Abraham Fraenkel to formulate a theory free of paradoxes. [1, 2]
• Zermelo set theory, set out in 1908 by Ernst Zermelo, serves as the ancestor of modern Zermelo-Fraenkel set theory. [1, 3]
• Zermelo-Fraenkel set theory combined with the axiom of choice is abbreviated as ZFC and acts as the standard foundation of mathematics. [2]
• The axioms prevent models from containing urelements and do not allow for the existence of a universal set. [2]

Where to go next

• whoWho were Ernst Zermelo and Abraham Fraenkel?
• what caused itWhat is Russell's paradox in set theory?
• how it workedWhat is the axiom of choice?
• compared withHow does Von Neumann-Bernays-Godel set theory extend ZF?
• the bigger pictureWhat is constructive set theory?
• an unexpected connectionHow does constructive set theory differ in its logic?While standard Zermelo-Fraenkel set theory relies on classical logic, constructive set theory rejects the principle of the excluded middle.

Sources

• [1]Zermelo–Fraenkel set theory (Wikidata Q191849) · CC0 1.0
• [2]Zermelo–Fraenkel set theory · CC BY-SA 4.0
• [3]Zermelo set theory · CC BY-SA 4.0
• [4]Union (set theory) · CC BY-SA 4.0
• [5]Constructive set theory · CC BY-SA 4.0