Zermelo–Fraenkel set theory

Lore & Background

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s, but paradoxes in naive set theory led to the desire for a more rigorous form. In 1908, Ernst Zermelo proposed the first axiomatic set theory. However, Abraham Fraenkel pointed out in a 1921 letter that this theory could not prove the existence of certain sets and cardinal numbers, such as aleph-omega and the set formed by iterating the power set operation on an infinite set. Additionally, one of Zermelo's axioms invoked a 'definite' property whose operational meaning was unclear.

In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a 'definite' property as one formulable in first-order logic with atomic formulas limited to set membership and identity. They also proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, along with the axiom of regularity first proposed by John von Neumann, to Zermelo set theory yields the theory ZFC.

Reader's Guide

Zermelo–Fraenkel set theory is the standard axiomatic system for set theory, serving as the most common foundation for mathematics. It formalizes a single primitive notion—that of a hereditary well-founded set—so that all entities in its universe are such sets, preventing models from containing urelements. Proper classes can only be treated indirectly, and the theory avoids Russell's paradox by not allowing a universal set or unrestricted comprehension. The axioms include the axiom of pairing, which states that given any two sets there is a new set containing exactly them, and other axioms describing set membership. A goal is that each axiom should be true if interpreted as a statement about the von Neumann universe (cumulative hierarchy).

The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results established the logical independence of the axiom of choice from the remaining ZF axioms and of the continuum hypothesis from ZFC. Gödel's second incompleteness theorem shows that the consistency of a theory such as ZFC cannot be proved within the theory itself. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension that allows explicit treatment of proper classes.

Did You Know?

• Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for 'choice'.
• The theory prevents its models from containing urelements (elements that are not themselves sets).
• The axiom of pairing implies that given any two sets a and b, there is a new set {a, b} containing exactly them.
• The consistency of ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.