Zermelo-Fraenkel Set Theory (ZF) is the standard form of axiomatic set theory, providing a rigorous foundation for much of modern mathematics. It addresses paradoxes in naive set theory by using a specific set of axioms to define what sets are and how they behave, including the Axiom of Choice in its extended form, ZFC.

Zermelo-Fraenkel Set Theory

Axiomatic set theory is a branch of mathematical logic that uses a formal system to define sets and their relationships, providing a foundation for much of modern mathematics. It addresses paradoxes and inconsistencies in naive set theory by introducing axioms that precisely dictate how sets can be constructed and manipulated.

axiomatic set theory

modern mathematics

paradoxes in naive set theory

specific set of axioms

what sets are

How sets behave

The Axiom of Choice is a foundational principle in set theory that asserts the ability to select a member from each set in a collection of non-empty sets, even when no explicit rule for selection is given. It is essential for many mathematical proofs but is independent of the standard Zermelo-Fraenkel set theory, meaning it can neither be proven nor disproven from the other axioms of set theory.

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