# Zermelo-Fraenkel

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In mathematics, **Zermelo-Fraenkel set theory** (**ZFC**) is the standard formal axiomatization of axiomatic set theory. It is commonly considered the foundation of modern mathematics.^{[1]} It was formulated by two logicians, Zermelo and Fraenkel.

The nine axioms in Zermelo-Fraenkel set theory are:

- Axiom of Empty Set
- Axiom of Extensionality
- Axiom of Unordered Pairing
- Axiom of Subset
- Axiom of Superset
- Axiom of Power
- Axiom of Infinity
- Axiom of Replacing
- Axiom of Foundation
- Axiom of Choice

Many authors take "Zermelo-Fraenkel" (ZF) to refer to these axioms without the axiom of choice, indicating the inclusion of choice by writing "ZFC". The Axiom of Choice is independent of (i.e. neither provable nor disprovable from) the Zermelo-Fraenkel axioms. Sometimes alternate axioms are added to ZF, for example the Axiom of Determinacy, which is incompatible with the Axiom of Choice.