Zermelo-Fraenkel
This is an old revision of this page, as edited by MarkGall (talk | contribs) at 20:26, September 30, 2009. It may differ significantly from current revision.
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". Mathematicians who find the Axiom of Choice questionable often replace it with the weaker Axiom of Determinacy.