Table of Contents - 3. ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
In this section we add the Axiom of Choice ax-ac, as well as weaker forms
such as the axiom of countable choice ax-cc and dependent choice ax-dc.
We introduce these weaker forms so that theorems that do not need the full
power of the axiom of choice, but need more than simple ZF, can use these
intermediate axioms instead.
The combination of the Zermelo-Fraenkel axioms and the axiom of choice is
often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is
the most commonly-accepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies about the
Axiom of Choice. The axiom of choice does not satisfy those who wish to have
a constructive proof (e.g., it will not satisfy intuitionistic logic). Thus,
we make it easy to identify which proofs depend on the axiom of choice or its
weaker forms.