Consider the following four theorems of pure mathematics. The Hahn-Banach Theorem of Analysis: If F is a bounded linear functional
defined on a subspace M of a Banach space B, there is an extension of F to a linear
functional G on B such that II Git = llFll .
The Nielsen-Schreier Theorem of Group Theory: If G is a free group and H
is a subgroup of G, then H is a free group.
The Tychonoff Product Theorem of General Topology! The product of any
family of compact topological spaces is compact.
The Zermelo Well-Ordering Theorem of Set Theory! Every set can be well-
ordered.
The above theorems have two things in common. Firstly they are all
fundamental results in contemporary mathematics. Secondly, none of them can be proved
without the aid of some powerful set theoretical assumption: in this case the Axiom
of Choice.