The axiom of choice has been one of the most hotly contested principles of 20th-century mathematics .
Every set can be well-ordered.
This means that the notion of size is well-defined for infinite sets. There are several other, more complicated formulations of the same principle, one of the most commonly used being known as Zorn's Lemma. Even once these definitions have been disentangled, there at first seems to be no problem. Some of the consequences of the axiom are much less desirable, being completely at odds with common sense. Besides, the axiom of choice is vastly useful.