The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics. It is now a basic assumption used in many parts of mathematics. In fact, assuming AC is equivalent to assuming any of these principles (and many others):

• Given any two sets, one set has cardinality less than or equal to that of the other set -- i.e., one set is in one-to-one correspondence with some subset of the other. (Historical remark: It was questions like this that led to Zermelo's formulation of AC.)
• Any vector space over a field F has a basis -- i.e., a maximal linearly independent subset -- over that field. (Remark: If we only consider the case where F is the real line, we obtain a slightly weaker statement; it is not yet known whether this statement is also equivalent to AC.)
• Any product of compact topological spaces is compact. (This is now known as Tychonoff's Theorem, though Tychonoff himself only had in mind a much more specialized result that is not equivalent to the Axiom of Choice.)

¶ 12:04 AM