Ony of my favorite topics in mathematics is infinity and it’s diverse forms. A nice result is that the Continuum Hypothesis (the hypothesis that there is no infinity between aleph_zero of the integers and aleph_1 of the reals) is independent of the currently accepted/used Axioms of set theory (ZFC) – which form the basics of mathematics, together with logic.
Independent means that CH can neither be proved true nor false within the ZFC axioms.
Here is a nice blog post explicating a little more:
EvolutionBlog : Paul Cohen Dead at 72
http://scienceblogs.com/mt/pings/37198
