"This statement is false." Is that statement true then? In 1931, Kurt Gödel published his first Incompleteness Theorem showing that no logical system can be entirely proven. The Incompleteness Theorems were (and still are) mind grenades in the world of mathematics and philosophy. In the video above, British mathematician and author Marcus du Sautoy does a wonderful job presenting the paradox for a lay audience.
And for more on this and other related head-spinning topics, there's always the classic 1979 tome Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter.