"Computers, Paradoxes and the Foundations of Mathematics," by Gregory J. Chaitin. American Scientist, March-April 2002, pages 164-171.
Chaitin's article is excerpted from a 1999 lecture that he delivered at the University of Massachusetts, Lowell. He traces the history of formalism and incompleteness, concluding with randomness and algorithmic information theory. A nice feature of the article is illustrations of Bertrand Russell, David Hilbert, Kurt Gödel and Alan Turing, which accompany descriptions of their contributions to the subject.