This article is an expanded version of a talk given at the International Symposium Celebrating the 100th Birthday of Kurt Gödel (Vienna, 2006). It seeks to trace the path which led this preeminent mathematical logician to discover one of the famous results of General Relativity, the rotating Gödel Universe. This universe has some remarkable properties, which gave the philosophers plenty to worry about. It allows a person to travel into his own past, with all the ensuing causal paradoxes; it allows no unique temporal ordering of events; and though Gödel's Universe is rigid and infinite, the Foucault pendulum planes everywhere in it rotate in unison, a clear affront to adherents of Mach’s Principle. We also discuss some lesser known precursors in the field, who just missed discovering Gödel’s universe. While the article gives all the necessary derivations in simplified form (for example, of the metric and its geodesics), much of it should be accessible to the general reader, who can simply skip most of the mathematics. [Reprinted, with permission, from Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, edited by Matthias Baaz, Christos H. Papadimitriou, Dana S. Scott, Hilary Putnam, and Charles L. Harper, Jr. (Cambridge U. P., New York, 2009).]

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