Gödel’s Incompleteness Theorems

Gödel’s first incompleteness theorem holds a special place in the history of mathematics. In a popular list of the “top 100” theorems of all time, it ranks 6 \({ }^{th}\) , following the irrationality of \(\sqrt {2}\) , Gauss’s fundamental theorem of algebra, the countability of the rationals, the Pythagorean theorem, and the prime number theorem. And for good reason: for those expecting a foundation of mathematics on a defined collection of axioms from which all theorems are derived, the result must have been as startling as the discovery of \(\sqrt {2}\) by the Pythagoreans, who believed that all numbers are rational.