Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.
But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream.
Hofstadter was too artistic, the Nagel and Newman book was too long, Wikipedia is too thorough or too simple. This presentation is just right; short enough to comprehend while including the important details.
Gödel’s Proof relies on the liar paradox, "this statement is false." Gregory Chaitin produced a similar result using the Berry paradox, demonstrated by the fifty-seven letter expression "the smallest positive integer not definable in under sixty letters."
Hagen von Eitzen expanded the predicate, giving two expressions of the Gödel sentence at the bottom of the page.