Gödel's Proof

This is a talk at the United States Military Academy at West Point Mathematics Seminar, January 20, 2010.
Slides

In 1931, Kurt Gödel showed that for any "reasonable" axiomatization of number theory, such as the Peano Axioms, there are always statements that are true but not provable from those axioms. Thus, there is no reasonable collection of axioms from which all truths of number theory can be proved; all such axiomatizations are incomplete. This monumental result, known as the First Incompleteness Theorem, was the first in a series of discoveries asserting the fundamental limitations of formal mathematics. At the time, the drive to formalize mathematical knowledge was led by the renowned mathematician David Hilbert who, in his famous Hilbert’s Program, strove to establish Mathematics as the science guaranteeing the full certainty of its conclusions through rigorous logical means. Together with the Second Incompleteness Theorem, Gödel’s work effectively ended any hope of carrying out Hilbert’s Program. In this talk, I will introduce the historical context of Hilbert’s Program and prove Gödel’s First Incompleteness Theorem.