On the Proof of Gödel's Incompleteness Theorems

Abstract: For millennia, but especially in the last century, mathematicians have had the hope of finding an axiomatic system that formalized areas like geometry and number theory. But can we really produce one that is both consistent and complete? Can mathematics truly be reduced to a finite set of axioms and inference rules from which we can deduce all the great theorems of history and all the ones to come? In 1931, a young German mathematician from the University of Vienna publishes “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, where he single-handedly revolutionized the field of mathematical logic by answering this very question; Kurt Gödel, among other things, proves that for an axiomatic system mirroring number theory to be consistent, it must be incomplete, meaning that it will contain true statements that we can’t prove to be true. Many mathematics enthusiasts are aware of this infamous and surprising result, but due to the great technical difficulties lying within Gödel’s highly specialized field, the proof in the original paper remains to be considerably inaccessible both to the average reader, and even to the non-specialized professional mathematician. This seminar thus aims to present the general idea of the proof of Gödel’s Incompleteness Theorems and its main repercussions for the field of mathematics, following the exposition of the book “Gödel’s Proof” by Ernest Nagel and James Newman.

Pre-requisites: None.