Title: The Good, the Bad and the Undecidable – An Excursion into Computability Theory-classes
Abstract: In 1900, David Hilbert put forth problems that motivate mathematics research to this day. Several of these problems dealt with finding an effective procedure to prove the existence of certain mathematical objects. This opened up many new questions as mathematicians debated exactly what constituted an effective procedure.
Later in the early 1920s, Hilbert put forward a new proposal for the foundations of classical mathematics which has come to be know as Hilbert’s Program. He called for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization is consistent. Unfortunately, in 1931 with his famous Undecidability and Incompleteness theorems, Kurt Gödel demonstrated that Hilbert’s program was doomed to failure.
We discuss how from these beginnings, Computability Theory has emerged as an independent subfield of mathematical logic and review some of the problems that are of interest to Computability Theorists today. We also examine some of the speaker’s current work in this field.
This talk will be accessible and interesting to both undergraduates and faculty alike.