Axiomatic number theory and Gödel's incompleteness theorems
Speaker: Eric Ottman (Syracuse)
Friday October 6, 2017, 1:00, Carnegie 109
Abstract: At the start of the 20th century, various mathematicians attempted to remove all paradoxes from mathematics by creating systems of "axiomatic" set theory and number theory, consisting (vaguely speaking) of a set of axioms and a strict set of rules on how these axioms may be manipulated to produce theorems. However, in 1931, Kurt Gödel proved that any sufficiently powerful such axiomatic system must be either incomplete or inconsistent - in other words, in any such system there must exist either a provable false statement or an unprovable true statement. In this talk, I will attempt to give a more precise description of what I mean by "axiomatic systems," and then to summarize the (constructive) proof of Gödel's result. It's a little more fun than it sounds and should be accessible to all graduate students.