Michael Cohen, NDSU

Math Club, September 19^th.

Title:  Diagonal Arguments and Paradoxes in Mathematics

Abstract:  Many students have heard the tale of Hilbert's hotel, which has infinitely many rooms (numbered 0, 1, 2, ...) and is currently filled to capacity, but by way of clever room re-assignment may still find space for quite a large amount of additional occupants.  As vast as is the hotel, however, there is a sharp limitation to its size- this was established by Georg Cantor in 1891 with a groundbreaking new proof method called a "diagonal argument."  We'll look at how diagonal arguments have been used in the 20th century to establish facts and make arguments that seem at first glance baffling, such as the solution of the halting problem (Is there an algorithm which determines the success or failure of all algorithms?) and Richard's paradox (Can we describe in English words a number which cannot be described in English words?).