Title: Finite and infinite
Speaker: David Farris, NCBS
Date: Friday, 1st March 2019
Time: 4 pm (Tea/Coffee at 3:45pm)
Venue: Simons Centre at NCBS, Ground Floor
Abstract:
Some sets are infinite, and some are more infinite than others. I'll first discuss a way to compare the size of infinite sets introduced by Georg Cantor. He showed in 1874 that the number of integers and number of real numbers are fundamentally distinct kinds of "infinities". I'll explain Cantor's argument, first by direct analysis, and then by a breathtakingly short and simple abstract argument that goes further to show there is an infinite number of different kinds of bigger and bigger infinities, which form a new kind of number system. With the exact same argument, I'll show that a "typical" real number cannot be computed, even theoretically!
The same arguments also give us Russell's Paradox, which shook the foundations of mathematics a century ago, and is also remarkable for being the most important theorem proved by a Bollywood actor: https://www.youtube.com/watch?v=BilHsf4FcG8.
For a preview, see this popular article about recent new work on size of different infinities: https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/