This is a fun talk I will give at Temple University for the mathematics undergraduates in the Senior Problem Solving forum. We’ll be exploring some of the best puzzles and paradoxes I know of that arise with large numbers and infinity. Many of these paradoxes are connected with deep issues surrounding the nature of mathematical truth, and my intention is to convey some of that depth, while still being accessible and entertaining.
Abstract: Are there some real numbers that in principle cannot be described? What is the largest natural number that can be written or described in ordinary type on a 3×5 index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every true statement provable? Does every mathematical problem ultimately reduce to a computational procedure? Is every sentence either true or false or neither true nor false? Can one complete a task involving infinitely many steps? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics.
This is a talk I shall give for the Mathematics Colloquium at Temple University, April 23, 2012.
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was successively generalized to larger and larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower. Building on this, I proved (1997) that every group has a terminating automorphism tower. After giving an account of this theorem, I will give an overview of work with Simon Thomas and newer work with Gunter Fuchs and work of Philipp Lücke, which reveal a set-theoretic essence for the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.
slides | list of my articles on automorphism towers | announcement poster