The purpose of this talk will be to demonstrate the existence of a well ordered chain of length $\epsilon_0 + 1$ in the finitely generated subgroups of Richard Thompson's group $F$. This requires the development of an apparently novel way of representing the ordinal $\epsilon_0$. This is joint work with Collin Bleak and Matt Brin.

This talk will be preceded by a closely related talk in the Topology Seminar the same day (1:30pm, 207 Malott). The talks are structured so as not to require knowledge from the other talk. The topology seminar talk will, however, spend more time motivating the main result and explaining group theoretic aspects of the construction.