$\epsilon_0$ is the second infinite ordinal which is closed under the operations of ordinal arithmetic. It plays an important role in understanding the consistency of Peano Arithmetic. This talk will give an introduction to ordinal arithmetic and explain how it can be used to measure the logical strength of theories such as Peano Arithmetic. I'll illustrate this with Goodstein's Theorem, the Finite Kruskal Theorem, and mention how this ordinal appeared recently in studying the subgroups of Richard Thompson's group $F$.