This talk concerns a connection between fractals and an interesting tame structure. If K is a Cantor subset of the real line (compact nowhere dense perfect subset) then (R,<,+,K) defines an isomorphic copy of the monadic second order theory of the successor function. This result is sharp as the monadic second order theory of the successor defines an isomorphic copy of (R,<,+,C) where C is the classical middle-thirds Cantor set. One can also show that if X is essentially any fractal subset of Euclidean space then (R,<,+,X) defines a Cantor subset of the real line, but I probably won't have time to say much about this. Joint work with Philipp Hieronymi.