  A basic problem in smooth dynamics is determining if a system can be
distinguished from its inverse, i.e., whether a smooth diffeomorphism $T$ is
isomorphic to $T^{-1}$. We show that this problem is sufficiently general that
asking it for particular choices of $T$ is equivalent to the validity of
well-known number theoretic conjectures including the Riemann Hypothesis and
Goldbach's conjecture. Further one can produce computable diffeomorphisms $T$
such that the question of whether $T$ is isomorphic to $T^{-1}$ is independent
of ZFC.
