We consider the computability of entropy and information in classical
Hamiltonian systems. We define the information part and total information
capacity part of entropy in classical Hamiltonian systems using relative
information under a computable discrete partition.
Using a recursively enumerable nonrecursive set it is shown that even though
the initial probability distribution, entropy, Hamiltonian and its partial
derivatives are computable under a computable partition, the time evolution of
its information capacity under the original partition can grow faster than any
recursive function. This implies that even though the probability measure and
information are conserved in classical Hamiltonian time evolution we might not
actually compute the information with respect to the original computable
partition
