  We investigate how undecidability enters into computations of classical
physical systems and contributes to the increase of entropy and loss of
information. In actual computation with finite bit of information capacity we
accept inconsistency to avoid undecidability, which in turn affects entropy of
the system. We apply the Shannon entropy to the discretized Liouvillian system.
It is shown that for any finite bit of information capacity information is
always lost or the entropy always increases for the probability density
following Hamiltonian dynamics, both in time forward and time backward
direction, thus showing information theoretical version of second law of
thermodynamics. This is due to the finiteness of information capacity and
incompressibility of probability distribution in Liouville's equation.
