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We investigate how undecidability enters into computations of classicalphysical systems and contributes to the increase of entropy and loss ofinformation. In actual computation with finite bit of information capacity weaccept inconsistency to avoid undecidability, which in turn affects entropy ofthe system. We apply the Shannon entropy to the discretized Liouvillian system.It is shown that for any finite bit of information capacity information isalways lost or the entropy always increases for the probability densityfollowing Hamiltonian dynamics, both in time forward and time backwarddirection, thus showing information theoretical version of second law ofthermodynamics. This is due to the finiteness of information capacity andincompressibility of probability distribution in Liouville's equation.

Undecidability, entropy and information loss in computations of classical physical systems