  We present the concept of the \emph{information efficiency of functions} as a
technique to understand the interaction between information and computation.
Based on these results we identify a new class of objects that we call
\emph{Semi-Countable Sets}. As the name suggests these sets form a separate
class of objects between countable and uncountable sets. In principle these
objects are countable, but the information in the descriptions of the elements
of the class grows faster than the information in the natural numbers that
index them. Any characterization of the class in terms of natural numbers is
fundamentally incomplete. Semi-countable sets define one-to-one injections into
the set of natural numbers that can be computed in exponential time, but not in
polynomial time. A characteristic semi-countable object is $\phi_{\Sigma}$ the
set of all additions for all finite sets of natural numbers. The class
$\phi_{\Sigma}$ codes the Subset Sum problem. This gives a natural and
transparant analysis of the separation between the classes $P$ and $NP$.
