  We reconsider some classical natural semantics of integers (namely iterators
of functions, cardinals of sets, index of equivalence relations), in the
perspective of Kolmogorov complexity. To each such semantics one can attach a
simple representation of integers that we suitably effectivize in order to
develop an associated Kolmogorov theory. Such effectivizations are particular
instances of a general notion of "self-enumerated system" that we introduce in
this paper. Our main result asserts that, with such effectivizations,
Kolmogorov theory allows to quantitatively distinguish the underlying
semantics. We characterize the families obtained by such effectivizations and
prove that the associated Kolmogorov complexities constitute a hierarchy which
coincides with that of Kolmogorov complexities defined via jump oracles and/or
infinite computations. This contrasts with the well-known fact that usual
Kolmogorov complexity does not depend (up to a constant) on the chosen
arithmetic representation of integers, let it be in any base unary, binary et
so on. Also, in a conceptual point of view, our result can be seen as a mean to
measure the degree of abstraction of these diverse semantics.
