  In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems
of a finitely-specified theory cannot be significantly more complex than the
theory itself}, for an appropriate measure of complexity. We show that the
measure is invariant under the change of the G\"odel numbering. For this
measure, the theorems of a finitely-specified, sound, consistent theory strong
enough to formalize arithmetic which is arithmetically sound (like
Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded
complexity, hence every sentence of the theory which is significantly more
complex than the theory is unprovable. Previous results showing that
incompleteness is not accidental, but ubiquitous are here reinforced in
probabilistic terms: the probability that a true sentence of length $n$ is
provable in the theory tends to zero when $n$ tends to infinity, while the
probability that a sentence of length $n$ is true is strictly positive.
