Complexity and Incompleteness

The theorems of a finitely-specified, sound, consistent theory which is strong enough to include arithmetic have bounded δ-complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. More precisely, according to Theorem 4.6 in [1], for any finitely-specified, sound, consistent theory strong enough to formalize arithmetic (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) and for any Gödel numbering g of its well-formed formulae, we can compute a bound N such that no sentence x with complexity δg(x) > N can be proved in the theory; this phenomenon is independent on the choice of the Gödel numbering.