TITLE: Gödel's Incompleteness Theorem and the complexity of arithmetic
QUESTION [13 upvotes]: In How complicated can structures be? Jouko Väänänen says:

The guiding result of mathematical logic is the Incompleteness Theorem of Gödel,
  which says that the logical structure of number theory is so complicated that it cannot be
  effectively axiomatized in its entirety. In other
  words, the theory is non-recursive, i.e. there
  is no Turing machine that could tell whether
  a sentence of number theory is true or not.

I've never seen Gödel's Incompleteness Theorem this way: that it's a matter of the overall complexity of the structure of the natural numbers that there are facts about them that cannot be proved. 
So I wonder whether I can take the quote above literally:

Can Gödel's Theorem be rigorously stated in terms
  of complexity? 

Somehow like this: "Every
system which exceeds complexity
threshold X is undecidable."
Or is it just a vague paraphrase, not to be taken too seriously?

REPLY [2 votes]: Hmm, nobody has mentioned Chaitin's incompleteness theorem.