Hal, Chaitin just says "you cannot prove 20 pound theorems with 10 pound
axioms". But the infinite cascade of all provable theorems of number
theory collectively does not convey more information than the axioms.
> But what if you declare theorems which say the same thing to be the
> _same theorem_, rewritten?
Since a theorem is a symbol string, different strings cannot be the same
theorem. Perhaps you mean you can easily derive certain theorems from
others stating "the same thing, rewritten". But this is vague - you can
derive all theorems from the axioms stating "the same thing, rewritten".