To prove the theorem, we proceed by contradiction and assume that an L-formula True(n) exists which is true for the natural number n in N if and only if n is the Gödel number of a sentence in L that is true in N. We
We assume True(n) reports the truth of a godel-encoded sentence.
could then use True(n) to define a new L-formula S(m) which is true for the natural number m if and only if m is the Gödel number of a formula phi(x) (with a free variable x) such that phi(m) is false when interpreted in N (i.e. the formula phi(x), when applied to its own Gödel number, yields a false statement). If we now consider the Gödel number g of the
We reuse godel's and turing's contradictions, and construct phi(x) and S(m) such that phi(phi) = false
formula S(m), and ask whether the sentence S(g) is true in N, we obtain a contradiction. (This is known as a [[Diagonal lemma|diagonal argument]].)
True(phi) .... I'm confused here, but I'm noting a flaw in Godel's counterargument -- this disproof of locally defined truth assumes the logic of Godel's diagonal argument that uses self-referential statements is valid. If this logic is not valid, then it could be possible to define truth within a given system. I want to understand better. It's notable that although all this is done in natural number arithmetic, this may not limit the bounds of the logic systems used if it can be described within and about natural number arithmetic.