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.
True(n): degodel(n) is true
We 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
S(m): degodel(m) == phi(x), phi(m) == false S(m) is true only if m is a statement that is false when passed itself maybe challenge is, define phi using True() phi(m) = (m != phi) True(x): x is a true statement maybe i'll look for descriptions of this proof eslewhere
number, yields a false statement). If we now consider the Gödel number g of the 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]].)