14 Sep
2023
14 Sep
'23
5:53 a.m.
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 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 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]].)