 Proof. Assume the existence of a consistent formal theory \(S\) extending first-order arithmetic and containing schema \(T\). We need to show that this assumption leads to a contradiction. The proof mimics the liar paradox. Apply the diagonal lemma to obtain a sentence \(\lambda\) satisfying \(\lambda \leftrightarrow \neg T \langle \lambda \rangle\) in S. The sentence \(\lambda\) expresses of itself that it is not true, so \(\lambda\) corresponds to the liar sentence. Instantiating schema \(T\) with the sentence \(\lambda\) gives us \(\lambda \leftrightarrow T\langle \lambda \rangle\). We now have that both \(\lambda \leftrightarrow \neg T\langle \lambda \rangle\) and \(\lambda \leftrightarrow T\langle \lambda \rangle\) hold in \(S\) (are provable in \(S)\), and thus \(T\langle \lambda \rangle \leftrightarrow \neg T\langle \lambda \rangle\) must hold in \(S\). This contradicts \(S\) being consistent. \(\Box\)