Diagonal Lemma - Statement of The Lemma

Statement of The Lemma

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let ψ be a formula in the theory T with one free variable. The diagonal lemma states that there is a sentence φ such that φ ↔ ψ(#(φ)) is provable in T.

Intuitively, φ is a self-referential sentence saying that φ has the property ψ. The sentence φ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence ψ(#(θ)). The sentence φ constructed in the proof is not literally the same as ψ(#(φ)), but is provably equivalent to it in the theory T.