# Diagonal Lemma justification

Given the diagonal lema stated as above:

Diagonal Lema. Let $\mathfrak{T}$ be a theory wich is capable of representing the primitive recursive functions, and a codification schema for formulas in $\mathfrak{T}$ such that $\ulcorner \phi \urcorner$ is the codification of $\phi$. For all formulas $\psi(x)$ where $x$ is it's only free variable, we have $\mathfrak{T} \vdash \delta \leftrightarrow \psi(\ulcorner \delta \urcorner)$.

Why is it called after Cantor's diagonal arugment? Giving the justification why the standard diagonalization technique for demonstrating that the set of reals isn't contable is straightforward, but for this lemma it isn't.

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Are you asking why it is named "Diagonal Lemma"?

It doesn't seem to be named directly after Cantor, but I suspect Cantor's argument was an early, influential use of arguments that make use of the diagonal elements of some structure.

The term "diagonal" is now often used to refer to the mapping $i \mapsto A(i,i)$ whenever $A(i,j)$ parametrizes some natural collection of objects (or just the image of this mapping). Another analogy is that of the diagonal of a matrix. The Diagonal Lemma can be viewed as constructing an object on a certain diagonal, hence the name.

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Diagonal Lema. Let $\mathfrak{T}$ be a theory wich is capable of representing the primitive recursive functions, and a codification schema for formulas in $\mathfrak{T}$ such that $\ulcorner \phi \urcorner$ is the codification of $\phi$. For all formulas $\psi(x)$ where $x$ is it's only free variable, there is a formula $\delta$ such that $\mathfrak{T} \vdash \delta \leftrightarrow \psi(\ulcorner \delta \urcorner)$.