User:IssaRice/Computability and logic/Diagonalization lemma

Rogers's fixed point theorem

Let $f$ be a total computable function. Then there exists an index $e$ such that ${\phi }_e\simeq {\phi }_{f(e)}$.

Diagonalization lemma

(semantic version)

Let $A$ be a formula with one free variable. Then there exists a sentence $G$ such that $G$ iff $A(⌜G⌝)$.

Define $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(x)$ to be $⌜C(⌜C⌝)⌝$ where $x=⌜C⌝$. In other words, given a number $x$, the function $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}$ finds the formula with that Godel number, then diagonalizes it (i.e. substitutes the Godel number of the formula into the formula itself), then returns the Godel number of the resulting sentence.

Let $B$ be $A(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(x))$, and let $G$ be $B(⌜B⌝)$.