User:IssaRice/Computability and logic/Diagonalization lemma
The diagonalization lemma, also called the Godel-Carnap fixed point theorem, is a fixed point theorem in logic.
Rogers's fixed point theorem
Let be a total computable function. Then there exists an index such that .
Define (this is actually slightly wrong, but it brings out the analogy better).
Consider the function . This is partial recursive, so for some index .
Now since . This is equivalent to by definition of . Thus, we may take to complete the proof.
Let be a formula with one free variable. Then there exists a sentence such that iff .
Define to be where . In other words, given a number , the function 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 be , and let be .
Then is , by substituting in the definition of .
We also have by definition of . By definition of , this is , so we have .
To complete the proof, apply to both sides of the final equality to obtain iff ; this simplifies to iff .
|Step||Rogers's fixed point theorem||Diagonalization lemma|
|Definition of diagonal function|
|Composition of given mapping with diagonal function ()|
|Naming the composition|
|Index of composition|
|Expanding using definition of diagonal|
|Composition applied to own index (i.e. diagonalization of the composition)|
|Explicitly showing previous composition|
|Definition of G||is||is|
|Leibniz law to previous row|
|Substituting definition of G|