in the following manner: "given any effective procedure to transform programs, there is always a program that, when modified by the procedure, does exactly what it did before", or: "it's impossible to write a program that changes the extensional behaviour of all programs".

Proof of the fixed-point theorem

The proof uses a particular total computable function , defined as follows. Given a natural number , the function outputs the index of the partial computable function that performs the following computation:
:Given an

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