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Full Idea
We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way.
Gist of Idea
Two functions are the same if they have the same extension
Source
Peter Smith (Intro to Gödel's Theorems [2007], 11.3)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.87
A Reaction
So there's only one way to skin a cat in mathematical logic.