why do the Computability theory choose the natural number as the object of study？

Question

I am wondering why the computable function is defined in the natural number set.Can people give me the answer or some resources that can solve my puzzle.

Answer 1

There's a few points to make, here:

First, the historical. The first real use of the idea of computable functions to prove a theorem was Goedel's use of them in his incompleteness theorem - that appropriate (primitive recursive) functions are representable. These functions, essentially, were functions of natural numbers, and so historically the first real context for computability is in the naturals.

Second, the applied. Any actual computing machine acts on finite strings of symbols in a finite alphabet - and we can biject these with the naturals in an intuitively-computable way. So we're "morally" still studying computability on the naturals.

Finally, the philosophical. A major factor in the mathematical significance of computability is the Church-Turing thesis; and, as a consequence, the fact that different models of computation yield the same notion of computable function. This fails wildly when we try to generalize to computability in broader contexts, e.g. computability of arbitrary sets, or of real numbers, or . . .

This is not to say that such "generalized computability theory" (usually called "generalized recursion theory," since most of the research on it predates the recursive/computable terminology change) isn't interesting; indeed, it's one of the things I'm most interested in! But computability on the naturals has a special role.