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.



why do the Computability theory choose the natural number as the object of study?
The natural numbers are often used as a background for (ordinary) computability theory because they are a simple and widely known system of finitary objects. Alternative possibilities would be finite strings of symbols from a fixed finite alphabet (which were, if I remember correctly, used as the background in Shoenfield's book "Recursion Theory") or hereditarily finite sets. The particular choice of background makes little difference, as any reasonable finitary background can be computably coded in any other.


