Can we use domains other than the naturals in computability theory?
The domain is finite strings of symbols from some alphabet &ndash; i.e., initial contents of Turing machine tapes. The natural numbers can be easily coded as finite strings: either in unary using length or in some higher base using symbols from the tape alphabet as digits. Other countable sets (e.g., integers and rationals) can be coded if, and only if, there is a computable bijection between that set and the naturals. However, you can't let your domain be any countable set: for example, if you want your domain to be the set of Turing machines that halt on every input, you couldn't even tell which Turing machine a given finite string codes.

There's no way to use an uncountable domain because the initial contents of the tape must be finite and there are only countably many finite strings over any finite alphabet.