Every finite integer is computable. We often represent non-integer numbers.
> your encodings will, unlike those in the lambda calculus, be completely arbitrary
Well, they /may/ be completely arbitrary. They may not be. The key is to choose encodings that are useful for the problem domain. Admittedly if the problem domain is "winning at the schoolyard game of saying a bigger number," those encodings may look arbitrary for most other purposes.
But there's actually an intent to my original comment besides being pedantic. The most general way to think of a 64-bit numeric representation is as a list of 2^64 numbers, feeding into a 2^64:1 mux, with the 64-bit string being the select bits of the mux. (Or, equivalently, a 2^64 entry x arbitrary width ROM with one number per entry, with the 64-bit string being the address input of the ROM. Same thing.) The two questions you must answer, then, are (a) which 2^64 numbers are most useful in your problem domain; and (b) are there hardware optimizations to reduce the (ridiculous) scale of the mux/ROM model that are so valuable that you're willing to make compromises on which numbers you select?
Every finite integer is computable. We often represent non-integer numbers.
> your encodings will, unlike those in the lambda calculus, be completely arbitrary
Well, they /may/ be completely arbitrary. They may not be. The key is to choose encodings that are useful for the problem domain. Admittedly if the problem domain is "winning at the schoolyard game of saying a bigger number," those encodings may look arbitrary for most other purposes.
But there's actually an intent to my original comment besides being pedantic. The most general way to think of a 64-bit numeric representation is as a list of 2^64 numbers, feeding into a 2^64:1 mux, with the 64-bit string being the select bits of the mux. (Or, equivalently, a 2^64 entry x arbitrary width ROM with one number per entry, with the 64-bit string being the address input of the ROM. Same thing.) The two questions you must answer, then, are (a) which 2^64 numbers are most useful in your problem domain; and (b) are there hardware optimizations to reduce the (ridiculous) scale of the mux/ROM model that are so valuable that you're willing to make compromises on which numbers you select?