Set you have set $S$, and are able to prove that any injective function mapping $S$ to the natural numbers cannot be a computable function. Can $S$ still be countable? And if so, does anybody know of an example of a countable set with no computable counting function?
2026-03-29 11:05:58.1774782358
Can there be a countable set with no computable counting function?
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By a "counting function" I assume you mean a computable bijective function from the set to the natural numbers.
If a set has such a counting function, and that counting function is computable, then the set itself is computably enumerable, by standard techniques.
So any non-c.e. set of natural numbers will be an example of a countable set which has no computable counting function.