New here. I'm wanting to know whether or not it's theoretically possible to construct a function which can map the set of all real numbers to the set of all computable functions, and whether or not this supposed function is even computable to begin with.
2026-03-28 11:15:17.1774696517
Mapping real numbers to functions
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The answer is indeed "no", but that's not because there are more computable functions than real numbers, but because there are more real numbers than computable functions. Citing Wikipedia: "Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions". If there are countably many computable functions, there is a bijection from natural numbers to that set. And there are uncountably many real numbers, so there cannot exist a bijection from real numbers to computable functions.