> But the mathematical question is about the existence of a mapping function, and it seems that whether or not the numbers in each set are finitely describable is irrelevant to this question:
If it's possible to construct the set at all, then there is by definition no function that can be described in a finite number of symbols that can select an element from the set (since if a finitely describable function can define a member of the set, it shouldn't be in the set - you'd define the element as being the result of the function).
I don't know what the scholarly view is on relying on functions that cannot be defined in any human language, including mathematics. I would intuitively lean towards not allowing them in proofs, but perhaps they're useful mathematically?
Another interesting aspect - these are numbers, so presumably there is a smallest one, but identifying that is definitionally ruled out too, since 'smallest element in the set of numbers indescribable in a finite sequence of symbols', is itself a finite sequence of symbols.
Consider the interval (0, 1). This set of real numbers contains no smallest element. It is not true that all sets of real numbers have or should have a least element. It is a property of the standard model of the nonzero integers that all subsets have a least element.
It should be noted that there are models of ZFC in which each real number is definable. You may be interested in reading the top answer on this mathoverflow.net question:
If it's possible to construct the set at all, then there is by definition no function that can be described in a finite number of symbols that can select an element from the set (since if a finitely describable function can define a member of the set, it shouldn't be in the set - you'd define the element as being the result of the function).
I don't know what the scholarly view is on relying on functions that cannot be defined in any human language, including mathematics. I would intuitively lean towards not allowing them in proofs, but perhaps they're useful mathematically?
Another interesting aspect - these are numbers, so presumably there is a smallest one, but identifying that is definitionally ruled out too, since 'smallest element in the set of numbers indescribable in a finite sequence of symbols', is itself a finite sequence of symbols.