> we can describe the set as a whole without any issues
But it does not mean the set is well defined. The description is not necessarily clear an unambigious. It might not be consistent under the generalizations we come up with.
> "the set of all undescribable primes"
Interesting example, because the set is trivially empty under the assumption that we can describe any finite natural number by writing down its decimal expansion. (Based on that you can either argue that the primes are a subset of the natural numbers or that the primes are countable.)
I see no reason to believe that "undescribable" is a well defined property so far.
> But it does not mean the set is well defined. The description is not necessarily clear an unambigious. It might not be consistent under the generalizations we come up with.
Well, obviously such a set must be defined relative to whatever system you're using to describe the numbers. But once you have it, you can simply declare that a number is in your set if and only if there does not exist a finite description in your system of choice. (In other words, you enumerate the countable many finite descriptions in your system, put them in the set, then define the new set as the complement.) How is that ambiguous?
> Interesting example, because the set is trivially empty under the assumption that we can describe any finite natural number by writing down its decimal expansion. (Based on that you can either argue that the primes are a subset of the natural numbers or that the primes are countable.)
Whoops, you're right, all integers are describable. Perhaps I should amend that to "the set of all undescribable real numbers with a prime integer part", or something along those lines.
> you can simply declare that a number is in your set if and only if there does not exist a finite description in your system of choice.
The problem I see is that we do not have such a rigurous system and instead can introduce new notation, and how we decide on the notation may change the contents of the set of undescribable numbers. If there is no ambiguity, because we are using a very limited description language that can not be extended, we may be too restrictive in the language so the set becomes meaningless, e.g., because we could describe a number within it outside of the system which represents a trivial choice function.
Well, typically, I'd say, "A proposition P(x) of first order logic describes a particular number iff, for all numbers x and y such that P(x) and P(y) are true, x = y." Then, your system of description is exactly as powerful as your system of logical axioms. But I think you're correct in that you can construct a number where it's undecidable whether it has such a description. That doesn't mean that the set of undescribable numbers is ill-defined (relative to your axiom system): it just means that you can't always answer whether it contains that number.
But it does not mean the set is well defined. The description is not necessarily clear an unambigious. It might not be consistent under the generalizations we come up with.
> "the set of all undescribable primes"
Interesting example, because the set is trivially empty under the assumption that we can describe any finite natural number by writing down its decimal expansion. (Based on that you can either argue that the primes are a subset of the natural numbers or that the primes are countable.)
I see no reason to believe that "undescribable" is a well defined property so far.