There are uncomputable numbers which can still be defined.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

See also this Numberphile video on numbers, including algebraic, computable, transcendental, and talking about numbers beyond those.

https://m.youtube.com/watch?v=5TkIe60y2GI

That's the point though, right? Without Choice, there need be no way to select a number or subset from the set, so there's no paradox.

It's not too mind-boggling: we can describe the set as a whole without an issues, we just can't uniquely describe any particular number in the set. That doesn't stop us from generalizing over all numbers in the set.

> 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.

I think the resolution is along the lines of this:

* ZFC does not have a unique model that satisfies the axioms.

* Each of those models have different "undescribable numbers" with different properties.

But it was a long time I touched model theory, and it was just an optional extra course at uni.

https://en.wikipedia.org/wiki/Model_theory

To me, the expression "The set of undescribable numbers" is meaningless. While discussing plausible-sounding, but meaningless, linguistic constructs like this, we descend to the intellectual level of chatGPT. My conjecture is that one can design a prompt to chatGPT such that the output would become a decent PhD thesis with no internal contradictions, but a complete BS otherwise.

I think if we take "description of a number" to mean "ZF formula that uniquely picks out that number", then that cannot be defined, because a formula picks out a number when it is true for that number and false for all others, but by Tarski[0], the truth predicate cannot be defined inside the logic itself. So "the set of all numbers which cannot be described" cannot be talked about using ZF.

However, there is a way around it, by taking as axiom that ZF is consistent, choosing some model M of ZF, and then talking about the set S of numbers inside M that cannot be described.

[0] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...

Bingo. The issue here is that ZF and ZFC are both first order theories. They can only talk about things that can be defined within them, not things that cannot be defined within them. The wiki article talks about that where it says:

> Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It is possible to define a formula True(n){\displaystyle True(n)} whose extension is T∗,{\displaystyle T^{*},} but only by drawing on a metalanguage whose expressive power goes beyond that of L.L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language L.L. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on.