[librexpr]

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...

[actually_a_dog]

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.