4 ms·
> 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 t
by ketzu 4y ago
> 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.
- LegionMammal978 4y agoWell, 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.