cousin_it is right about the undefinability of definability in ZFC.
Even the undefinable reals are taken to be in the “domain of discourse”. They remain “accessible” in some infinitary sense (e.g. corresponding to some infinite bit string expansion) even if they aren’t finitely definable.
Also, the predicate of being a real number in ZFC is definable. So the real numbers are definable in aggregate.
The physics case is about references like “the objective world” which are singular and aren’t any kind of infinitely long name.
In the case of the reals, a skeptical perspective will note that there exists a countable model of ZFC (by Löwenheim-Skolem), hence reals “could be” a countable set from the perspective of ZFC. Our formal systems may, thus, lack the capacity of referencing truly uncountable sets.
There are a few things to say about this:
cousin_it is right about the undefinability of definability in ZFC.
Even the undefinable reals are taken to be in the “domain of discourse”. They remain “accessible” in some infinitary sense (e.g. corresponding to some infinite bit string expansion) even if they aren’t finitely definable.
Also, the predicate of being a real number in ZFC is definable. So the real numbers are definable in aggregate.
The physics case is about references like “the objective world” which are singular and aren’t any kind of infinitely long name.
In the case of the reals, a skeptical perspective will note that there exists a countable model of ZFC (by Löwenheim-Skolem), hence reals “could be” a countable set from the perspective of ZFC. Our formal systems may, thus, lack the capacity of referencing truly uncountable sets.