Let me give a more concrete example. Take the set of uncomputable real numbers. You can write out a formal definition of that set, you can prove that it is infinite and uncountable. But you can't produce any element of it. So you can't actually pick an arbitrary element of it, since you can't know what any of the elements are. Is there even any general rule you can use to pick out a single element? – you can't pick the smallest or largest element, since for every element there will be a smaller or larger one; nor can you pick the element closest to zero, since for every element there will be one closer to zero, and zero itself is not an element.
I suppose, if one had a Turing machine with an oracle for the halting problem, it could produce some elements of this set. And then you could ask, what is the shortest program which generates an element of this set. And then if you had multiple shortest programs, you could order them lexicographically. So that's a way you could pick out a single element in principle, even though it is impossible in practice.
Another example would be the set of numbers that are first-order undefinable. (Meaning, numbers for which there does not exist any sentence in first-order logic which is true only for that number). There is no way using first-order logic to pick out a unique element of that set. But I suppose there is some number which is second-order definable but not first-order definable, and hence we could use a sentence of second order logic to uniquely locate a member of the set of first-order undefinable numbers.
As a generalisation – you can define a set such that you can't pick out any individual member of it using certain resources. But it seems like there is always some way to pick out a unique member of such a set using more resources (a higher Turing degree, higher order logic, etc). Can you define a set so you can't pick out an individual member of it no matter how much resources you use? I think the answer to that is "no".
I suppose, if one had a Turing machine with an oracle for the halting problem, it could produce some elements of this set. And then you could ask, what is the shortest program which generates an element of this set. And then if you had multiple shortest programs, you could order them lexicographically. So that's a way you could pick out a single element in principle, even though it is impossible in practice.
Another example would be the set of numbers that are first-order undefinable. (Meaning, numbers for which there does not exist any sentence in first-order logic which is true only for that number). There is no way using first-order logic to pick out a unique element of that set. But I suppose there is some number which is second-order definable but not first-order definable, and hence we could use a sentence of second order logic to uniquely locate a member of the set of first-order undefinable numbers.
As a generalisation – you can define a set such that you can't pick out any individual member of it using certain resources. But it seems like there is always some way to pick out a unique member of such a set using more resources (a higher Turing degree, higher order logic, etc). Can you define a set so you can't pick out an individual member of it no matter how much resources you use? I think the answer to that is "no".