This is an interesting question that gets to the heart of the constructivism debate, which I won't touch on here. As for the integer example, the integers are

This is an interesting question that gets to the heart of the constructivism debate, which I won't touch on here. As for the integer example, the integers are all easily (finitely) constructed (successors to 0 and their additive opposites), so it isn't a consistent concept to say "This is an unconstructable integer." Sets, on the other hand, are not necessarily always constructable, unless you only take the constructivist axioms.