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.
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.