Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one mor

Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one more than any member of that set. Since the set is a constructive set, if you have anything you claim to be an integer, you must be able to either prove that it is 0 or one more than an integer. The number you named isn't zero, so presumably it must be more than some integer ("one less than the number you named"). But if I then ask you whether that number is one more than an integer, and then the same about that one, and so on, at some point you will be unable to answer me. Since you can't provide the evidence that qualifies something as an integer, there's no reason for me to think it is an integer.