[ex3xu]

Thank you for the clarification. Following up on this, I found a blog post [0] that explains why you don't get bizarre consequences like the Banach-Tarski paradox this way:

> [The constructivist method] avoids the basic problem with the axiomatic problem. You can’t have something ridiculous like the set of all sets that don’t include themselves, because the only way to show that a set exists is to show a process to create it – and there’s no way to create a paradoxical set! You don’t get nonsense like [the Banach-Tarski paradox], because the division of the spherical topological space into the non-metric subspaces is impossible: you can’t show a process that produces it. In the world of intuitionism and constructivism, existence proofs that don’t produce concrete examples don’t exist!

[0]: http://www.goodmath.org/blog/2014/08/18/the-constructivist-r...