Most paradoxes can be "resolved" by just saying "Don't do that." Problem is, there is a lot of mathematics that's widely used and which depends on the reals.

Most paradoxes can be "resolved" by just saying "Don't do that." Problem is, there is a lot of mathematics that's widely used and which depends on the reals. Pretty much all of calculus, for example. Discarding the reals is pretty ambitious.