The standard delta-epsilon type of reasoning crucially depends on existence of arbitrary reals. Many geometry proofs / lines of reasoning crucially depend on the ability to position a point on a line at an arbitrary distance from another point.
All numbers ever written are rationals and thus countable. But those endless irrational numbers make a lot of ways of reasoning simpler, or possible at all.
All numbers ever written are rationals and thus countable. But those endless irrational numbers make a lot of ways of reasoning simpler, or possible at all.