> I was taught there is countably and uncountably infinite. Integers are countably infinite because the number of integers between any two numbers if finite. Real numbers are uncountably infinite because there are infinite numbers between any two real numbers.

This is unsound: a set is countable if there's a mapping that assigns an integer to each element of that set. There are an infinite number of rational numbers between any pair of rational numbers, but you can pretty easily construct a mapping between integers and rationals. Imagine representing a rational x/y as a point on the plane, and draw a square spiral around the origin. Every rational number lays on that spiral, and there's a unique "shortest arc-length of the spiral from the origin" for each rational.