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

>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. So this isn't exactly right, although I suppose the argument for integers isn't exactly wrong. When we get to the rational numbers however, they are countable but there are an infinite number of rational numbers between any two rationals in the typical way of thinking about "between" numbers. Based on your argument above, this would mean that rationals are uncountable. A better way to think about this is that the set of natural numbers is the first (and smallest) infinite set you can construct. This is the set of all counting numbers so we call it countable. We then say two sets are the same size (cardinality) if you can create a one-to-one mapping between the two that covers both sets (a bijection). This way you exactly pair one element of one set with one element of another. You can do this with the natural numbers and integers by just alternating positive and negative (so 0 -> 0, 1 -> 1, 2 -> -1, 3 -> 2, etc.). All the sets that you can do this sort of mapping with the natural numbers are considered countable. You can't do this sort of mapping from the natural numbers to the reals (see cantor's diagonalization argument) so since you can't "count" the reals, the set is "uncountable."