The set of integers is a countable infinite set. That is, the integers can be arrange in ordinal number (first, second, third, ...) order. For example {0, 1, -1, 2, -2, 3 , -3, 4, ...}.
The real numbers cannot be so arranged. In set theory this means that the set of reals has a higher cardinal number than the set of integers. In plain language, this means that the set of integers is smaller than the set of real numbers.
Of course, saying that the real numbers cannot be so arranged means that one would have to prove a negative - never an easy thing to do.
There is a large mathematical literature on this point. There is no way to do it justice here.
