The "number" of real numbers in $[0,1]$ is uncountably infinite, just as the "number" of real numbers in $[0,10]$ is uncountably infinite. However, my intuition would tell me the second interval has many more real numbers in it than the first interval. And what about the "number" of real numbers in $(-\infty, \infty)$? Are there the "same number" of real numbers in all those intervals?