According to mathematicians who follow Cantor's idiocy, the set of all square numbers is the same size as the set of counting numbers. In fact they go even further and declare that the set of rational numbers is the same size too. They have a fundamental problem with their definition of the infinity symbol. By declaring that twice infinity is the SAME infinity, they can't actually compare the sizes of different infinite values. Consider some value N which we will soon increase towards infinity. N² is much bigger than N, which is bigger than sqrt(N), which is bigger than log10(N). Put N = 10^10. N² has 19 digits, N has 10 digits, log10(N)=10. These values do no get closer as we approach infinity, they diverge more. It is useful to compare different infinites and they have set themselves up to fail.
It's time we ditched this Cantor-induced insanity: http://lesliegreen.byethost3.com/articles/new_maths.pdf