"A German mathematician active in the mid-19th century, Georg Cantor (1845-1918), realised that infinities came in different sizes; more precisely he showed that the infinite set of all integers was smaller than the infinite set of all real numbers, and gave the labels Aleph Null to the smaller, and Aleph One to the larger set."

Um, well, the problem with Cantor's work is that the concept of size doesn't apply to infinite sets. Cantor's "proof" that Aleph one is larger than Aleph null (the Diagonalization Theorem) is a constructive proof, and constructive proofs only work on finite sets. Nevertheless, the vast majority of mathematicians buy into this BS because it leads to some very useful mathematics. Much like physicists use renormalization (subtracting infinities, and worse - dividing infinity by infinity and calling it 1) because it leads to answers that agree with experiment. Odd how mathematicians criticize physicists for playing footloose and fancy free with math, but then do it themselves.

A small percentage of mathematicians (google for "ultrafinitism") say no way, Jose, it's bogus. Go ahead and do the math that results since it's useful, but don't try to justify it with specious arguments. Infinity is not a number and if you treat it as a number, it results in nonsense. Infinity isn't even a mathematical object and therefore it doesn't make sense to perform mathematical operations (addition, multiplication, etc) on it.

Whenever some "transfinite" advocate claims that the set of integers is "countable", my response is "Okay, go count them. Come back and let me know when you're done."