Those are very conventional mathematics, and most core concepts have been settled for a century now. Cantor in particular paved the way toward a better understa

Those are very conventional mathematics, and most core concepts have been settled for a century now. Cantor in particular paved the way toward a better understanding of many related concepts in that area. As for the significance of those differences: I can think about two areas where this is fundamental (but there are many more). First, the natural numbers (0, 1, ...) vs rationals vs real numbers is really big in terms of "how many" numbers you have. You can prove than the set of natural numbers and the set of rational numbers is roughy the same size (they are both countable). Now, as you may know, there are some irrational numbers, like square root (2), pi, e, etc... But how many ? The answer is "many more than rationals". One way to define the real numbers (rationals + irrationals) is as limits of a set of rational numbers - i.e. for any real number, you can find a set of rationals which get arbitrarily close to an irrational (we say that rational numbers are dense for the set of real numbers). Another field where you see the difference: topology vs probability. Roughly speaking, probability is a function which for a set gives you a number between 0 and 1. The key property of probability is that P(A U B U ....) = P(A) + P(B) + ... where A, B, ... are disjoint sets. Another way to look at it is that probability is a special case of the notion of set measure, which is the mathematical way of talking about volume, length, etc... There is a key restriction when defining measure, that is you cannot take arbitrary unions of sets, like say as many as real numbers - actually, you can prove that you cannot build measures if you want arbitrary (uncoutable) unions. This is a naive way to describe the Banach-Tarski paradox, which says that you can split a ball into two new balls which are identitical to the original one.