Cardinal numbers

Kobe

If there is a one-one relation between the elements in two sets, the sets have the same cardinal number. The sets do not have to be finite. The cardinal number of the set of algebraic number is alef-null, for example. There is not an one-to-one relation between the elements in the set of non-algebraic numbers and the set of natural numbers. Accordingly, the cardinal number of the first set, the transcendentals, must be greater. This might surprise those of us, who know only two: pi and e. The transfinite cardinal numbers in every-day sense. They are infinite. Adding numbers to infinity leaves us with infinity, which is not very interesting. What set theory showed was relations between various infinte sets. There are different infinities, depending of the starting point, which might surprise some. "There is nothing infinite in the real world, except the phantasy of an idiot". I am not sure, if this medium is a part of the real world. Let's suppose it is not, or rather say I am an idiot. Georg Cantor's achievements are dubious. Was it mathematics he dealt with, or was it logic? Maybe we should look into the logic of arithmetics. I should say it is about answering the question: are the two enteties a and b equal? How do we prove equality? If it is not possible to prove equality, we can not be sure of the opposite, inequality, eighter. This was not fairly understood until the 20th century.