The only way to think about infinities IMHO is to (for example) prove “for all N, P(N) is true”. That will make it really clear what you are talking about when

The only way to think about infinities IMHO is to (for example) prove “for all N, P(N) is true”. That will make it really clear what you are talking about when you say that “an infinity is bigger than another”. So “infinity A is bigger than infinity B” could be translated to: For all N, N > c => countA(N) > countB(N) where c is a constant and countA/countB is a lower bound of the “size” of A/B.