Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for ?actual infinity.? The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. T. Jech: "Set Theory", Stanford Encyclopedia of Philosophy (2002) http://plato.stanford.edu/entries/set-theory/
There are only countably many names. An uncountable set of names cannot be well-ordered - because it does not exist. A set of numbers cannot be well-ordered unless all the numbers have names. This seems to contradict Cantor's diagonal argument - but only if infinite set are complete.
Conclusion: Infinities do not come come in different sizes. In fact mathematicians have never had use for actual infinity because they could not. All they could is to believe that they had use for actual infinity, i. e., for numbers that have no names and cannot be used. That's called mathelogy.