What Does it Really Mean to Have Different Kinds of Infinities? Can someone explain to me how there can be different kinds of infinities?
I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me.
Any help would be appreciated.

This is an answer to the following question marked as duplicate which redirects here: "I've known for some time that infinitary numbers can be different in order, such as the integers (countable), and the real numbers (uncountable). I read that you can always find a higher order of infinity given any order of infinity. Since infinity is the limit of the natural numbers under the successor function, I would like to know if there is a similar concept for orders of infinity under taking power-sets, if there is a sort of "super-infinity", a limit to the orders of infinity."
Yes, there is such a concept: the smallest strongly inaccessible cardinal. Roughly, it is the smallest uncountable infinity that can not be reached by taking either unions or power sets of infinities under it, see here http://en.wikipedia.org/wiki/Limit_cardinal. Existence of such cardinals is widely believed to be independent of the standard axioms of set theory (ZFC), in other words it can neither be proved nor disproved from them. However, there are many works, where people postulate existence of strongly inaccessible cardinals and see what they can derive from it. 
Of course, even with such a postulate you still don't get the "infinity of all infinities", such a concept is self-contradictory according to the Russel paradox, but the smallest strongly inaccessible cardinal is in a similar relation to the ones under it regarding power sets as the countable cardinal is regarding successors and unions. 
