As far as I know, there are two different notions to the word "infinity" in Mathematics.
First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of elements, the set is said to be an "infinite set"; this notion of infinity deals with "how many".
Second notion of infinity deals with "how large" rather than "how many" - here, infinity is simply a "infinitely large number", that is, one that is greater than any numbers.
I have heard that some mathematician proved that one infinity can be larger than other infinity, but I have also heard that the proof only has to do with the first notion of infinity.
What about in terms of the second notion of infinity - say you have two infinitely large numbers (again, this is different than having a set with infinite number of elements). Can one infinitely large number be greater than other infinitely large number?