There is a math field cold Set Theory which covers problems exactly like this. From their point of view infinity + infinity is infinity and has the same size.
They use aleph-null as the cardinality of natural numbers = the size of all natural numbers, kind of infinity. As long as you can map set of numbers to natural numbers it still has a size of aleph-null. Doubling set of size aleph-null will give you only a set of size aleph-null since the numbers can still be mapped to the set of natural numbers.
There is such a thing as countably infinite sets (the size of natural numbers) and uncountable sets.
If you look at all real numbers, you will find out that there are many more real numbers and they can no longer be mapped to the natural set. The size of it is 2 to the power of aleph-null. Here you go, that's a completely different "infinity" that all natural numbers.
Tell DD that doubling won't change anything, but 2^ will do the trick
I hope I didn't mess it up too much. It has been a while since I took Theory of Sets.
