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I don't think it does. At least in mathematical sense.... Something can be infinite and growing. But it's been a long time since school so I may be off with my language.
I am not a mathematician so I am not able to well explain this concept, but there are different sizes infinities. "Infinity" is not one thing.
But there are infinite sets that contain infinite sets, so therefore there are infinities that are larger than other infinities.
It is possible that the infinite universe is expanding into another, larger, infinite space.
Or really anything is possible, we don't know.
Numberphile has a pretty good video on different sized infinities. Newpipe is bugging out lately so I can't dig up the video but if you're interested you should be able to find it easily enough
Yes exactly. What you're describing is cardinality. For example the number of natural numbers (counting numbers) is a smaller infinity than the number of real numbers (rational numbers + irrational numbers).
This can be proven rigorously.