Infinity could be defined as something that resides outside of space and where time doesn't flow and never did. If space and time have a beginning then they could only come from something that doesn't have a beginning. The concept of beginning is tied to time, if time didn't exist, then whatever it came from, is infinity. The universe could be infinite but it isn't infinity. Infinity is everything that is.
It's difficult for us to understand because we are inside of the universe. Our thoughts are limited by space and time, we can't think outside of them because we can't imagine any existence outside of them. For us, nothing can exist outside of space and time. Yet, if we see that time had a start, something before it must not have had one. The same goes for space, if it had a start then it must have been that it didn't exist before, this is hard to express: there must be a place where space and time do not exist at all, a place that isn't a place where time doesn't flow?
Some fields of science advance the universe came from infinite potential or possibilities. They seem to support the concept of infinity.
Let's say that one possibility (the one we exist in for example) out of all the infinite possibilities is taken out to put the universe into it. It could be expressed very simply:
∞ - 1
Since even though something was taken out of it, infinity itself didn't change. So it's logical to say:
∞ - 1 = ∞
Where does the 1 exist? Now we can see that whatever was taken out of infinity is still part of it. The 1 that was taken out can only exist inside of infinity.
Physics looks at this from the inside out, inside the 1 looking out at infinity. Maybe that's why infinities come up everywhere and no one can figure out the mass of the universe. These infinities may not be flaws or errors, they could be that infinity is showing through there, only we don't account for it because we are looking from the inside.
The paradox is how could the universe not dissolve back into infinity. Since it is part of infinity there is no particular identification for that specific potential
Imagine the universe is a drop of water in the ocean. It has its own identity, well, because it exists and there is something in it that notices the existence. How is it possible for it to maintain it's identity or existence inside the ocean? it automatically mixes with the water and all trace of the individual drop is gone forever. In this scenario, the universe exists inside infinity. You could argue the degree of existence it has. One way it could exist without disappearing in infinity is if it's shielded from 'realizing' it is inside and part of it.
Another example with water, if you want to make a hole in it, you have to keep it spinning. As soon as you stop the hole is gone. Maybe infinity has the same kind of property, the universe has to keep spinning so its own space would continue to exist inside of infinity. Maybe, the reason we can't put our finger on the mass of the universe or explain why so many infinities are showing up all the time is because the universe has a finite mass that is growing inside of infinity.
Here is a link to an essay I wrote about the same. It may have some additional info for those interested.