I don't think the universe is infinite in size for a few reasons.
-The universe is seen to be expanding, if something is infinitely big, how can it expand into something bigger?
-We only see stars up to a certain "distance" away. If the universe is infinitely big, why is there a sudden stop in stellar material?
-The universe is thought to have started from the big bang. If this started from an infinitely small space, what did it expand into to become infinitely big?
I do agree that the universe is of an unimaginable scale, but it is not infintely large, it has a finite size. What is outside of the universe is another question.
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- It can still expand if you look at particle densities. If particles all move away from each other then there is expansion, even if it happens everywhere in an infinite universe.
- This is because the universe has a finite age. Light from very distant objects just didn't have the time to reach us yet.
- The big bang didn't expand the universe inside space. Rather, it created space itself, and as of now space is still expanding.
The current theory shows 3 different models of the universe. In each one, space and time starts at the big bang, and from there space expands, taking everything with it. A common analogy is a balloon with dots on it, being inflated. The universe (the balloon) expands, so all the dots move away from each other. As such, the universe is really expanding in 4 dimensions (beat that!), which explains why there is no common 'centre point' for the expansion, just like on the balloon. Everything moves away from everything else, no matter where you are.
The first model is that of a 'closed' universe. A closed universe is finite in size, but still unbounded. This would mean that you could keep going and never reach any kind of border. This is just like the balloon analogy. If you keep going, you eventually end up where you started. Space itself wraps around just like the surface of a balloon (or the Earth) does.
Second model is the 'flat' universe. Space is just as it says, flat. There's no wrapping around this time, so you can go on forever. This is what we're accustomed to, and this is probably very nearly what our universe is like. A flat universe is unbounded and infinite. Anything sent off into space, no matter where, will eventually slow down, but never stop completely. This applies to everything in the universe, so the universe itself will continue to expand forever. It will get closer and closer to its final size, but never actually reach it. Instead it will slow down more and more until it's going so slow you can't notice. But it'll never stop.
Third model is an open universe. It's similar to the flat model, but here there's no way to stop the expansion. Everything will continue to fly away from everything else, and in the end you just have a very big, cold, dark and empty universe. Anything sent into space with enough velocity will just keep going forever. It will not turn around or slow down to a near stop. Sure it will slow down a bit (after all, the whole universe will be pulling it back once it reaches the 'edge') but it will keep going.