“Two things are infinite, the universe and human stupidity, and I am not yet completely sure about the universe.” -Frederick S. Perls, quoting Einstein
The Universe we can see and access is certainly a big place. We see that it goes on for 46 billion light years in all directions, full of stars, galaxies, matter and radiation wherever we look, consistent with an origin in a hot Big Bang. But beyond what we can see, there ought to be more Universe just like our own, originating from either the same Big Bang, or possibly, if inflation is correct, from other Big Bangs at later or earlier times.
What are the prospects, then, for the Universe being either finite or infinite? Thanks to the cosmic microwave background, we can place lower limits on the size of the unobservable Universe, and thanks to the ideas of eternal inflation, we have very, very large numbers for what’s possible as far as size goes. But there’s a long way from very large to infinite, and determining whether that’s true is a very difficult prospect.