I can think of at least three ways in which the universe could be infinite off the top of my head:
- My closet is infinite because I can keep sticking things in it. (Is there infinite space between or around the mass in the universe?)
- My set is infinite because I can find a one-to-one correspondance with the natural numbers. (Is there an infinite number of things that can be counted?)
- My volume is infinite because given any subset of my volume, you can divide it. (Is the universe infinitely divisable?)
Each of those types of infinite could be satisfiable:
- If you're counting particles, then look no further than your nearest vacuum.
- If you're measuring space, then just wait long enough.
- If you believe Gorgonzola, then you can fit infinity in the palm of your hand.
...Or maybe they're not:
- If you +1 to your count for every particle and -1 for every antiparticle then your vacuum contains nothing (see the Law of Conservation of Mass).
- Waiting an infinite amount of time before counting is cheating.
- Our computers are virtual machines in the real machine that is the universe so I don't believe Gorgonzola. :)
Is the universe infinite? That's an ambiguous question.