The universe is in all likelihood not infinite. Assuming the big bang theory is mostly correct, then we have a finite span of time and a finite (though variable) rate of expansion. Hence, at any point in time t < ∞ the volume must be finite. Sure, it will keep getting bigger faster than we'll be able to explore it (unless we somehow manage to exceed the speed of light reliably, which is unlikely), but it won't create new planets or anything as it goes along. Planets will continue to form and break apart from matter which already exists, and it will be very much finite. Now, big, yes, the universe will be very big. But very big is a different manner of beast from infinite.
If you restrict the definition of "the universe" to the area that contains planets, then of course the volume is finite. There doesn't need to be matter present for there to be spacetime. There just needs to be an observer.
However, if we can't see it, it might as well be infinite. The lack of an observer removes the restriction on the possibilities. The cat is either alive or dead.
Let us consider an "open" universe. Perhaps it truly is infinite, in every sense of the word. In that case, we cannot reach the end of space-time, even if we travel at an infinite speed. The box is closed, and the cat is again either alive or dead.
But what if the universe is finite? Unless we travel faster than light, we will never catch up with the "edge" of the universe. We could go for an infinite amount of time in any one direction, as long as our speed is less than
c. Thus, the universe is infinite in the sense that we will never reach its edge, and it makes no real difference whether it is truly infinite or not. The cat is both alive and dead.
Let us say that we can travel at faster-than-light speed, and catch up to the edge of the universe. Just say that we try to "pass" this point of expansion. What happens? Does our spacecraft refuse to maintain its speed? Do we pop out of existence as we transcend the limits of the universe? If we do, then there is no longer an observer, and the edge of the universe is unreachable, and therefore again infinite. The cat is dead, but is magically resurrected as soon as we peek in the box.
But what if we "pass" the edge of the universe, and do not pop out of existence? Where are we? Past the point of creation? No, rather we have
created a new portion of the universe, simply by
observing it. By being at that specific point, we have expanded the possibilities and
created more universe. But what's that in front of us? If we speed up a little more, we can reach MORE universe! We could increase our speed without bound, and as long as we still
are, the universe is infinite, unless we can somehow escape the confines of dimensionality. The cat is neither alive nor dead, and not both, but suspended in a sort of limbo.
However, if there is no observer, the restrictions on probability are removed. Adding a self-aware consciousness to the picture is essentially assigning bounds to an integral. If there is no observer, anything is possible. Thus, the only way that we can prove that there is a finite universe is if we are simultaneously aware of each and every point within it, which would require for us to be present everywhere.
Oh, by the way, I just looked in the box, and realized that you forgot to put the cat in.