big bang theory - How can the universe be infinite?

I think the source of confusion between the two concepts - the Big Bang singularity and an infinite universe - is the misconception that the universe began as a finite expanse originally. This misconception easily arises from analogies using present-day logic and numbers that were not applicable in the early universe. For example, I've heard it said that shortly after the Big Bang, the entire observable universe was the size of a grapefruit, but that explanation neglects to mention that grapefruits would have been much larger then.

The problem is that space is where we can measure how large something is, but space expands, so something that is a certain distance away currently was a lot closer a long time ago, even if neither object has moved in the normal sense. As an analogy to help illustrate the effect:

You and I are standing on a preposterously large deflated balloon. You set down a meter stick, make a mark on the balloon at each end and we each stand on one mark and are now a meter apart. Then I turn on a pump and start inflating the balloon. As the balloon inflates, the surface stretches out and you and I appear to get farther from each other, when though we're not 'moving' (e.g. walking away from each other): now we have conflicting sets of information to consider; according to the marks on the balloon surface we're still one meter apart, but according to the meter stick in your hand (which is not expanding) the distance is greater than that.

Note that while I called the balloon "preposterously large," it could have been infinitely large and still behave the same way. I point this out because I've seen in comments on other answers that you don't see how space could be both infinite and expanding - that if it's expanding, then it must have been previously finite. That is incorrect: in fact, because infinity is the quality of unboundedness, something that is infinitely large can always get bigger, because by definition there is no upper bound on its size.

Note also that if you recorded the earlier analogy in reverse, it would appear that space was shrinking such that a several-meters distance between us reduced over time to one meter. If you continue shrinking the universe in such a manner, it eventually becomes the case that there is zero distance between us. And if you apply that to a scenario where there are people infinitely distributed across the balloon, all of them would come closer together as the balloon deflated, until there was zero distance between any two people... in theory, at least, since real human beings have size. Energy and space don't have size, however, so at the point of the Big Bang, space was still infinite (since an infinite/unbounded space cannot shrink to become finite/bounded) but the distance between any two points in space was zero.

So if you could go back in time to the Big Bang you'd see an infinite ocean of energy, since all the energy was "shoulder-to-shoulder" (infinitely dense) but it rapidly expands (and therefore cools) to the point that basic particles can form, then later matter and molecules. Of course since your size would depend on the metric of space, it wouldn't necessarily look like space was expanding, but simply like the energy and matter were cooling down. In fact we still see this as an effect of spatial expansion in the redshift of light from distant sources: the light "cools down" or loses energy along the way because it is stretched out on its journey through space.