In cosmology, the term the Big Bang has several subtly different meanings. It means the moment in time, it means the “explosion” event, and it means the all the events and states of the early universe. How early is early? Could be lots of things. Maybe just the first fraction of a second, maybe the first few minutes (until atomic nuclei stopped forming), or maybe all the way up to the time of recombination (estimated to be around age 370,000 years).
In the early universe, the part of the universe that corresponds to what is now our observable universe was very small. For a given age, we can estimate how small. But we can’t say very much about the size of the quasi-hypothetical greater universe that evidently exists beyond the observable universe. That could even be infinite, even at the moment of the Big Bang (assuming such a moment existed…).
We can (and should) define the Big Bang’s moment in time to be “time zero”, though this leads to a bit of a problem, because we cannot actually extrapolate all the way back to that moment. The known laws of physics break down when we get back the moment which, by naive extrapolation, we would assess the universe to be about 10−43 seconds (the Planck time) old. Before that moment, time as we know it didn’t even work right.
I imagine the expansion of the early universe being something like this:
I say that unless we can figure out some more powerful laws of physics, the exact moment of the Big Bang didn’t truly exist. The events of the early universe certainly happened, but there really never was a time when the universe had literally zero size. The universe did not actually start out as a singularity.
It was very close to being a singularity, and calling it a singularity ought to be close enough, in most cases. But on the other hand, there’s a big difference between something being infinite, and something being not infinite. I think we should be careful about glossing over the distinction. For one thing, it gives ammunition to those who would deny that the events of the early universe even happened.