There is a theorem due to Borde, Guth, and Vilenkin which might be taken as evidence for a beginning of time.
Roughly speaking, this theorem says that in any expanding cosmology, spacetime has to be incomplete to the past. In other words, the BGV theorem tells us that while there might be an "eternal inflation" scenario where inflation lasts forever to the future, inflation still has to have had some type of beginning in the past. BGV show that "nearly all" geodesics hit some type of beginning of the spacetime, although there may be some which can be extended infinitely far back to the past.
If we assume that the universe was always expanding, so that the BGV theorem applies, then presumably there must have been some type of initial singularity.
The fine-print (some readers may wish to skip this section):
[BGV do not need to assume that the universe is homogeneous (the same everywhere on average) or isotropic (the same in each direction on average). Although the universe does seem to be homogeneous and isotropic so far as we can tell, they don't use this assumption.
More precisely, let be the Hubble constant which says how rapidly the universe is expanding. In general this is not a fully coordinate-invariant notion, but BGV get around that by imagining a bunch of "comoving observers", one at each spatial position, and defining the Hubble constant by the rate at which these observers are expanding away from each other. The comoving observers are assumed to follow the path of geodesics, i.e. paths through spacetime which are as straight as possible, that is without any acceleration.
Now let us consider a different type of geodesic—the path taken by a lightray through spacetime. Now if the average value along some lightlike geodesic is positive, then BGV prove that it must reach a boundary of the expanding region in a finite amount of time. In other words, these lightlike geodesics reach all the way back to some type of "beginning of time" (or at least the beginning of the expanding region of spacetime which we are considering).
We can also consider timelike geodesics, describing the motion of particles travelling at less than the speed of light. For nearly all timelike geodesics, if then that geodesic also begins at a beginning of time. However, the theorem only applies to geodesics which are moving at a finite velocity with respect to the original geodesics which we used to define . The original set of observers is allowed to extend back infinitely far back in time.
As an example of this, one can consider a spacetime metric of the following form:
If we set the "scale factor" to be exponentially inflating:
then such a universe extends infinitely far to the past from the perspective of an observer who remains at a fixed value of . But nevertheless, observers travelling at a finite velocity relative to those hit a beginning of time (or else exit the region of spacetime where this metric is valid).
Since the BGV theorem only refers to the average value of the expansion, it applies even to cosmologies which cyclically oscillate between expanding and contracting phases, so long as there is more expansion (during the expanding phases) then there is contraction (during the contracting phases).
On the other hand, in certain cases even an expanding cosmology may have 0 average expansion, due to the fact that we are averaging over an infinite amount of time. So the BGV theorem does not rule out e.g. a universe where the scale factor approaches some constant value in the distant past.]
The fine print is now over.
All right, everyone who skipped the details section is back, yes?
The BGV theorem is sometimes referred to as a "singularity theorem", but it is not really very closely connected to the others, because it doesn't use an energy condition or any other substantive physical assumption. It's really just a mathematical statement that all possible expanding geometries have this property of not being complete.
Carroll correctly observes that the BGV theorem relies on spacetime being classical:
So I’d like to talk about the Borde-Guth-Vilenkin theorem since Dr. Craig emphasizes it. The rough translation is that in some universes, not all, the space-time description that we have as a classical space-time breaks down at some point in the past. Where Dr. Craig says that the Borde-Guth-Vilenkin theorem implies the universe had a beginning, that is false. That is not what it says. What it says is that our ability to describe the universe classically, that is to say, not including the effects of quantum mechanics, gives out. That may be because there’s a beginning or it may be because the universe is eternal, either because the assumptions of the theorem were violated or because quantum mechanics becomes important.
It is quite true that the BGV theorem is proven only for classical metrics, although I see no particular reason to believe that its conclusion (if the universe is always expanding, than it had an edge) breaks down for quantum spacetimes.
However, Carroll's secondary point that the assumptions of the theorem might not hold seems even more devastating. It says that there must be a beginning if the universe is always expanding. So maybe have it contract first, and then expand. That's an easy way around the BGV theorem, and (as Carroll points out) there are a number of models like that. On this point I agree with Carroll that the BGV theorem is not by itself particularly strong evidence for a beginning.