Correct interpretation of the Borde Guth Vilenkin (BGV) theorem? I recently watched this debate between Sean Carroll and William Laine Craig on the topic of theism. At 8:29, Dr. Craig presents his interpretation of the BGV Theorem, and claims it implies the universe must have had a beginning, and time cannot extend infinitely into the past. At 26:28, Dr. Carroll explains why Craig's argument misrepresents the BGV theorem. 
Now I would take a physicists (Dr. Carroll) word any day in this argument. But supposedly, after the debate Dr. Craig emailed Vilenkin about this issue, and Vilenkin allegedly claimed that Craig did not misinterpret the theorem. 
I would put a summary of Carroll's and Craig's arguments, but I am afraid I would butcher them so I just mentioned the timestamps and left the link to the video. 
 A: Having watched only the referenced sections of the video (the arguments concerning the BGV theorem, and short periods surrounding those marks), I want to note that to the best of my knowledge both Craig and Carroll make themselves guilty of misrepresenting some information to the public, for example Craig's quote of Hawking (dealt with by Carroll) and Carroll bringing up the infinite-time result of the Schrödinger equation (because it might be taken as indicative that it is a relevant result, which I find highly questionable seeing as it can be taken as indication of the Schrödinger equation's incompatability with certain observed events within the universe).
Concerning the BGV theorem Craig is correct in saying that the result holds for any geodesic observer with a non-comoving congruence of time-like geodesic test particles which is on average (over the past) expanding (in a generalized sense, within general relativity), in the sense that such an observer geodesic is necessarily past-incomplete. That is not what he says, of course, but to my ears his statement of this is as good as one can expect. That does not imply a creation event. What it does mean is that if e.g. a light-ray passes through a fiducial "cloud" of objects (without interacting), such as can e.g. be taken to be the case locally in any cosmological model I have seen, and observes an overall expansion, then that geodesic does end somewhere in the past. I am fairly certain that this is not an in any way controversial result.
Carroll is correct in that the BGV theorem thus does not include all models, but it includes a wider range than Carroll's "some" may make it sound like (depending on interpretation, of course).
Ultimately, both the BGV theorem and the quoted paper on the generalized second law may be interpreted as supporting the idea of some sort of "creation event," in the sense that they do not conradict it and serves to narrow down other possibilities. However, in the same way they could then be interpreted as supporting any theory they do not contradict. This is often the natural, but certainly not scientific, response of someone who has themselves formed a theory already. They most definitely cannot be taken as proof either way. 
A: Dr. Craig rejects Professor Vilenkin's naturalistic model of the Universe's origins:
http://www.reasonablefaith.org/the-ultimate-question-of-origins-god-and-the-beginning-of-the-universe
Ask yourself how Dr. Craig, who is not a physicist, can appeal to Vilenkin being correct on some things but not others?
Also watch Dr. Craig reject the number zero in his debate with the philosopher Peter Millican:
https://www.youtube.com/watch?v=fEw8VzzXcjE
Beginning around the 1:18:00 mark, Craig also rejects all of transfinite arithmetic, never telling his audience, of course, that one of the authors of the BGV theorem, Arvind Borde, is a distinguished professor of mathematics.
