I think it's around 3:20 that you get something like this:
Philosophically, the idea of an infinite past seems absurd. Just think about it. If the universe never had a beginning then the series of past events in the universe is infinite. But, mathematicians recognize that the existence of an actually existing number of things leads to self-contradictions. For example, "What is infinity minus infinity?
I can't tell you how many times I've heard some variant on this in class or in coffee shops from kids yammering on and on about actual and potential infinities. Having taken three semesters of Calculus (during which time I completed Calc I and II!), I think I'm almost qualified to weigh in on this one.
Craig likes to argue that there must have been only a finite number of moments prior to this one on the grounds that there cannot be an actual infinity of anything. To bring out the absurdity of an actual infinite collection, he asks us to consider the following example:
Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? ... But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds!
Maybe this is too quick, but isn't an answer to Craig's question that the mathematicians and the cleaning lady are both right in their own way? If the guest in room #1 is, say, Wes and he checks out at 9:00 a.m., then when the cleaning lady says at 10:00 a.m. "There is one less person in the hotel now than there was earlier this morning, his name was Wes" I think she speaks truthfully. No mathematician would deny this. If the mathematician says that "There are not fewer members in the set of guests in HH at 8:59 a.m. than there are in the set of guests at 10:00 a.m." I think the mathematician speaks truthfully. Here the mathematician is working from the idea that one set has fewer members than another only if you can't put the members of the first set into 1-1 correspondence with the members of the second. No cleaning lady would deny this.
We can do the same thing with numbers. If you have the set of primes and then kick out the smallest prime, there's a sense in which there is one less member in the first set than the second (i.e., 2 is contained in the first set only). There is a sense in which there is not one less member in the first set than the second because if you wanted to have a dance you could put the members of the sets into 1-1 correspondence.
Now, if I understand Craig's view, it is that everything I've said about the sets of numbers is correct but you can't have "in reality" collections of things that are infinite because that would lead to absurdities. But, to the extent that I've defused the absurdities in the mathematical case I think I've done so in the previous case with Hilbert's Hotel. The appearance of contradiction is really due to an ambiguity exposed. Close the hotel down.