Crucial to the Kalaam argument is showing that the universe has only a finite past. A standard approach is:

If the universe has an infinite past, there is an actual infinite.

If an actual infinite is possible, Hilbert’s Hotel is possible.

Hilbert’s Hotel is impossible.

So, the universe doesn’t have an infinite past.

Apart from the real challenge, which is defending (3), there are at least three distracting difficulties here.

First, one needs to defend the often not very clear distinction between a potential infinite and an actual infinite.

Second, holding on to the argument while believing in an infinite future afterlife requires a very particular theory of time: the growing block theory. For consider the two alternatives: eternalism and presentism. On eternalism, future infinites are no less actual than past ones, and so an infinite future is just as much a problem as an infinite past. On presentism, neither past nor future infinites are actual, and premise (1) fails.

Third, the conditional in (2) is dubious. Not all actual infinites are of the same sort. An actual infinity of past events does not seem to be close enough to Hilbert’s Hotel—a present infinity of rooms—to allow inferring the possibility of the Hotel from the possibility of the past infinity.

Here is an alternative:

If the universe has an infinite past, a simultaneous infinity of objects is possible.

If a simultaneous infinity of objects is possible, Hilbert’s Hotel is possible.

HIlbert’s Hotel is impossible.

So, the universe doesn’t have an infinite past.

Advantages: We don’t need any murky distinction between actual and potential infinites, just the much clearer notion of a simultaneous infinite.[note 1] An infinite future is not an issue, and any theory of time can be plugged in. Finally, the move from a simultaneous infinity of objects to Hilbert’s Hotel in (6) is much more plausible than the corresponding move in (2). For if a simultaneous infinity of objects is possible, surely they could be shaped like rooms.

The one disadvantage, of course, is that (5) needs an argument. Here it is. If an infinite past is possible, surely it’s possible that at infinitely many past times an object came into existence that has not yet ceased to exist. But if that happened, there are now infinitely many objects, a simultaneous infinity.

Still, one might worry. How can we argue for the possibility of an object coming into existence at infinitely past times, given an infinite past? Well, we can imagine a law of nature on which when two particles collide they are annihilated into four particles, with correspondingly smaller individual mass-energy, and we can imagine that these particles by law cannot otherwise disappear. We can then suppose that there have always been such particles, and that during each of a past infinity of years there was at least one collision. Then that very plausibly possible story implies that there is a present infinity of particles.

I think the difficulties in arguing for (5) are far outweighed by the advantages of the simultaneous infinity formulation of the argument.”