If the Hotel Can Have an Infinite Number of Rooms, Why Can't the Universe Have an Infinite Number of Days?

In my previous post, wherein I attempted to provide an undercutting defeater of Craig's Hilbert's Hotel argument for the impossibility of an actual infinite number of things, I allowed for the conception of a hotel, the set of whose rooms is infinite, and has a cardinality of aleph-null. And yet, I also provided a small argument at the end that I believe may better perform the same function as Hilbert's Hotel in the greater Kalam. But as I thought about what I had done, a possible reply came to mind: If the Hotel can have an infinite number of rooms, why can't the universe have an infinite number of days? I do not have a polished answer, buy I here journal my initial musings.

In the Hotel conception, I said that God built the hotel by the power of His word, and sustains it by His power. I don't actually believe that Hilbert's Hotel exists in this world, but I can conceive of it pretty good, and I think I have illustrated such a conception in a satisfactory way. At any rate, if one claims that he does not know whether he can conceive of such a hotel or not, I think I have made it more difficult to say that it is impossible to conceive of it, such that an argument for the impossibility of such a hotel, or any actual infinite number of things, is not as easy as pointing to the abnormal phenomena such a hotel might give rise to.

Nevertheless, it might appear incoherent for me to posit such a hotel and at the same time argue for the impossibility of an actual infinite number of days in the universe. After all, why can't God simply just create such a beginningless universe and sustain it by the power of His word? Such a question might be unfair for an atheistic infinite universe argument to utilize at face value, but really it is just a call for internal coherence on my part. Well, I believe that there is a fundamental difference between the Hotel and the number of days in the universe.

In the case of the hotel, the rooms are instantly built by God, such that there is no temporal transgression of an infinite spatio-temporal distance. One minute the Hotel is not, the next minute, God speaks such a structure into existence. The set A of all Hotel rooms is at T1 empty, and then the set B of all Hotel rooms is all at once at T2 infinite, and of the cardinality aleph-null.

But in the case of the days in the universe, they cannot exist all at once. Rather, this moment exists and naught else. Thus, by "days" we mean to denote a segment of temporal transgression made by temporally bound objects. There does not exist, "now", all moments in time. This view, Presentism, is something I hold but do not care to explicate and defend here. I am interested in researching it further, and perhaps writing on it someday. My Philosophy of Religion professor, Thomas Crisp, is also a presentist. I am not sure whether my argument for the beginning of the universe is altered whether or not Presentism is accepted, but I will go forward as if it is.

This difference is all that is necessary to establish why it is that a hotel with an infinite number of rooms is conceivable, but a universe whose temporal past is not. For, as was shown in my argument in the previous post about the Hotel, there is no room with the number -∞ (negative infinity). Nor is there a room with the number 'infinity' on its threshold. For infinity is the state of a set, not a number unto itself. Any given number is finite, while infinite sets of finite numbers may be considered.

Similarly, if 'now' is labeled "0", and yesterday "-1", then it is seen that there cannot be a day with the label "-∞". Any given day in history will have a finite label. But why can't the set of all past days be infinite? Because of the impossibility of transgressing an infinite spatio-temporal distance. Just as the present must transgress the moments in today to get to tomorrow, so it would have had to transgress the past in order to get to today. This is what was illustrated in my thought experiment by the runners who attempted to begin their race infinitely far away. They will begin by crossing room numbers 0 through -10, and then they will transgress rooms -10 through -20, and continue running for eternity, without ever finding a starting line from which to run back to room 0. This is a textbook example of a potential infinite. At any given time, the runners will be in front of a room with a finite number, and the cardinality of the set of rooms they will have transgressed will be ever-increasing. But a potentially infinite set will never become an infinite set. In the same way that even God, if He counts one number at a time, will never reach infinity. He will be counting forever, and the set of numbers He utters will be potentially infinite. For at any given time, He will be uttering a finite number. Not even God can transgress an infinite spatio-temporal distance.

The universe cannot have existed from infinity past because it is impossible to cross an infinite distance, or an infinite amount of time. The Hotel can have an infinite number of rooms, because its construction was instantaneous. This would be like God thinking of an infinite set of numbers all at once - something He is no doubt capable of doing.