Part 5 of my Mapping the Kalam series.
Premise 2 of Craig & Sinclair’s Kalam Cosmological Argument is:
The universe began to exist.
They support this premise with evidence from physical cosmology, and also from the philosophy of mathematics. For the latter, they provide this supporting argument:
2.11. An actual infinite cannot exist.
2.12. An inﬁnite temporal regress of events is an actual inﬁnite.
2.13. Therefore, an infinite temporal regress of events cannot exist.
Last time, I made some clarifications and pointed out that…
…a Realist might say that there is an actually infinite number of mathematical objects, and because mathematical objects really exist, this disproves premise 2.11. But to do this, the Realist is going to have to rebut the arguments for Anti-Realism coming from Conventionalists, Deductivists, Fictionalists, Structuralists, Constructibilists, and Figuralists.
Craig & Sinclair leave the Realist with this rather difficult task, and move on to support premise 2.11 “by way of thought experiments that illustrate the various absurdities that would result if an actual inﬁnite were to be instantiated in the real world.”
The first such thought experiment is that of Hilbert’s Hotel.
Imagine a hotel with a finite number of rooms. Say, 500 rooms. And all the rooms are taken. A new guest arrives and asks for a room, and the owner says, “Sorry, all the rooms are taken.” End of story.
But now let’s imagine a hotel with an infinite number of rooms, and suppose once more that all the rooms are taken. A new guest asks for a room, and the owner says, “But of course! Come on in.” The owner then shifts the person in room #1 to room #2, the person in room #2 to room #3, and so on – into infinity. He then places the new guest in room #1.
How did this happen? All the rooms were full, and yet the guest checked in to room #1. What is more, we added a new guest, didn’t lose any guests, and yet there are the same number of guests! Their number is, specifically, infinite.
It gets stranger. The next day, an infinity of new guests arrive, asking for rooms. “But of course!” says the owner, who shifts the person in room #1 into room #2, the person in room #2 to room #4, the person in room #3 to room #6, and so on. He moves each person to the room that is numbered double his original room number. Because doubles of integers are always even, every person in the hotel is in an even-numbered room. The infinite number of new guests now check in to the odd-numbered rooms. And yet, before they came, all the rooms were occupied! And yet the number of guests in Hilbert’s Hotel is the same as before: their number is infinite.
And this can be repeated an infinite number of times. Each time, the hotel is full when new guests arrive, and yet the guests check in, and after they check in the number of guests in the hotel remains the same as before.
And we’re not done yet. Suppose the guest in room #1 checks out. Are there any fewer guests in the hotel? According to set theory, no. There are still an infinite number of guests in the hotel. Suppose an infinite number of guests check out – say, all those in odd-numbered rooms. After this, there are still an infinite number of guests in the hotel. But the owner doesn’t like a half-empty hotel – that looks bad! So he shifts each guest to the room that has a number half that of his current room, and the hotel is now completely full without adding any guests.
But the owner can’t always keep his hotel full with these maneuvers. Let’s say that the infinite number of guests in any room numbered higher than #3 checks out. Now Hilbert’s Hotel has an very finite number of guests: three! And yet, the same number of guests checked out this time as had checked out when everyone in an odd-numbered room checked out. Both times, the number of departing guests was infinite. And yet in the first case, the hotel still had an infinite number of guests, and in the second case it’s guest count was reduced to three.
Craig & Sinclair conclude:
Hilbert’s Hotel is absurd. But if an actual inﬁnite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual inﬁnite is not metaphysically possible.1
Next time, we’ll look at some objections that have been raised to Craig’s use of Hilbert’s Hotel to support the proposition that “an actual infinite cannot exist.”
- Craig & Sinclair continue: “Partisans of the actual infinite might concede the absurdity of a Hilbert’s Hotel but maintain that this case is somehow peculiar and, therefore, its metaphysical impossibility warrants no inference that an actual inﬁnite is metaphysically impossible. This sort of response might seem appropriate with respect to certain absurdities involving actual infinities; for example, those imagining the completion of a so-called supertask, the sequential execution of an actually inﬁnite number of definite and discrete operations in a ﬁnite time. But when it comes to situations involving the simultaneous existence of an actually inﬁnite number of familiar macroscopic objects, then this sort of response seems less plausible. If a (denumerably) actually inﬁnite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert’s Hotel. Since nothing hangs on the illustration’s involving a hotel, the metaphysical absurdity is plausibly attributed to the existence of an actual inﬁnite. Thus, thought experiments of this sort show, in general, that it is impossible for an actually inﬁnite number of things to exist in reality.” [↩]