An Argument for Finitism, Part 1

Finitism is the view that only finitely many things exist. Different types of finitism can be identified based on what we mean by "exist." Metaphysical finitism holds that an infinite number of things cannot be instantiated in mind-independent reality.

Instantiation of something means that there is an instance (or an example) of that thing. Mind-independent reality is whatever exists "out there," in the real world, independently of human thought (it leaves open the question of whether reality is ultimately dependent in some way on a divine mind). Anything that can be instantiated in mind-independent reality is said to be metaphysically possible.

Actual Versus Potential Infinity

An assessment of finitism requires a clear understanding of the difference between actual and potential infinity. When a set (a collection of things) is actually infinite, it contains an infinite number of members. A potential infinite is a collection that is finite but growing. Notice that growing implies change over time. An actually infinite collection need not change, but by definition a potentially infinite collection is changing. So it is technically not possible to have a potentially infinite set because at any particular time, the potentially infinite set will just be a finite set. Potential infinity involves a process. As a finite collection adds new members one-at-a-time, it approaches infinity as a limit, but it never quite gets there.

Think about the natural numbers, which are all the positive integers (1, 2, 3, 4. . .). On the one hand, you can consider the set of natural numbers as a whole. This would be an actually infinite collection. The natural numbers just go on and on to infinity. On the other hand, you can consider the process of counting the natural numbers: "One, two, three, four. . . ." In this case, the numbers you count are a potentially infinite collection. At any time, you will only have counted a finite amount of numbers. The more you count, the closer you get (in a sense) to infinity, but you can never count all the way to infinity. This is because there is no "last" natural number prior to infinity. For any natural number you specify, there will always be a higher number.

People often use the figure-eight symbol (∞) to represent infinity. What ∞ actually represents, as far as I know, is potential infinity. Sets that are actually infinite are (again, as far as I know) represented by aleph numbers. Sets that are actually infinite can have different sizes, or cardinalities. The smallest infinite set would be the set of all natural numbers, which is represented with an aleph-zero (ℵ₀). So I will use ℵ₀ to represent infinity in the following discussion.

Finitism denies the existence of actually infinite collections. If metaphysical finitism is true, it means that it is impossible for an actually infinite number of things to be instantiated in reality.

Hilbert's Hotel

The best reason to accept metaphysical finitism is that, if an actually infinite number of things could exist in reality, it would result in certain absurdities. These absurdities can be identified by constructing thought experiments involving infinity and paradoxes.

Hilbert's Hotel is a famous thought experiment meant to illustrate the strange, counterintuitive nature of actual infinities. We are invited to imagine a hotel with an infinite number of rooms. Each room is currently occupied. Even so, the proprietor can always accommodate any new guests by shuffling the current guests around in different ways. For instance, if one new guest shows up, the proprietor can just have each guest move to the room with the number that comes directly after their current room number. The guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on, and the new guest takes room 1. Importantly, after the new guest checks in, the hotel still has the same number of guests.

Now suppose that an infinity of new guests shows up. The proprietor can accommodate them too, by having all the current guests move to the room with the number that is twice their current room number. The guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6, and so on. In this way, all the odd-numbered rooms are cleared, and all the new guests can check in. In fact, the proprietor could do this an infinite number of times, yet each time, the hotel will still have the same number of guests. This seems strange because prior to the arrival of each new guest, all the rooms in the hotel are occupied. The appearance of new empty rooms almost seems like magic.

So far, everything that I've described is consistent with the way things work with infinite sets in mathematics. Mathematically, if you add any number to infinity, the total will be infinity.

ℵ₀ + 1 = ℵ₀

ℵ₀ + ℵ₀ = ℵ₀

There is nothing logically contradictory about this. But part of the argument from Hilbert's Hotel is that this would be an absurd scenario in real life. That is, it would be absurd to say that a fully occupied hotel could accommodate an infinite number of new guests, and it would be absurd to say that the number of guests in the hotel is the same no matter how many new guests check in.

However, the argument from Hilbert's Hotel is not finished. In fact, we've not yet seen what makes it such a powerful argument. What happens when guests start to check out of the hotel? If all of the guests check out, then there will be no guests left in the hotel. If all the guests in rooms 4 and higher check out, there will be three guests left. If all the guests in the even-numbered rooms check out, there will be an infinite number of guests left. And here's the problem: in each case, the same exact number of guests checked out, yet there was a different number of guests remaining.

ℵ₀ - ℵ₀ = 0

ℵ₀ - ℵ₀ = 3

ℵ₀ - ℵ₀ = ℵ₀

This results in a real logical contradiction. In mathematical operations, ℵ₀ - ℵ₀ is undefined, which puts sort of a prohibition on using subtraction in this context. But the problem is that this prohibition has no impact on real life. In real life, guests have a habit of checking out of hotels. So if it were possible to have an actually infinite number of things in real life, it would result in logical impossibilities. This gives us good reason to deny that an actually infinite number of things could exist in real life.

Thus we have a powerful argument for metaphysical finitism:

1. If Hilbert's Hotel is logically impossible, then finitism is true.
2. Hilbert's Hotel is logically impossible.
3. Therefore, finitism is true.

In the next post, I will consider two common objections to finitism.