Against Against The Infinite

Infinity is problematic, but it's not impossible

Mark Zuckerberg would approve of the infinite. It moves fast and breaks things.

What exactly does it break? Seemingly everything. It generates a ton of paradoxes with eccentric names like “the Grim Reaper paradox.” Worst of all, it seems to break ethics and probabilistic reasoning.

Suppose we go throughout the universe and discover that each galaxy is filled with mostly miserable people. Each galaxy has 1,000 miserable people and only 1 happy person. This seems like bad news! Let’s call—and hopefully this doesn’t bias the jury—this world HELL.

Consider another universe where each galaxy has 1,000 happy people and just 1 miserable person. This world seems good, assuming the happy people are as happy as the miserable people are miserable. Let’s call this world HEAVEN. It seems like HEAVEN is better than HELL.

Here’s another plausible principle: moving people around in ways that don’t affect their welfare at all doesn’t improve the quality of a world. If we just shuffle around the planets in a way that doesn’t improve anyone’s life, the world has not gotten better.

These, however, seem to conflict. Let’s give everyone a label—say 1M is the first miserable person, 2M is the second miserable person, 1H is the first happy person, and 2H is the second happy person. So HEAVEN has, in galaxy 1, 1H, 2H, 3H, 4H, and 5H, and 1M, while HELL has, in galaxy 1, 1M, 2M, 3M, 4M, 5M, and 1H.

But surely each of the people in HEAVEN could take a vacation to their places in HELL. 1M, 2M, 3M, 4M, and 5M could all move to galaxy 1, 6M, 7M, 8M, 9M, and 10M could all move to galaxy 2, and so on. But once this happens, HEAVEN transforms into HELL without anyone being made worse off. Thus, one can’t think HEAVEN>HELL without thinking shifting people around improves a world.

Worse, these principles seem to infect probabilistic reasoning as well! Suppose that each galaxy has 5 red-shirted people and 1 blue-shirted person. It seems I should think I probably have a red shirt. Let’s call the world where this is true REDLAND.

In contrast, consider a world where each galaxy has 5 blue-shirted people and 1 red-shirted person. Call this BLUELAND. It seems the odds I have a red shirt are higher in REDLAND than BLUELAND. It also seems like my credence in having a red shirt shouldn’t change if people move around. If no one changes their shirt color, but the planets they’re on are simply reshuffled, my credence in having a red shirt shouldn’t change. But these conflict for the same reason.

Let’s call 1R the first red-shirted person, 2R the second, and so on. 1B is the first blue-shirted person, 2B is the second blue-shirted person. REDLAND could be turned into BLUELAND by having 1B, 2B, 3B, 4B, and 5B move to galaxy 1, 6B, 7B, 8B, 9B, and 10B move to galaxy 2, and so on, and having 1R move to galaxy 1, 2R move to galaxy 2, etc.

This is really weird. Infinity conflicts with extremely plausible principles. Worse, it makes it totally unclear how to think about your odds of having various properties in an infinite world. If there are ℵ₀ people with every property in the universe, how do I probabilistically reason about my odds of having different properties? Every way of doing it will either require giving up on the idea that you’re likelier to have a red shirt in REDLAND than BLUELAND, or will require thinking that shuffling people around affects your probability of being them. Maybe you could also modify the math, but the prospects there look rather dismal (I think there may be a solution in this vicinity that I’m sympathetic to, but it’s still quite rough).

Here’s a natural thought at this point: maybe this whole infinity stuff is bullshit. If infinity produces lots of horrendous paradoxes in a dozen different domains, why not just think you can’t have infinite amounts of stuff? Stuff existing truly without end, without limit, seems like it’s maybe impossible, especially when you consider the fact that it generates tons of horrifying paradoxes.

I used to be a bit sympathetic to this as I expressed about a year ago. Maybe you just can’t have infinite things, and then the puzzles go away. But I’ve since started to think this is very implausible, for a host of reasons, and so in case anyone with the mindset of my younger self is reading, I thought I’d lay them out. In short, then, I intend to argue it’s possible for there to be infinite things.

A first reason to suppose this is simply because of the modal intuition that infinite space seems possible. Generally, if something seems possible you should think it is possible. The reason I think there’s a possible world with a horned horse is that it seems possible. But intuitively, it just seems like space could never end or there could be an infinite number of blocks. Now, this isn’t totally decisive—maybe it could be overturned by sufficiently strong reasons—but it’s certainly a reason.

Second, how many numbers are there? It seems the answer is infinity. But if this is right, there’s something that’s infinite in number. Of course, you can get out of this by either being a nominalist (and denying that numbers are real in any important sense) or suggesting that only concrete things must be finite, but abstract things can be infinite. I find nominalism unattractive, and it’s certainly a cost to a theory that rejects the infinite if it requires nominalism (one of many costs, as we’ll see). The second option introduces arbitrariness—why is infinity possible in the abstract realm but not the concrete realm? It’s not clear.

Third, we have some evidence from physics that space is infinitely big. Now, the evidence isn’t overwhelming or decisive, but it’s always bad news for a theory that it needs to reject our best theory of physics. It also just seems inconceivable that space would have an edge.

Fourth, the view that the infinite is impossible requires space to have a smallest unit. But we have no evidence that space has a smallest unit and it seems impossible in principle for there to be a smallest unit of space. Can’t you just consider the right half of the smallest unit of space?

Fifth, as I’ve argued elsewhere, the self-indication assumption is the best theory of anthropics. On this view, if a theory predicts there are N times more people with my experiences than another theory, that theory predicts my existence N times as well. For example, if a coin gets flipped that creates one person if it comes up heads and 10 if tails, if I’m created from this, I should think tails is 10x likelier than heads. But if this is right, from your existence, you get infinitely strong evidence that infinite people really do exist. But if infinite people actually exist, then infinite people existing must be possible.

Sixth, what about the future? It seems the future could be infinite. God doesn’t have to have heaven end after 3289473289 years so that it doesn’t go forever. But this means the view is committed to rejecting B-theory, according to which the future really exists. But B-theory is pretty plausible and accepted by many smart people, so it’s a cost to the theory if it requires giving up B-theory.

Seventh, the view requires that theism is false. If God exists, his power and knowledge are infinite. But if, as I’ve argued elsewhere, we have good reason to believe in God, then we have good reason to give up on this argument.

Eigth, it doesn’t even avoid many of the paradoxes. Because the view holds that a potential infinite is possible—that the future can be infinite as long as there’s no particular day on which there will be infinite people—it still runs into similar paradoxes (though admittedly it avoids some of the gnarlier ones). To see this, let’s consider two examples:

First, imagine that God creates one unhappy person. Each year, every unhappy person becomes happy but has five unhappy children. This goes on forever. Is this scenario good? Answer: it’s paradoxical.

Argument for it being good: each person spends only one year unhappy and then infinity years unhappy.

Argument for it being bad: every year, five times as many people are unhappy as happy.

So we either have to give up on the principle that if every year almost everyone is miserable, such that each year has more suffering than well-being, the world is bad or on the principle that reality is good if it’s good for everyone. But these are both super plausible.

A second class of paradoxes that still arise for this view is the Saint Petersburg paradox and the Pasadena game, which I discuss in section 6. I won’t go into detail about these, as I’ve already discussed them and they get very complicated, but they’re still sort of troubling.

Therefore, while this view avoids some of the paradoxes, it still has paradoxes of its own. It pays a major price and doesn’t even avoid all the paradoxes. Because the view runs afoul of lots of different plausible judgments across many different domains, I think it’s almost surely false. The solution to infinite paradoxes resides elsewhere.