I find the idea that an actual infinity is impossible very counterintuitive, but sometimes arguments can establish something very counterintuitive. Actually, even if the following argument does not show that actual infinities are impossible, it will, I think, show that one cannot make any probabilistic inferences in an infinite multiverse. And that’s an interesting conclusion in its own right. (That said, I have some technical qualms about the argument that I can’t articulate.)

Begin with the Parity Principle: If it is almost certain (i.e., if it has probability one) that (a) the basic properties Q and R have exactly the same distribution at t1, and that (b) x is a substance existing at t1 and a member of basic kind K, and no other information is available about x, then the probability that x has Q is equal to the probability that x has R. (Here, I allow such probability values as “undefined”, “inscrutable” as well as intervals, vague values.)

For my argument I will need the assumption that it is possible to have indiscernibles–objects that have all the same basic properties. I actually think this assumption is false, but I am hoping that this assumption can be relaxed.

The possibility of indiscernibles and the Parity Principle are going to be the only potentially controversial assumptions in the argument. If you trust me on this point, you can stop reading the argument, and just argue against these two assumptions. Though you might want to read on to see how exactly I understand the “same distribution” condition in the Parity Principle.

In my earlier post, I gave a Grim Reaper based argument against an infinite past. Here I want to give two more arguments. Unlike the earlier argument, these two arguments are not going to be useful for arguing for the existence of God, since they make use of premises that the atheist is likely to deny (in one case, a version of the Principle of Sufficient Reason, and in the other, the existence of God). But they are useful in a broader sense, namely they help show what might be wrong with an infinite past.

Argument 1. If there is an infinite past, we could imagine that each January 1 in the infinite past somebody looks around and checks if there are any rabbits. If there are, she does nothing. If there aren’t, she makes a breeding pair. Of course, once a breeding pair of rabbits exists, there will be rabbits forever. Nobody and nothing but one of these potential rabbit-makers makes a rabbit. The setup entails that there have always been rabbits, and the rabbits have not been made by anybody or anything, contrary to a causal version of the Principle of Sufficient Reason.

Argument 2. If there is an infinite past, the following scenario should be possible. The universe contains nothing but bobs, and at no time is there more than one. A bob is an asexually reproducing person who lives for a century. At the end of the century he dies, but at the end of his existence he has a choice whether to reproduce or not, and can choose either way. If he freely chooses to reproduce, a new bob comes into existence out of the old bob’s body after death. So, this is a universe where every bob has always chosen to reproduce, though they could have chosen otherwise. But now consider the following very plausible Thesis:

(*) Necessarily, if a world contains at least one contingent being, then there exists something in that world determined into existence by God’s will.

But the story in Argument 2 seems to violate (*), since each bob’s existence is partly dependent on the free choice of the preceding bob. Maybe God has determined, then, not the fact that there is a bob, but that there is some initial infinite sequence of bobs, without determining which initial infinite sequence there is. But even that there is an initial infinite sequence of bobs already depends on bob-made choices.

Argument 2 won’t impress theological compatibilists.

[Cross-posted on my own blog.]

A Grim Reaper (GR) timed to go off at t0 is an entity which does the following at exactly t0. If Fred is not alive at t0, the GR does nothing at t0. If Fred is alive at t0, the GR instantaneously annihilates Fred. (If instantaneous action is not logically possible, one can complicate the situation by allowing shorter and shorter time intervals for these actions.) The GR Paradox then is this scenario. Fred is alive at 11:00 am today, and that he does not die today unless killed by a GR and he does not get resurrected today. There are infinitely many GRs, timed to go off in a staggered way at the respectively times t1,t2,… where tn is equal to 11:00 am + 1/n minutes. Well, by 11:02 am, Fred is certainly dead, since it is impossible that he survive a time at which a GR is timed to go off. But when was he killed? He wasn’t killed by the 11:00 am + 1 minute GR, because if he were alive just before 11:01 am, then he would have been alive at 11:00 am + 1/2 minute, when another GR went off, and he can’t survive a GR going off. It seems that none of the GRs could have killed him, because before each, there was another. So we have a contradiction: he both was and was not killed. Somebody has suggested that Fred is killed by the mereological sum of all the GRs, but that’s mistaken in the present setting because the GRs check if Fred is already dead before they do anything, so in the present setting, none of them actually do anything–and if they don’t do anything, how can they kill Fred?

The Kalaam argument needs the premise that there couldn’t be a backwards infinite sequence of events. Here is an argument for this:

- If there could be a backwards infinite sequence of events, Hilbert’s Hotel would be possible.
- If Hilbert’s Hotel were possible, the GR Paradox could happen.
- The GR Paradox cannot happen.
- Therefore, there cannot be a backwards infinite sequence of events.

Actually, one could make steps 1 and 2 into a single step, but this is more fun, and, if it works, establishes the interesting corollary that Hilbert’s Hotel couldn’t exist.

Argument for (1): If there could be a backwards infinite sequence of events, there could be a backwards infinite sequence of events during each of which a hotel room is created, none of which are destroyed. An infinite number of hotel rooms would then be the result.

Argument for (2): If Hilbert’s Hotel were possible, each room in it could be a factory in which a GR is produced. Moreover, it is surely possible that the staff in room n should set the GR to go off at 11 am + 1/n minutes. And that would result in the GR Paradox.

The argument for (3) was already given at the beginning of the post.

For about two years, I’ve smelled this argument coming, but I think my vanity has kept me from seeing it. I still have to confess that I have a really hard time accepting the corollary that Hilbert’s Hotel couldn’t exist–that corollary seems extremely counterintuitive to me. I wish I had some good way out.

On the other hand, establishing a major premise of an argument for the existence of God is a very happy outcome.

I am not completely convinced by the following argument, but let’s try it.

Let p be a positive real number. A p-widget is a device that on its back has written down a positive integer (perhaps in very small numerals), and that is physically necessitated to behave as follows: As soon as a p-widget w is made, it makes a copy of itself–another p-widget. The amount of time it takes to make a copy of itself is n^{-p} years, where n is the number on w’s back. Moreover, while making the copy, w inscribes n+1 on the copy’s back (all within that n^{-p} year period). Finally, a p-widget does not perish once made.

So, once a p-widget comes into existence with 1 written on its back, it makes a copy of itself in 1^{-p} years and the copy has 2 written on its back. The copy then takes 2^{-p} years to make a copy, which has 3 written on its back. And so on. However, it seems that the enemies of supertasks and actual infinities should not object to a p-widget if p is less than or equal to 1. The reason for that is that if a p-widget is produced where p does not exceed 1, then although the production times for subsequent p-widgets do get smaller and smaller, nonetheless there is no supertask or actual infinity involved–at any given time, there are only finitely many p-widgets. The reason for that is that if p does not exceed 1, then the amount of time for infinitely many p-widgets to come into existence is 1^{-p}+2^{-p}+3^{-p}+… and this is equal to infinity if p is less than or equal to 1.

On the other hand, if p>1, then this infinite series adds up to a finite number, and so after a finite amount of time, there will be infinitely many p-widgets. For instance, if the first 2-widget has 1 written on its back, then there would be an infinite number of 2-widgets after pi^{2}/6 years. This, of course, the enemy of supertasks and actual infinities will claim to be impossible. So the initial difficulty for the enemy of supertasks and actual infinities is that a 0.9-widget and a 1-widget could be made, but a 1.1-widget cannot. That seems problematic–why should there be this intrinsic logical limit on how much faster new widgets can be produced?

But there is a further move I want to make. If the only objection is to supertasks and infinities, then the opponent of supertasks and actual infinities should not object to a world that contains a 2-widget–as long as God miraculously intervenes to stop the reproduction of 2-widgets before the pi^{2}/6 years are up. For if God does so intervene, then no paradox ensues.

Now imagine a world that contains only God and physical stuff and a time sequence lasting at least two years (the magic number pi^{2}/6 is approximately 1.64493), including initially a 2-widget with 1 written on its back. It is now metaphysically necessary that if such a world is actual, then God miraculously intervenes at some time in the first pi^{2}/6 years. But that seems really, really strange: Why would God be necessitated to miraculously intervene? There is something very odd about the answer: “He must intervene to prevent a supertask or actual infinity.” (One could perhaps imagine a case where an essentially omnibenevolent, omnipotentent and omniscient being would have to intervene to prevent an evil. But to prevent a supertask or an actual infinity?)