Monday, February 2, 2015

Betting on paradoxical sets

Suppose that a point z will be uniformly randomly chosen on the surface of a sphere S and you are asked to place bets as to which set z is in. Then, plausibly:

  1. If two sets A and B are equivalent under rotations about the center of the sphere, you should accept this offer: get three dollars if z is in A and pay two dollars if z is in B.
But now consider a paradoxical decomposition of the whole sphere S, by a version of the Banach-Tarski Paradox[note 1]. In this, the sphere is partitioned into two subsets C and D, each of which can be decomposed into a finite number of subsets that can be rotated to form the whole sphere. Applying (1) to each set in the decomposition of C and its rotation, you will accept a sequence of deals that adds up to:
  1. If z is in C, you get three dollars and if z is in S you pay two dollars.
Repeating this with D's decomposition, you get a sequence of deals that adds up to:
  1. If z is in D, you get three dollars and if z is in S you pay two dollars.
But of course if z is in C or D, it is also in S, and if it's in S, then it's in exactly one of C or D. It follows that the deal adds up to:
  1. No matter what, you get three dollars and you pay four dollars.
So, repeated application of (1) yields an unacceptable conclusion.

One might say that this is an artifact of the fact that there is no finitely additive rotation-invariant probability measure on the sphere. But I think the above formulation is a little bit more telling. I make no reference to probabilities here. All I assume is (1), which is a very intuitive rationality judgment, namely that when one has two equivalent scenarios, one should accept an unequal bet between them that is in one's favor.

What to conclude? One conclusion might be that a single application of (1) is fine, but the sequence of applications needed to yield (4) is not.

My own conclusion, however, is that it is metaphysically impossible to have a betting scenario like the above. But why not? What's wrong with it? Well, one possibility is that space is necessarily discrete, but that doesn't seem very plausible to me.

My own preference, however, is to conclude that it is impossible to have anything causally depend on whether a random point (or a particle or the like) is in one of these weird sets that are found in the paradoxical decomposition of the sphere. Why is that? I think it's because it would in effect be a violation of causal finitism, the thesis that no event can causally depend on infinitely many things. But the full story here requires significant amounts of work to complete.


Heath White said...

My own preference, however, is to conclude that it is impossible to have anything causally depend on whether a random point (or a particle or the like) is in one of these weird sets

I don't understand this theorem at all, but...

*couldn't* you make exactly the bet you described in this post? And *wouldn't* the outcome of that bet be something which causally depended on whether a random point was in one of those weird sets?

Alexander R Pruss said...

I don't think you can have a causal process that's sensitive to whether a point is in one of those weird sets. So no causal process could generate a payoff for it.

IanS said...
This comment has been removed by the author.
IanS said...

Professor Pruss:

Another question about causal finitism. It seems to rule out all the standard textbook “with probability 1” results. Examples: in an infinite sequence of independent fair coin tosses, with probability 1 the number of heads is infinite; with probability 1 the limiting proportion of heads exists and equals ½. These do not seem obviously paradoxical. Of course, for a strictly finitist point of view, they should be rejected. But I’m wondering whether you endorse this.

Alexander R Pruss said...


No: you can have an infinite sequence of coin tosses. You just can't have a single event (here, "event" means something a bit more robust than just a subset of the sample space) that depends on the whole sequence. Thus you can't put the sequence into a supertask and have a light that lights up afterwards iff the limiting proportion was 1/2.

IanS said...

That makes sense. To make sure I understand: B-T decompositions and “with probability 1” theorems are fine as pure mathematics. We can’t apply them is this world, for obvious practical reasons. Causal finitism says they cannot be applied in any possible world.

Alexander R Pruss said...


For those of us who are theists, it is somewhat weird that even God cannot arrange such bets. But if causal finitism (or at least its generalization, explanatory finitism) is true, it is logically impossible for him to do so, and God isn't expected to be able to do the logically impossible.

And probabilistic paradoxes aren't the only thing ruled out by causal finitism. We also get to rule out Grim Reapers and (less importantly) Thomson's Lamp. And we also rule out that weird paradox (not mine) that if coin flips had been going on for an infinitely long time, then you would be able to guess all but finitely many of the coin flips before they happen. (Guessing technique: Say two sequences of coin flips are equivalent iff they differ only in finitely many places. By AC, choose a representative in each equivalence class. Whenever you've seen an infinite number of coin flips--and that's always!--you then know which equivalence class is the relevant one, and you guess the next flip according to the chosen representative. It's guaranteed that you will miss only finitely many guesses. But that shouldn't be possible, given the independence of the flips!)