It is often assumed that one couldn’t finitely specify a nonmeasurable set. In this post I will argue for two theses:
It is possible that someone finitely specifies a nonmeasurable set.
It is possible that someone finitely specifies a nonmeasurable set and reasonably believes—and maybe even knows—that she is doing so.
Here’s the argument for (1).
Imagine we live an uncountable multiverse where the universes differ with respect to some parameter V such that every possible value of V corresponds to exactly one universe in the multiverse. (Perhaps there is some branching process which generates a universe for every possible value of V.)
Suppose that there is a non-trivial interval L of possible values of V such that all and only the universes with V in L have intelligent life. Suppose that within each universe with V in L there runs a random evolutionary process, and that the evolutionary processes in different universes are causally isolated of each other.
Finally, suppose that for each universe with V in L, the chance that the first instance of intelligent life will be warm-blooded is 1/2.
Now, I claim that for every subset W of L, the following statement is possible:
- The set W is in fact the set of all the values of V corresponding to universes in which the first instance of intelligent life is warm-blooded.
The reason is that if some subset W of L were not a possible option for the set of all V-values corresponding to the first instance of intelligent life being warm-blooded, then that would require some sort of an interaction or dependency between the evolutionary processes in the different universes that rules out W. But the evolutionary procesess in the different universes are causally isolated.
Now, let W be any nonmeasurable subset of L (I am assuming that there are nonmeasurable sets, say because of the Axiom of Choice). Then since (3) is possible, it follows that it is possible that the finite description “The set of values of V corresponding to universes in which the first instance of intelligent life is warm blooded” describes W, and hence describes a nonmeasurable set. It is also plainly compossible with everything above that somebody in this multiverse in fact makes use of this finite description, and hence (1) is true.
The argument for (2) is more contentious. Enrich the above assumptions with the added possibility that the people in one of the universes have figured out that they live in a multiverse such as above: one parametrized by values of V, with an interval L of intelligent-life-permitting values of V, with random and isolated evolutionary processes, and with the chance of intelligent life being warm-blooded being 1/2 conditionally on V being in L. For instance, the above claims might follow from particularly elegant and well-confirmed laws of nature.
Given that they have figured this out, they can then let “Q” be an abbreviation for “The set of all values of V corresponding to universes wehre the first instance of intelligent life is warm-blooded.” And they can ask themselves: Is Q likely to be measurable or not?
The set Q is a randomly chosen subset of L. On the standard (product measure) understanding of how to probabilistically make sense of this “random choice” of subset, the event of Q being nonmeasurable is itself nonmeasurable (see the Sawin answer here). However, intuitively we would expect Q to be nonmeasurable. Terence Tao shares this intuition (see the paragraph starting “Intuitively”). His reason for the intuition is that if Q were measurable, then by something like the Law of Large Numbers, we would expect the intersection of Q with a subinterval I of L to have a measure equal to half of the measure of I, which would be in tension with the Lebesgue Density Theorem. This reasoning may not be precisifiable mathematically, but it is intuitively compelling. One might also just have a reasonable and direct intuition that the nonmeasurability is the default among subsets, and so a “random subset” is going to be nonmeasurable.
So, the denizens of our multiverse can use these intuitions to reasonably conclude that Q is nonmeasurable. Hence, (2) is true. Can they leverage these intuitions into knowledge? That’s less clear to me, but I can’t rule it out.
For this to work, the universes would have to be objectively chancy. But does objective chance make sense? I have no firm view, but many have doubted it. “God does not play dice …”
ReplyDeleteThe universes form a continuum in the sense that they are indexed by the real parameter V. Is it metaphysically possible to have a continuum of discrete things?
In what sense has a set been specified? Maybe in the mind of God, who could presumably look over all the universes. But a denizen of a particular universe could know nothing specific about the other universes. So she could know almost nothing specific about the membership of Q.
What exactly does it mean to “finitely specify” a set? Does a specification that depends on an infinite number of random outcomes count as finite?
The sense of specification is this: "Give a finite definite description in a natural language."
ReplyDeleteOne can leverage all sorts of things in specifying a set in this sense. For instance, I can let S be the set of all the heights of philosophers who have ever lived.
I’m not sure that is how I would read the claim that you cannot finitely specify a nonmeasurable set. I would gloss it as saying that there can be no mathematically self-contained finite description. On this view, invoking an infinite multiverse as a lookup table would be cheating.
ReplyDeleteIt’s not directly on topic, but (if I am thinking straight), the argument given by Sawin applies equally to uncountability. (Note that as with nonmeasurability, uncountability of Q does not depend on the outcomes in any countable subset of universes.) So “Q is uncountable” would be nonmeasurable in the natural product measure. Intuitively, you would surely expect Q to be uncountable. But, as with nonmeasurability, it is not obvious how you would go about checking this.
Yes, there is an ambiguity between "specify" and "mathematically specify". I think we can specify things without mathematically specifying them. For instance, if my body had perfectly precise boundaries, I could specify a real number by saying that it is equal to my diameter (i.e., supremum of distances between two points in me). But that real number, most likely, would not be one I could *mathematically* specify.
ReplyDeleteIt's easy to give examples of nonmeasurable subsets of a uncountable product measure. I think you're right that the event of Q being uncountable is an example. It's a really nice example, actually. I think what these cases show is that the product measure's sigma algebra is much too coarse to match our intuitions.