Monday, September 11, 2017

Supertasks and empirical verification of non-measurability

I have this obsession with probability and non-measurable events—events to which a probability cannot be attached. A Bayesian might think that this obsession is silly, because non-measurable events are just too wild and crazy to come up in practice in any reasonably imaginable situation.

Of course, a lot depends on what “reasonably imaginable” means. But here is something I can imagine, though only by denying one of my favorite philosophical doctrines, causal finitism. I have a Thomson’s Lamp, i.e., a lamp with a toggle switch that can survive infinitely many togglings. I have access to it every day at the following times: 10:30, 10:45, 10:52.5, and so on. Each day, at 10:00 the lamp is off, and nobody else has access to the machine. At each time when I have access to the lamp, I can either toggle or not toggle its switch.

I now experiment with the lamp by trying out various supertasks (perhaps by programming a supertask machine), during which various combinations of toggling and not toggling happen. For instance, I observe that if I don’t ever toggle the switch, the lamps stays off. If I toggle it a finite number of times, it’s on when that number is odd and off when that number is even. I also notice the following regularities about cases where an infinite number of togglings happens:

  1. The same sequence (e.g., toggle at 10:30, don’t toggle at 10:45, toggle at 10:52.5, etc.) always produces the same result.

  2. Reversing a finite number of decisions in a sequence produces the same outcome when an even number of decisions is reversed, and the opposite outcome when an odd number of decisions is reversed.

(Of course, 1 is a special case of 2.) How fun! I conclude that 1 and 2 are always going to be true.

Now I set up a supertask machine. It will toss a fair coin just prior to each of my lamp access times, and it will toggle the switch if the coin is heads and not toggle it if it is tails.

Question: What is the probability that the lamp will be on at 11?

“Answer:” Given 1 and 2, the event that the lamp will be on at 11 is not measurable with respect to the standard (completed) product measure on a countable infinity of coin tosses. (See note 1 here.)

So, given supertasks (and hence the falsity of causal finitism), we could find ourselves in a position where we would have to deal with a non-measurable set.

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