There are definitely well-defined measures on any set (e.g. pick one atomic outcome to have probability 1 and the rest 0); there’s just not only one, and picking exactly one would be arbitrary. But the same is true for any set of outcomes with at least two outcomes, including finite ones (or it’s at least often arbitrary when there’s not enough symmetry for equiprobability).
For the question of how many people will exist in the future, you could use a Poisson distribution. That’s well-defined, whether or not it’s a reasonable distribution to use.
Of course, trying to make your space more and more specific will run into feasibility issues.
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
If you’re not taking a measure theoretic approach, and instead using propositions (which I guess, it should be assumed that you are, because this approach grounds Bayesianism), then using infinite sets (which clearly one would have to do if reasoning about all possible futures) leads to paradoxes. As E.T. Jaynes writes in Probability Theory and the Logic of Science:
It is very important to note that our consistency theorems have been established only for probabilities assigned on finite sets of propositions … In laying down this rule of conduct, we are only following the policy that mathematicians from Archimedes to Gauss have considered clearly necessary for nonsense avoidance in all of mathematics. (pg. 43-44).
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
using infinite sets (which clearly one would have to do if reasoning about all possible futures)
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.
There are definitely well-defined measures on any set (e.g. pick one atomic outcome to have probability 1 and the rest 0); there’s just not only one, and picking exactly one would be arbitrary. But the same is true for any set of outcomes with at least two outcomes, including finite ones (or it’s at least often arbitrary when there’s not enough symmetry for equiprobability).
For the question of how many people will exist in the future, you could use a Poisson distribution. That’s well-defined, whether or not it’s a reasonable distribution to use.
Of course, trying to make your space more and more specific will run into feasibility issues.
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
If you’re not taking a measure theoretic approach, and instead using propositions (which I guess, it should be assumed that you are, because this approach grounds Bayesianism), then using infinite sets (which clearly one would have to do if reasoning about all possible futures) leads to paradoxes. As E.T. Jaynes writes in Probability Theory and the Logic of Science:
(Vaden makes this point in the podcast.)
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.