Like, I agree that you can have gambles closed under finite but not countable gambling and the math works. But it seems like reality is a countably-additive sort of a place. E.g. if these different outcomes of a lottery are physical states of some system, QM is going to tell you to take some infinite sums. I’m just generally having trouble getting a grasp on what the world (and our epistemic state re. the world) would look like for this finite gambles stuff to make sense.
Note that you can take infinite sums, without being able to take all possible infinite sums.
I suspect it looks like you have a prior distribution, and the allowable probability distributions are those that you can get to from this distribution using finitely many bits of evidence.
Hm, what would that last thing look like?
Like, I agree that you can have gambles closed under finite but not countable gambling and the math works. But it seems like reality is a countably-additive sort of a place. E.g. if these different outcomes of a lottery are physical states of some system, QM is going to tell you to take some infinite sums. I’m just generally having trouble getting a grasp on what the world (and our epistemic state re. the world) would look like for this finite gambles stuff to make sense.
Note that you can take infinite sums, without being able to take all possible infinite sums.
I suspect it looks like you have a prior distribution, and the allowable probability distributions are those that you can get to from this distribution using finitely many bits of evidence.