Technical nitpick: I don’t think it’s the fact that the set of possible futures is infinite that breaks things, it’s the fact that the set of possible futures includes futures which differ infinitely in their value, or have undefined values or can’t be compared, e.g. due to infinities, or conditional convergence and no justifiably privileged summation order. Having just one future with undefined value, or a future with +∞ and another with −∞ is enough to break everything; that’s only 1 or 2 futures. You can also have infinitely many futures without things breaking, e.g. as long as the expectations of the positive and negative parts are finite, which doesn’t require bounded value, but is guaranteed by it.
Both actually! See section 6 in Making Ado Without Expectations—unmeasurable sets are one kind of expectation gap (6.2.1) and ‘single-hit’ infinities are another (6.1.2)
Technical nitpick: I don’t think it’s the fact that the set of possible futures is infinite that breaks things, it’s the fact that the set of possible futures includes futures which differ infinitely in their value, or have undefined values or can’t be compared, e.g. due to infinities, or conditional convergence and no justifiably privileged summation order. Having just one future with undefined value, or a future with +∞ and another with −∞ is enough to break everything; that’s only 1 or 2 futures. You can also have infinitely many futures without things breaking, e.g. as long as the expectations of the positive and negative parts are finite, which doesn’t require bounded value, but is guaranteed by it.
If a Bayesian expected utility maximizing utilitarian accepts Cromwell’s rule, as they should, they can’t rule out infinities, and expected utility maximization breaks. Stochastic dominance generalizes EU maximization and can save us in some cases.
Both actually! See section 6 in Making Ado Without Expectations—unmeasurable sets are one kind of expectation gap (6.2.1) and ‘single-hit’ infinities are another (6.1.2)
When would you need to deal with unmeasurable sets in practice? They can’t be constructed explicitly, i.e. with just ZF without the axiom of choice, at least for the Lebesgue measure on the real numbers (and I assume this extends to Rn, but I don’t know about infinite-dimensional spaces). I don’t think they’re a problem.
You’re correct, in practice you wouldn’t—that’s the ‘instrumentalist’ point made in the latter half of the post