What do you think about the St Petersburg problem now?

I think your arguments re infinities conflict with Occam's razor, and the principle of indifference should be applied across models within the same complexity (or submodels), otherwise you will assign 0 or too little credence to simpler models that are special cases, e.g. parameter value=0 or effectively eliminating some type of feature. There are infinitely many ways the universe could be more complex, and arbitrarily more complex, than you'd guess.

A flat universe effectively has one fewer parameter and is simpler. So it shouldn't get 0 credence. Among flat universes, the bounded/finite ones also have extra parameters for the boundaries or shape of the universe (compared to something that looks like R^3), so the infinite one shouldn't get 0 credence.

(There probably are many ways for the universe to be infinite spatially in its shape, too, but those are more complex than R^3.)

EDIT:

As an aside, there have been many cases where quantities in physics were assumed to be 0, but then turned out to be just small values, like the mass of neutrinos. It often makes sense to round a quantity to 0 for simplicity, but a sufficiently small value would explain exactly the same empirical evidence.

Note that I never suggested to assign 0 probability to anything. I think this leads to further examples to illustrate my point:

Would you assign exactly 0 probability to photons having exactly 0 mass?

Exactly 0 probability to there being no additional fundamental force, because it could just be vanishingly weak? And shouldn't this get you to infinitely many fundamental forces? For any finite set of fundamental forces, you could posit another one and just say it's very weak.

Exactly 0 probability to there not being ghosts, because their effects could just be very small or rare?

I think consistently applying your arguments suggest you should assign 0 probability in these cases, and so your models blow up in complexity and you become too credulous, contrary to Occam's razor.

I like the principle of indifference. However, I think infinities are unfalsifiable in principle. In this case, does it make sense for me to attribute probabilities to them? If I did, they would be just metaphysical priors that can never be updated by any evidence.

St Petersburg-like lotteries, defined in terms of your Bayesian credences, don't require assigning positive probability to any possible infinities out there in the world.

I'll leave the rest in a footnote, because it's not that relevant to the point I've been making so don't plan to go further with it, but I already wrote it and it may be of interest to you.^[1]

1. ^{^}
   1. I think falsifiability for a Bayesian could just mean we can imagine evidence that would warrant assigning very very low credences to the hypothesis. Actual infinities are often falsifiable in principle in this way, in specific cases. I think the probability you should assign to you being infinite in spatial size is extremely small. Similarly for the Earth being infinite in size.
   2. In the case of the spatial extent of the universe being infinite, it seems hard to falsify now only because all the evidence we've gathered so far is in fact consistent with it (or no less consistent than with a bounded/finite universe), and it remains one of the standard models used by experts. My own view is that it’s more likely than not infinite in spatial extent, because it’s the simplest model consistent with the evidence.
   3. I think our credences that the universe is infinite in spatial extent should increase with our credences that the universe is globally flat (0 global curvature), which has been measurable. If we had good evidence of nonzero global curvature, that would be decent (but not overwhelming) evidence that the universe is finite/bounded, because it would rule out the most plausible infinite models of the universe.
   4. I agree that we can sometimes compare actions with infinite expected cost-effectiveness. I'm most partial to expansionist views of some kind, as the most complete (given precise probabilities).

I object to relying on infinities (not arbitrarily large finities) to guide decisions because they do not explain more empirical evidence than arbitrarily large finities.

But do they deserve to be privileged at the exclusion of the infinite EV distributions if and when the latter are at least as consistent with the evidence? Why? Shouldn't you use some principle of indifference or symmetry here?

If there's no finite upper bound on what the value the specific probability distribution can take, how can you be 100% confident it is not a probabilistic mixture with a distribution with infinite or undefined EV (but finite for every actual value), like 0.0001% probability to it being drawn from something like a St. Petersburg lottery?

The expected value of the St. Petersburg lottery is 1 + 1 + ... = +inf. It involves finite terms, but infinitely many terms. I meant to relate f(x) = x in my comment to the expected value of the St. Petersburg lottery. If this involved an arbitrarily large number of terms, its expected value would be arbitrarily large, but not infinite. 

I don't understand what you're objecting to. Are you saying all probability distributions should only take finitely many possible values? If so, do you have an upper bound on what the total value should be over all of the future and possible existence, that you're 100% confident in, under your course of action if where you do nothing? (It doesn't have to be the least upper bound / tight.)

Below is how Claude thinks Adam (the author of the article) would object to your comments. The objections make sense to me. Any reactions?

Claude is dumb (at least without further critique and verification, and usually with), and your prompt basically put it on the task of defending the position, not actually assessing the arguments fairly. So it turned up bad arguments.

I doubt the author would respond this badly.

Taking bet A doesn’t require any commitment. My argument just uses backward induction (+ignoring statewise incomparability), which you should generally use in sequential choice situations, or else you'll be worse off in many situations, even with sharp probabilities.

It allows unsharpness. Having unsharp probabilities does not require sequential decisions to be made independently.

that very concession is what Elga reads as victory: the imprecision has stopped doing the one thing it was introduced to do.

The argument against unsharp probabilities is defeated. We just have to treat them in certain ways. The summary of the paper here missed one way we could treat them, and claimed too much against another (if we accept commitments or resolute choice in other cases).

Solution to which problem? I am not sure what is supposed to be problematic.

That if you use backward induction on acting rationally at each step, you will be worse off. You will predict later that you'll change your mind, unless you can force your future self to honor a commitment (or plan) you'd no longer want to keep when it actually comes time to honor it.

EDIT: my bad, the problem is that if you don't use commitments, you could be worse off. Using backward induction in the Sally argument actually works fine, doesn't leave you (or Sally) worse off and doesn't require any commitment.

However, I do not think it can be infinite. The function f(x) = x can take an arbitrarily large value, but not an infinite value (its range is the set of real numbers).

St Petersburg doesn't require any state to have infinite value. Its value is (canonically) 2^n with probability 1/2^n for each n at least 1. Always finite actual value, but infinite expected value.

Why does Parfit's hitchhiker pose a problem?

Because the same kind of solution is available to someone with unsharp probabilities in Elga's scenario, if you're treating them fairly.

The St. Petersburg paradox involves an infinite expected payoff, and I reject infinite worlds.

It doesn't require an infinite world, only that you can't be 100% confident in any finite upper bound on your impact that you specify, and that there are infinitely many ways that the world could be (due largely to not full certainty about physics).

(But also 0% to infinite worlds seems epistemically immodest, doesn't treat the evidence on each side fairly, and is poorly argued, imo. But I don’t want to rehash this.)

In Sally's case, her money is the only thing that matters.

Why can't the fact that she'd pick a dominated sequence or regret it if she rejects both bets matter to her after rejecting bet A?

I understand one should accept bet A based on that strategy. However, unsharp probabilities are supposed to allow for accepting or rejecting A?

They don't have to in every case. If it was A in isolation, and no other decisions, then yes, both rejecting and accepting should be permissible. But that's not the case presented to us.

As a rational actor with no useful information

mood

3. SEQUENCE — Sequences of actions can be assessed for rationality independently of their parts: each of "reject A" and "reject B" can be individually permissible while the sequence "reject-A-then-reject-B" is impermissible. Elga turns the same Sally argument on it. SEQUENCE makes rejecting Bet B fine when no Bet A preceded it but irrational when it would complete the bad sequence—so it, too, imposes different requirements across two situations Sally can see are identical in everything she cares about. Hence SEQUENCE fails.

I don't think this is a strong argument. There are other cases where you should make commitments that you would later be inclined to break, like Parfit's hitchhiker, and St. Petersburg lotteries with unbounded utility functions. The latter is an argument that unbounded utility functions are irrational, based on similar logic.

Furthermore, "imposes different requirements across two situations Sally can see are identical in everything she cares about". What if I do care about the differences? Or, is this any worse than picking numbers to ensure precision for no better reason than that they occured to you? Because that's what it takes to produce arbitrarily precise probabilities if you fix what information is available to you in realistic settings.

Also, here's another way someone with unsharp probabilities might handle this situation. In summary, I should accept bet A at the start to rule out the possibility of picking a dominated sequence:

1. If I accept bet A at the start, then the probability that I pick the dominated sequence (rejecting both) is 0.
2. If I reject bet A at the start and if I can't guarantee that I will accept bet B next, then there's some chance that I pick the dominated sequence.

If I compare 1 and 2 statewise, then 1 > 2 with some probability, and 1 and 2 are incomparable otherwise. In other words, either 1 beats 2, or I have no decisive reasons favouring either and I can ignore those cases. So I decide on the cases where 1 beats 2 and accept bet A at the start.