Do you assign probability 1 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator? (well, computer maybe. For most calculators testing this proposition would be somewhat tedious)

If yes, would you willing to pay me $10 if you ever found out it was not prime?

Conversely

Do you assign probability 0 to the proposition that 182547553 is prime? Right now, without doing an experiment on a calculator?

If yes, would you willing to pay me $10 if you ever found out it was prime?

EDIT: Actually, I suppose this counts as "doing it again", even though I'm not Peter de Blanc. I think that makes me a bad bad person.

I suggest you look up the concept of "subjective Bayesian". Probabilities are states of knowledge. If you don't know an answer, it's uncertain. If someone who doesn't know anything you don't can look over your odds and construct a knowably losing bet anyway, or construct a winning bet that you refuse, then you are Dutch-bookable.

Also, considering that you have apparently been reading this site for years and you still have not grasped the concept of subjective uncertainty, and you are still working with a frequentist notion of probability, nor yet have you even grasped the difference, I would suggest to you in all seriousness that you seek enlightenment elsewhere.

(Sorry, people, there's got to be some point at which I can express that. Downvote if you must, I suppose, if you think such a concept is unspeakable or unallowed, but it's not a judgment based on only one case of incomprehension, of course.)

If physics is deterministic then conditional on the state of the world at the time you make the bet, the probability of heads is either 0 or 1. The only disanalogy with your example is that you may not already have sufficient information to determine how the coin will land (which isn't even a disanalogy if we assume that the person doesn't know what the Collatz conjecture says). But suppose you did have that information - there would be vastly more of it than you could process in the time available, so it wouldn't affect your probability assignments. (Note: The case where the Collatz conjecture turns out to be true but unprovable is analogous to the case where the laws of physics are deterministic but 'uncomputable' in some sense.)

Anyway, the real reason why I want to resist your line of argument here is due to Chaitin's number "omega", the "halting probability". One can prove that the bits in the binary expansion of omega are algorithmically incompressible. More precisely: In order to deduce n bits' worth of information about omega you need at least n - k bits' worth of axioms, for some constant k. Hence, if you look sufficiently far along the binary expansion of omega, you find yourself looking at an infinite string of "random mathematical facts". One "ought" to treat these numbers as having probability 1/2 of being 1 and 1/2 of being 0 (if playing games against opponents who lack oracle powers).

Excellent - now you've explained what you mean by "Perfect Bayesian" it all makes sense! (Though I can't help thinking it would have saved time if you'd said this earlier.)

Still, I'm not keen on this redefinition of the word 'metaphysics', as though your philosophy of mathematics were 'obviously correct' 'received wisdom', when actually it's highly contentious.

Anyway, I think this is a successful attack on a kind of "Bayesian absolutism" which claims that beings who (explicitly or implicitly) assign consistent probabilities to all expressible events, and update their beliefs in the Bayesian manner, can actually exist. That may be a straw man, though.

The obvious solution:

Y is information. It is not information about the world, but it is information - information about math.

Ok, I understood that, but I still don't see what it has to do with Dutch books.

The trouble is that if the Collatz conjecture is true then {"Collatz conjecture is true"} constitutes an exhaustive set of outcomes, whose probability you think is only 1/2.

And if the Collatz conjecture is false then {"Collatz conjecture is false"} constitutes a single outcome, whose probability I think is 1/2. As of now I don't know which premise (if true, if false) is actually the case, so I represent my uncertainty about the premises with a probability of 0.5. Representing uncertainty about an outcome (even an outcome that we would know if we were logically omniscient) does not make you dutch-bookable; incorrectly determining your uncertainty makes you dutch-bookable.

"Should PB accept bet X that PB will reject bet X?"

Even easier. No. "I'll bet you a dollar you won't take this bet." "So if I take it, I lose a dollar, and if I don't take it, I lose nothing? I reject the bet. Having rejected the bet, you will now proceed to point out that I would have won the bet. I will now proceed to show you that taking the bet closes off the path where I win the bet (it satisfies the failure condition), and rejecting the bet closes off the path where I win the bet (can't win a bet you didn't take), so my choices are between losing the bet and not taking the bet. Out of the only two options available to me, losing the bet costs me a dollar, not taking the bet costs me nothing. Quod erat demonstrandum."