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prase comments on 5 Axioms of Decision Making - Less Wrong
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If you answered 1 to the first, and anything but 0 or 1 to the second, you're inconsistent. If you're unwilling to answer the second, you just broke your axioms.
If (basic laws = axioms and inference rules), meaning of "true" needs clarification.
0.64 (Here, by "true" I mean "can be proven in Peano arithmetics".)
Then you're enitrely inconsistent, since P(Collatz sequence for k converges) is either 0 or 1 for all k by basic laws of mathematics, and P(Collatz conjecture is true) equals product of these, and by basic laws of mathematics can only be 0 or 1.
Why had you chosen Collatz conjecture to illustrate the fact (which already has been discussed several times) that uncertainty about mathematical statements introduces inconsistency of some sort? I am equally willing to put p = 0.1 to the statement "last decimal digit of 1543! is 7", although in fact this is quite easy to check. Just I don't want to spend time checking.
If for consistency you demand that subjective probabilities assigned to logically equivalent propositions must be equal (I don't dispute that it is sensible to include that to definition of "consistent"), then real people are going to be inconsistent, since they don't have enough processing power to check for consistency. This is sort of trivial. People hold inconsistent beliefs all the time, even when they don't quantify them by probabilities.
If you point to some fine mathematical problems with "ideal Bayesian agents", then I don't see how it is relevant in context of the original post.
Edit: by the way,
sounds frequentistish.
What probabilities are are you willing to assign to statements:
Bayesian probabilities don't give you any anchoring to reality, they only give you consistency.
If you're willing to abandon consistency as well, they give you precisely nothing whatsoever.
Probabilities are a tool for talking about uncertainty, they are not uncertainty, to think otherwise is a ridiculous map-territory confusion.
As ad hominem attacks go, that's an interesting one.
If there's one possible universe where Collatz conjecture is true/false, it is true/false is all other possible universes as well. There are no frequencies there, it's just pure fact of logic.
Updated. (Didn't occur to me it would be so easy.)
It is unnecessarily black-and-white point of view on consistency. I can improve my consistency a lot without becoming completely consistent. In practice we all compartmentalise.
I did certainly not dispute that (if I understand correctly what you mean, which I am not much sure about).
The point was, subjective probability is a degree of belief in the proposition; saying "it must be either 0 or 1 by laws of mathematics" rather implies that it is an objective property of the proposition. This seems to signal that you use a non-subjectivist (not necessarily frequentist, my fault) interpretation of probability. We may be then talking about different things. Sorry for ad hominem impression.