What if X(Y is consistent)=0.5? Then Y(2+2=5) = 0.5, and Y might or might not be inconsistent.
Another solution is of course to let X be incomplete, and refuse to assign X(Y is consistent). In fact, that would be the sensible thing to do. X can never be a function from ″all″ statements to probabilities, it’s domain should only include statements strictly smaller than X itself.
If Y(2 + 2 = 5) = 0.5, Y is still blatantly inconsistent, so that won’t help.
I think your second point might be right, though. Isn’t it the case that the language of first-order arithmetic is not powerful enough to refer to arbitrary probability assignments over its statements? After all, there are an uncountable number of such assignments, and only a countable number of well-formed formulas in the language. So I don’t see why a probability assignment X in a model that includes Peano arithmetic must also assign probabilities to statements like “Y is consistent”.
If you let X be incomplete like twanvl suggests, then you pretty much agree with my position of using probability as a useful tool, and disagree with Bayesian fundamentalism.
Getting into finer points of what is constructible or provable in what language is really not a kind of discussion we could usefully have within confines of lesswrong comment boxes, since we would need to start by formalizing everything far more than we normally do. And it wouldn’t really work, it is simply not possible to escape Goedel’s incompleteness theorem if you have something even slightly powerful and non-finite, it will get you one way or another.
I’m possibly being obtuse here, but I still don’t see the connection to the incompleteness theorem. I don’t deny that any consistent theory capable of expressing arithmetic must be incomplete, but what does that have to do with the argument you offered above? That argument doesn’t hinge on incompleteness, as far as I can see.
And it wouldn’t really work, it is simply not possible to escape Goedel’s incompleteness theorem if you have something even slightly powerful and non-finite, it will get you one way or another.
This is slightly exaggerated. The theory of real numbers is non-finite and quite powerful, but it has a complete axiomatization.
If you let X be incomplete like twanvl suggests, then you pretty much agree with my position of using probability as a useful tool, and disagree with Bayesian fundamentalism.
How is a distribution useful if it refuses to answer certain questions? I think I’m misunderstanding something you said, since I think that the essence of Bayesianism is the idea that probabilities must be used to make decisions, while you seem to be contrasting these two things.
What if X(Y is consistent)=0.5? Then Y(2+2=5) = 0.5, and Y might or might not be inconsistent.
Another solution is of course to let X be incomplete, and refuse to assign X(Y is consistent). In fact, that would be the sensible thing to do. X can never be a function from ″all″ statements to probabilities, it’s domain should only include statements strictly smaller than X itself.
If Y(2 + 2 = 5) = 0.5, Y is still blatantly inconsistent, so that won’t help.
I think your second point might be right, though. Isn’t it the case that the language of first-order arithmetic is not powerful enough to refer to arbitrary probability assignments over its statements? After all, there are an uncountable number of such assignments, and only a countable number of well-formed formulas in the language. So I don’t see why a probability assignment X in a model that includes Peano arithmetic must also assign probabilities to statements like “Y is consistent”.
If you let X be incomplete like twanvl suggests, then you pretty much agree with my position of using probability as a useful tool, and disagree with Bayesian fundamentalism.
Getting into finer points of what is constructible or provable in what language is really not a kind of discussion we could usefully have within confines of lesswrong comment boxes, since we would need to start by formalizing everything far more than we normally do. And it wouldn’t really work, it is simply not possible to escape Goedel’s incompleteness theorem if you have something even slightly powerful and non-finite, it will get you one way or another.
I’m possibly being obtuse here, but I still don’t see the connection to the incompleteness theorem. I don’t deny that any consistent theory capable of expressing arithmetic must be incomplete, but what does that have to do with the argument you offered above? That argument doesn’t hinge on incompleteness, as far as I can see.
This is slightly exaggerated. The theory of real numbers is non-finite and quite powerful, but it has a complete axiomatization.
How is a distribution useful if it refuses to answer certain questions? I think I’m misunderstanding something you said, since I think that the essence of Bayesianism is the idea that probabilities must be used to make decisions, while you seem to be contrasting these two things.