One subtlety that has been bothering me: In the draft’s statement of Theorem 2, we start with a language L and a consistent theory T over L, then go on considering the extension L’ of L with a function symbol P(.). This means that P(.) cannot appear in any axiom schemas in T; e.g., if T is ZFC, we cannot use a condition containing P(.) in comprehension. This seems likely to produce subtle gaps in applications of this theory. In the case of ZFC, this might not be a problem since the function denoted by P(.) is representable by an ordinary set at least in the standard models, so the free parameters in the axiom schemas might save us—though then again, this approach takes us beyond the standard models...
I think not having to think about such twiddly details might be worth making the paper a tad more complicated by allowing T to be in the extended language. I haven’t worked through the details, but I think all that should change is that we need to make sure that A(P) is nonempty, and I think that to get this it should suffice to require that T is consistent with any possible reflection axiom a < P(“phi”) < b. [ETA: …okay, I guess that’s not enough: consider ((NOT phi) OR (NOT psi)), where phi and psi are reflection axioms. Might as well directly say “consistent with A_P for any coherent probability distribution P” then, I suppose.]
I think including a condition with P in comprehension will require more subtlety—for example, such an axiom would not be consistent with most coherent distributions P . If I include the axiom P(x in S) = P(x not in x), then it better be the case that P(S in S) = 1⁄2.
We can still do it, but I don’t see how to factor out the work into this paper. If you see a useful way to generalize this result to include axioms of L’, that would be cool; I don’t see one immediately, which is why it is in this form.
Sorry, we’re miscommunicating somewhere. What I’m saying is that e.g. given a set A of statements, I want the axiom asserting the existence of the set
%20%3E%201/2\}), i.e. the comprehension axiom applied to the condition
%20%3E%201/2). I don’t understand how this would lead to
%20=%20\mathbb{P}(x%20\notin%20x)); could you explain? (It seems like you’re talking about unrestricted comprehension of some sort; I’m just talking about allowing the condition in ordinary restricted comprehension to range over formulas in L’. Maybe the problem you have in mind only occurs in the unrestricted comprehension work which isn’t in this draft?)
Consider my proposed condition that ”T is consistent with A_mathbb{P} for any coherent distribution mathbb{P}”. To see that this is true for ZFC in the language L’, choose a standard model of ZFC in L and, for any function mathbb{P} from the sentences of L’ to
, extend it to a model in L’ by interpreting
) as
); unless I’m being stupid somehow, it’s clear that the extended model will satisfy ZFC-in-L’ + A_mathbb{P}.
It seems to me that the only parts of the proof that need to be re-thought are the arguments that (a) mathcal{A} and (b)
) are non-empty. Perhaps the easiest way to say the argument is that we extend (a) T or (b) TA_mathbb{P′} to some arbitrary complete theory
I understand what you are saying. You are completely right, thanks for the observation. I don’t have time to muck with the paper now, but it looks like this would work.
One subtlety that has been bothering me: In the draft’s statement of Theorem 2, we start with a language L and a consistent theory T over L, then go on considering the extension L’ of L with a function symbol P(.). This means that P(.) cannot appear in any axiom schemas in T; e.g., if T is ZFC, we cannot use a condition containing P(.) in comprehension. This seems likely to produce subtle gaps in applications of this theory. In the case of ZFC, this might not be a problem since the function denoted by P(.) is representable by an ordinary set at least in the standard models, so the free parameters in the axiom schemas might save us—though then again, this approach takes us beyond the standard models...
I think not having to think about such twiddly details might be worth making the paper a tad more complicated by allowing T to be in the extended language. I haven’t worked through the details, but I think all that should change is that we need to make sure that A(P) is nonempty, and I think that to get this it should suffice to require that T is consistent with any possible reflection axiom a < P(“phi”) < b. [ETA: …okay, I guess that’s not enough: consider ((NOT phi) OR (NOT psi)), where phi and psi are reflection axioms. Might as well directly say “consistent with A_P for any coherent probability distribution P” then, I suppose.]
I think including a condition with P in comprehension will require more subtlety—for example, such an axiom would not be consistent with most coherent distributions P . If I include the axiom P(x in S) = P(x not in x), then it better be the case that P(S in S) = 1⁄2.
We can still do it, but I don’t see how to factor out the work into this paper. If you see a useful way to generalize this result to include axioms of L’, that would be cool; I don’t see one immediately, which is why it is in this form.
Sorry, we’re miscommunicating somewhere. What I’m saying is that e.g. given a set A of statements, I want the axiom asserting the existence of the set
Consider my proposed condition that ”T is consistent with A_mathbb{P} for any coherent distribution mathbb{P}”. To see that this is true for ZFC in the language L’, choose a standard model of ZFC in L and, for any function mathbb{P} from the sentences of L’ to
It seems to me that the only parts of the proof that need to be re-thought are the arguments that (a) mathcal{A} and (b)
I understand what you are saying. You are completely right, thanks for the observation. I don’t have time to muck with the paper now, but it looks like this would work.