If one is talking about one’s preferences over number of apples, then the statement that it is a total preorder, is a weaker statement than the statement that more is better. (Also, you know, real number assumptions all over the place.) If one is talking about preferences not just over number of apples but in general, then even so it seems to me that the complete class theorem seems to be making some very strong assumptions, much stronger than the assumption of a total preorder! (Again, look at all those real number assumptions.)

Moreover it’s not even clear to me that the complete class theorem does what you claim it does, like, at all. Like it starts out assuming the notion of probability. How can it ground probability when it starts out assuming it? And perhaps I’m misunderstanding, but are the “risk functions” it discusses not in utility? It sure looks like expected values of them are being taken with the intent that smaller is better (this seems to be implicit in the definition of r(θ), that r(θ) is measured by expected value when T isn’t a pure strategy). Is that mistaken?

(Possible source of error here: I can’t seem to find a statement of the complete class theorem that fits neatly into Savage/VNM/Cox/etc-style formalism and I’m having some trouble translating it to such, so I may be misunderstanding. The most sense I’m making of it at the moment is that it’s something like your examples for why probabilities must sum to one—i.e., it’s saying, if you already believe in utility, and something almost like probability, it must actually be probability. Is that accurate, or am I off?)

(Edit: Also if you’re taking issue with the preorder assumption, does this mean that you no longer consider VNM to be a good grounding of the notion of utility for those who already accept the idea of probability?)

I can’t make sense of this comment.

If one is talking about one’s preferences over number of apples, then the statement that it is a total preorder, is a weaker statement than the statement that more is better. (Also, you know, real number assumptions all over the place.) If one is talking about preferences not just over number of apples but in general, then even so it seems to me that the complete class theorem seems to be making some very strong assumptions, much stronger than the assumption of a total preorder! (Again, look at all those real number assumptions.)

Moreover it’s not even clear to me that the complete class theorem does what you claim it does, like, at all. Like it starts out assuming the notion of probability. How can it ground probability when it starts out assuming it? And perhaps I’m misunderstanding, but are the “risk functions” it discusses not in utility? It sure looks like expected values of them are being taken with the intent that smaller is better (this seems to be implicit in the definition of r(θ), that r(θ) is measured by expected value when T isn’t a pure strategy). Is that mistaken?

(Possible source of error here: I can’t seem to find a statement of the complete class theorem that fits neatly into Savage/VNM/Cox/etc-style formalism and I’m having some trouble translating it to such, so I may be misunderstanding. The most sense I’m making of it at the moment is that it’s something like your examples for why probabilities must sum to one—i.e., it’s saying, if you already believe in utility, and something almost like probability, it must actually be probability. Is that accurate, or am I off?)

(Edit: Also if you’re taking issue with the preorder assumption, does this mean that you no longer consider VNM to be a good grounding of the notion of utility for those who already accept the idea of probability?)