So this post is basically just collecting together a bunch of things you previously wrote in the Sequences, but I guess it’s useful to have them collected together.
I must, however, take objection to one part. The proper non-circular foundation you want for probability and utility is not the complete class theorem, but rather Savage’s theorem, which I previously wrote about on this website. It’s not short, but I don’t think it’s too inaccessible.
Note, in particular, that Savage’s theorem does not start with any assumption baked in that R is the correct system of numbers to use for probabilities[0], instead deriving that as a conclusion. The complete class theorem, by contrast, has real numbers in the assumptions.
In fact—and it’s possible I’m misunderstanding—but it’s not even clear to me that the complete class theorem does what you claim it does, at all. It seems to assume probability at the outset, and therefore cannot provide a grounding for probability. Unlike Savage’s theorem, which does. Again, it’s possible I’m misunderstanding, but that sure seems to be the case.
Now this has come up here before (I’m basically in this comment just restating things I’ve previously written) and your reply when I previously pointed out some of these issues was, frankly, nonsensical (your reply, my reply), in which you claimed that the statement that one’s preferences form a partial preorder is a stronger assumption than “one prefers more apples to less apples”, when, in fact, the exact reverse is the case.
(To restate it for those who don’t want to click through: If one is talking solely about one’s preferences over number of apples, then the statement that more is better immediately yields a total preorder. And if one is talking about preferences not just over number of apples but in general, then… well, it’s not clear how what you’re saying applies directly; and taken less literally, it just in general seems to me that the complete class theorem is making some very strong assumptions, much stronger than that of merely a total preorder (e.g., real numbers!).)
In short the use of the complete class theorem here in place of Savage’s theorem would appear to be an error and I think you should correct it.
[0]Yes, it includes an Archimedean assumption, which you could argue is the same thing as baking in R; but I’d say it’s not, because this Archimedean assumption is a direct statement about the agent’s preferences, whereas it’s not immediately clear what picking R as your number system means as a statement about the agent’s preferences.
So this post is basically just collecting together a bunch of things you previously wrote in the Sequences, but I guess it’s useful to have them collected together.
I must, however, take objection to one part. The proper non-circular foundation you want for probability and utility is not the complete class theorem, but rather Savage’s theorem, which I previously wrote about on this website. It’s not short, but I don’t think it’s too inaccessible.
Note, in particular, that Savage’s theorem does not start with any assumption baked in that R is the correct system of numbers to use for probabilities[0], instead deriving that as a conclusion. The complete class theorem, by contrast, has real numbers in the assumptions.
In fact—and it’s possible I’m misunderstanding—but it’s not even clear to me that the complete class theorem does what you claim it does, at all. It seems to assume probability at the outset, and therefore cannot provide a grounding for probability. Unlike Savage’s theorem, which does. Again, it’s possible I’m misunderstanding, but that sure seems to be the case.
Now this has come up here before (I’m basically in this comment just restating things I’ve previously written) and your reply when I previously pointed out some of these issues was, frankly, nonsensical (your reply, my reply), in which you claimed that the statement that one’s preferences form a partial preorder is a stronger assumption than “one prefers more apples to less apples”, when, in fact, the exact reverse is the case.
(To restate it for those who don’t want to click through: If one is talking solely about one’s preferences over number of apples, then the statement that more is better immediately yields a total preorder. And if one is talking about preferences not just over number of apples but in general, then… well, it’s not clear how what you’re saying applies directly; and taken less literally, it just in general seems to me that the complete class theorem is making some very strong assumptions, much stronger than that of merely a total preorder (e.g., real numbers!).)
In short the use of the complete class theorem here in place of Savage’s theorem would appear to be an error and I think you should correct it.
[0]Yes, it includes an Archimedean assumption, which you could argue is the same thing as baking in R; but I’d say it’s not, because this Archimedean assumption is a direct statement about the agent’s preferences, whereas it’s not immediately clear what picking R as your number system means as a statement about the agent’s preferences.