You should check out Abram’s post on complete class theorems. He specifically addresses some of the concerns you mentioned in the comments of Yudkowsky’s posts.
Also, my inner model of Jaynes says that the right way to handle infinities is not to outlaw them, but to be explicit and consistent about what limits we’re taking.
You should check out Abram’s post on complete class theorems. He specifically addresses some of the concerns you mentioned in the comments of Yudkowsky’s posts.
So, it looks to me like what Abrams is doing—once he gets past the original complete class theorem—is basically just inventing some new formalism along the lines of Savage. I think it is very misleading to refer to this as “the complete class theorem”—how on earth was I supposed to know that this was what was being referred to when “the complete class theorem” was mentioned, when it resembles the original theorem so little (and it’s the original theorem that was linked to)? -- and I don’t see why it was necessary to invent this anew, but sure, I can accept that it presumably works, even if the details aren’t spelled out.
But I must note that he starts out by saying that he’s only considering the case when there’s only a finite set of states of the world! I realize you weren’t making a point about bounded utility here; but from that point of view, it is quite significant...
Also, my inner model of Jaynes says that the right way to handle infinities is not to outlaw them, but to be explicit and consistent about what limits we’re taking.
I don’t really understand what that means in this context. It is already quite explicit what limits we’re taking: Given an action (a measurable function from states of the world to outcomes), take its expected utility, with regard to the [finitely-additive] probability on states of the world. (Which is implicitly a limit of sorts.)
I think this is another one of those comments that makes sense if you’re reasoning backward, starting from utility functions, but not if you’re reasoning forward, from preferences. If you look at things from a utility-functions-first point of view, then it looks like you’re outlawing infinities (well, unboundedness that leads to infinities). But from a preferences-first point of view, you’re not outlawing anything. You haven’t outlawed unbounded utility functions, rather they’ve just failed to satisfy fundamental assumptions about decision-making (remember, if you don’t have P7 your utility function is not guaranteed to return correct results about infinite gambles at all!) and so clearly do not reflect your idealized preferences. You didn’t get rid of the infinity, it was simply never there in the first place; the idea that it might have been turned out to be mistaken.
You should check out Abram’s post on complete class theorems. He specifically addresses some of the concerns you mentioned in the comments of Yudkowsky’s posts.
Also, my inner model of Jaynes says that the right way to handle infinities is not to outlaw them, but to be explicit and consistent about what limits we’re taking.
So, it looks to me like what Abrams is doing—once he gets past the original complete class theorem—is basically just inventing some new formalism along the lines of Savage. I think it is very misleading to refer to this as “the complete class theorem”—how on earth was I supposed to know that this was what was being referred to when “the complete class theorem” was mentioned, when it resembles the original theorem so little (and it’s the original theorem that was linked to)? -- and I don’t see why it was necessary to invent this anew, but sure, I can accept that it presumably works, even if the details aren’t spelled out.
But I must note that he starts out by saying that he’s only considering the case when there’s only a finite set of states of the world! I realize you weren’t making a point about bounded utility here; but from that point of view, it is quite significant...
I don’t really understand what that means in this context. It is already quite explicit what limits we’re taking: Given an action (a measurable function from states of the world to outcomes), take its expected utility, with regard to the [finitely-additive] probability on states of the world. (Which is implicitly a limit of sorts.)
I think this is another one of those comments that makes sense if you’re reasoning backward, starting from utility functions, but not if you’re reasoning forward, from preferences. If you look at things from a utility-functions-first point of view, then it looks like you’re outlawing infinities (well, unboundedness that leads to infinities). But from a preferences-first point of view, you’re not outlawing anything. You haven’t outlawed unbounded utility functions, rather they’ve just failed to satisfy fundamental assumptions about decision-making (remember, if you don’t have P7 your utility function is not guaranteed to return correct results about infinite gambles at all!) and so clearly do not reflect your idealized preferences. You didn’t get rid of the infinity, it was simply never there in the first place; the idea that it might have been turned out to be mistaken.