The extended non-negative reals really don’t do what the OP was looking for. They won’t even allow you to trade 1 life to save 10.000 lives, let alone have a hierarchy of values, some of which are tradable against each other and some of which are not.
Indeed, they certainly don’t. My point here isn’t “here is how you fix the problem with limits while still getting the things OP wanted”. My point here is “here is how you fix the problem with limits”. I make no claim that it is possible or desirable to get the things OP wanted. But yes I suppose it is possible that there may be some way to do so without completely screwing up limits, if we use a weird notion of limits.
Going back to the (extended) reals that do nothing interesting doesn’t strike me as a meaningful way of “fixing the problem with limits” in this context, when everybody knows that limits work for those… It doesn’t really fix any problem at all, it just says you can’t do certain things (namely, go beyond the (extended) reals) because that makes the problem come up.
Yes, that’s kind of my point. I’m not trying to do what the OP wanted and come up with a system of infinities that work nicely for this purpose. I’m trying to point out that there are very good reasons that we usually stick to the extended reals for this, that there are very real problems that crop up when you go beyond it, and that become especially problematic when you jump to the end and go all the way to the surreals.
I’m not trying to fix problems raised in the original post; I’m trying to point out that these are serious problems that the original post didn’t acknowledge—and the usual way we fix these is just not going beyond the extended reals at all so that they don’t crop up in the first place, because these really are serious problems. The ultimate problem here is coming up with a decision theory—or here just a theory of utility—and in that context, fixing the problem by abandoning goals that aren’t satisfiable and accepting the trivial solution that is forced on you is still fixing the problem. (Depending on just what you require, sticking to the extended reals may not be totally forced on you, but it is hard to avoid and this is a problem that the OP needs to appreciate.)
The point isn’t “this is how you fix the problem”, the point is “take a step back and get an appreciation for the problem and for what you’re really suggesting before you go rushing ahead like that”. The point isn’t “limits work in the extended reals”, the point is “limits work a lot less well if you go beyond there”. I personally think the whole idea is misguided and utilities should be bounded; but that is a separate argument. But if the OP really does want a viable theory along the lines he’s suggesting here even more than he wants the requirements that force the extended reals on us, then he’s got a lot more work to do.
The extended non-negative reals really don’t do what the OP was looking for. They won’t even allow you to trade 1 life to save 10.000 lives, let alone have a hierarchy of values, some of which are tradable against each other and some of which are not.
Indeed, they certainly don’t. My point here isn’t “here is how you fix the problem with limits while still getting the things OP wanted”. My point here is “here is how you fix the problem with limits”. I make no claim that it is possible or desirable to get the things OP wanted. But yes I suppose it is possible that there may be some way to do so without completely screwing up limits, if we use a weird notion of limits.
Going back to the (extended) reals that do nothing interesting doesn’t strike me as a meaningful way of “fixing the problem with limits” in this context, when everybody knows that limits work for those… It doesn’t really fix any problem at all, it just says you can’t do certain things (namely, go beyond the (extended) reals) because that makes the problem come up.
Yes, that’s kind of my point. I’m not trying to do what the OP wanted and come up with a system of infinities that work nicely for this purpose. I’m trying to point out that there are very good reasons that we usually stick to the extended reals for this, that there are very real problems that crop up when you go beyond it, and that become especially problematic when you jump to the end and go all the way to the surreals.
I’m not trying to fix problems raised in the original post; I’m trying to point out that these are serious problems that the original post didn’t acknowledge—and the usual way we fix these is just not going beyond the extended reals at all so that they don’t crop up in the first place, because these really are serious problems. The ultimate problem here is coming up with a decision theory—or here just a theory of utility—and in that context, fixing the problem by abandoning goals that aren’t satisfiable and accepting the trivial solution that is forced on you is still fixing the problem. (Depending on just what you require, sticking to the extended reals may not be totally forced on you, but it is hard to avoid and this is a problem that the OP needs to appreciate.)
The point isn’t “this is how you fix the problem”, the point is “take a step back and get an appreciation for the problem and for what you’re really suggesting before you go rushing ahead like that”. The point isn’t “limits work in the extended reals”, the point is “limits work a lot less well if you go beyond there”. I personally think the whole idea is misguided and utilities should be bounded; but that is a separate argument. But if the OP really does want a viable theory along the lines he’s suggesting here even more than he wants the requirements that force the extended reals on us, then he’s got a lot more work to do.