Could you elaborate? Do you mean in the cases where an infinite number of possible worlds satisfy one’s background knowledge?
Yes, precisely.
And what do you mean by
regard some measure on the set of possible worlds as fundamental
Well, I’ll explain this. If there are finitely many possible worlds, you can get away with just saying that each possible world has exactly the same probability, in the absence of observations. If there are infinitely many, however, this is mathematically impossible; you must decide on some function that, given a set of some possible worlds, gives the probability of that set. Such a function is called a probability measure.
To see why it’s necessary to decide on such a function, suppose that there is some number X, and the only thing that you know about X is that it’s a real number. What’s the probability that X lies between 0 and 1? Between 1 and 2? Between Graham’s number, and Graham’s number plus one? It’s simply impossible to assign them all equal probabilities (at least, if you insist that your probabilities be real numbers).
I doubt that physically implementable agents ever actually compute an infinite number of possible worlds, but if we do, then MSF theory has to deal with it, or be likely wrong. One way of dealing with this is to do bayesian algebra with the hyper real or surreal numbers (I’m guessing) but I’m not sure that you can do that mathematically, or that “P(a)=x where x is a hyperreal (or surreal number)” makes any sense. Another argument you could make is that, for some reason, some particular probability measure is privileged, or that some particular probability measure is the one you should use when dealing with an infinite number of worlds. Though I guess we shouldn’t be too surprised if the probabilities in the mind don’t actually take on real values, we could in theory just use “high”, “low”, “higher than”, “lower than”, “necessary” and “impossible”.
But this is definitely something worth thinking about if I, or anyone else for that matter, wants to take MSF theory as a serious hypothesis, and not just a dandy intuition pump.
One way of dealing with this is to do bayesian algebra with the hyper real or surreal numbers (I’m guessing) but I’m not sure that you can do that mathematically, or that “P(a)=x where x is a hyperreal (or surreal number)” makes any sense.
“Nonstandard analysis” is not a substantive departure from the standard real number system; it is simply an alternative language that some people like for aesthetic reasons, or can sometimes be useful for “bookkeeping”. Basically, if nonstandard analysis solves your problem, there was already a solution in terms of standard analysis. Robinson’s “hyperreals” essentially just replace the concept of a “limit”, and do not represent a fundamentally “new kind of number” in the way that, say, Cantor’s transfinite ordinals do.
Conway’s surreals are a different story. However, if the sorts of problems you’re talking about could be solved simply by saying “oh, just use that other number system over there”, they would have been solved long ago (and everybody would probably be using that other number system).
Yes, precisely.
Well, I’ll explain this. If there are finitely many possible worlds, you can get away with just saying that each possible world has exactly the same probability, in the absence of observations. If there are infinitely many, however, this is mathematically impossible; you must decide on some function that, given a set of some possible worlds, gives the probability of that set. Such a function is called a probability measure.
To see why it’s necessary to decide on such a function, suppose that there is some number X, and the only thing that you know about X is that it’s a real number. What’s the probability that X lies between 0 and 1? Between 1 and 2? Between Graham’s number, and Graham’s number plus one? It’s simply impossible to assign them all equal probabilities (at least, if you insist that your probabilities be real numbers).
I doubt that physically implementable agents ever actually compute an infinite number of possible worlds, but if we do, then MSF theory has to deal with it, or be likely wrong. One way of dealing with this is to do bayesian algebra with the hyper real or surreal numbers (I’m guessing) but I’m not sure that you can do that mathematically, or that “P(a)=x where x is a hyperreal (or surreal number)” makes any sense. Another argument you could make is that, for some reason, some particular probability measure is privileged, or that some particular probability measure is the one you should use when dealing with an infinite number of worlds. Though I guess we shouldn’t be too surprised if the probabilities in the mind don’t actually take on real values, we could in theory just use “high”, “low”, “higher than”, “lower than”, “necessary” and “impossible”.
But this is definitely something worth thinking about if I, or anyone else for that matter, wants to take MSF theory as a serious hypothesis, and not just a dandy intuition pump.
“Nonstandard analysis” is not a substantive departure from the standard real number system; it is simply an alternative language that some people like for aesthetic reasons, or can sometimes be useful for “bookkeeping”. Basically, if nonstandard analysis solves your problem, there was already a solution in terms of standard analysis. Robinson’s “hyperreals” essentially just replace the concept of a “limit”, and do not represent a fundamentally “new kind of number” in the way that, say, Cantor’s transfinite ordinals do.
Conway’s surreals are a different story. However, if the sorts of problems you’re talking about could be solved simply by saying “oh, just use that other number system over there”, they would have been solved long ago (and everybody would probably be using that other number system).