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).
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).