Ok, that makes sense. As I told JoshuaZ, I need to retire and lick my wounds at the very least. I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
Sorry, what do you mean by this? We’re talking about the cardinality of the set the measure is on; this sounds like you’re talking about the cardinality of its target space? (Where values of measures are somehow generalized appropriately… let’s not worry about how.) It’s easy to put an order on, say, Q[t] so as to make t infinitesimal but I don’t see what that has to do with this. Or is that not what you meant?
We’re talking about the cardinality of the set the measure is on.
So am I. But I may be confused about what cardinality even means in Nelson’s internal set theory.
Let me give a simple example of the kind of thing I am thinking about. Consider the space of ordered pairs (a,n) where a is either 0 or 1 and n is a non-negative integer, i.e. an element of {1,2,...}. To each such pair with a=0, associate the measure M(0,n)= 1/2^n. To each such pair with a=1 associate the “infinitesimal measure” M(1,n)=M(0,n)/omega where omega is taken to be indefinitely large.
So, the total measure of this space is 1 unit, and all but an infinitesimal portion of that total measure is associated with the portion of the space with a=0.
I claim that in some sense P(a=1) = 0 but P(n=2 | a=1) = 1⁄4.
The analogy here is that the assertion a=1 corresponds to the assertion that God exists. The probability is infinitesimal, yet Bayesian updating is possible (in some sense). And yet the space of all events is countable.
Yes. Definitely. Sorry that was unclear. And infinitesimal measures result in probabilities which are zero in some sense, but not exactly zero in a different sense.
Ok, that makes sense. As I told JoshuaZ, I need to retire and lick my wounds at the very least. I seem to recall that in Nelson’s version of non-standard analysis, there could be infinitesimals even in systems of countable cardinality, but I need to check that and decide whether it matters in this case.
Sorry, what do you mean by this? We’re talking about the cardinality of the set the measure is on; this sounds like you’re talking about the cardinality of its target space? (Where values of measures are somehow generalized appropriately… let’s not worry about how.) It’s easy to put an order on, say, Q[t] so as to make t infinitesimal but I don’t see what that has to do with this. Or is that not what you meant?
So am I. But I may be confused about what cardinality even means in Nelson’s internal set theory.
Let me give a simple example of the kind of thing I am thinking about. Consider the space of ordered pairs (a,n) where a is either 0 or 1 and n is a non-negative integer, i.e. an element of {1,2,...}. To each such pair with a=0, associate the measure M(0,n)= 1/2^n. To each such pair with a=1 associate the “infinitesimal measure” M(1,n)=M(0,n)/omega where omega is taken to be indefinitely large.
So, the total measure of this space is 1 unit, and all but an infinitesimal portion of that total measure is associated with the portion of the space with a=0.
I claim that in some sense P(a=1) = 0 but P(n=2 | a=1) = 1⁄4.
The analogy here is that the assertion a=1 corresponds to the assertion that God exists. The probability is infinitesimal, yet Bayesian updating is possible (in some sense). And yet the space of all events is countable.
Ah, so by “there could be infinitesimals” you meant “there could be things of infinitesimal measure”.
Yes. Definitely. Sorry that was unclear. And infinitesimal measures result in probabilities which are zero in some sense, but not exactly zero in a different sense.