And stick to my example of the square for the time being.
No. That example was a metaphor, it is reasonable to explore how its features correlates back to features of the question of interest, which is if it makes sense for you to assign infinitesimal probability to propositions about God.
If you had a uniform prior over the square, and then I inform you that my “random point” is on the edge, have I provided you with a ‘finite’ or an ‘infinite’ amount of evidence? A case could be made, I think, for either answer.
Then I would update my differential probability distribution using prior conditional probablities that you would make such a claim given that your “random point” is any particular point in the square. This could cause me to conclude that your point is very close to the border with high probability, but not to concentrate all my probablity onto the border itself, which would require that I had infinite information about under which conditions you would make such a statement.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Ah! But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at? If so, I’m not sure exactly how the hypothesis that some kind of god exists can be expressed properly in any axiomatizable logic.
Yes, but in a countable measure space the measure is determined entirely by the measures on the points, hence there is no problem with making the interpretation “probability 0 = impossible”, and this sort of weirdness does not occur.
Countability is not precisely the condition needed to avoid this, but it’s certainly a sufficient condition.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Measure theory tends to be a lot simpler with countable sets.
But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at?
No, although, if you answer that the space of propositions is countable, then I would argue that all propositions in that space should have a real probability between 0 and 1.
I would like you to answer the question, rather than speculating on hidden meaning in the question, so that I can know what kind of probability distributions we should be talking about.
Ok, I don’t know the cardinality of the space we are talking about, but since I have trouble imagining a language permitting an uncountable number of sentences, lets assume that the space is countable. What are the consequences of that?
If the space is countable, then as long as you can order the propositions in some way, say by complexity, you can assign non-zero probability to every proposition so the total adds up to 1, so you don’t have the same excuse you have in the case of predicting which point in a continuous space is special for using infinitesimal probabilities.
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.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
The two are related. Most relevantly, if my set is countable then I must have some singletons with non-zero measure. Moreover, the subset of points who have zero measure itself has zero measure, so they don’t matter at all. It is only in higher cardinality sets that you can have a collection of points each with zero measure that still have positive measure.
Ok, I can see that this tends to rule out my use of the unit-square analogy to justify my suggestion that the probability of the God Hypothesis is infinitesimal. I’m going to have to look more closely at the math, and in particular at my references for non-standard analysis to see whether any of my intuitions can be saved.
No. That example was a metaphor, it is reasonable to explore how its features correlates back to features of the question of interest, which is if it makes sense for you to assign infinitesimal probability to propositions about God.
Then I would update my differential probability distribution using prior conditional probablities that you would make such a claim given that your “random point” is any particular point in the square. This could cause me to conclude that your point is very close to the border with high probability, but not to concentrate all my probablity onto the border itself, which would require that I had infinite information about under which conditions you would make such a statement.
You have not answered my question about if the proposition about God is part of an uncountable space. The rest of this only matters if your answer is yes.
If by “uncountable”, you mean of cardinality greater than aleph-nought, then I think that you are using the wrong mathematical machinery. It is measure theory that we are concerned with here, not cardinality.
Ah! But perhaps you are suggesting that I can only formulate a countable number of sentences in my logic and hence that I should be using some kind of Solomonoff prior which necessarily forces a finite prior for the God Hypothesis—assuming that I can express it. Is that what you are getting at? If so, I’m not sure exactly how the hypothesis that some kind of god exists can be expressed properly in any axiomatizable logic.
Yes, but in a countable measure space the measure is determined entirely by the measures on the points, hence there is no problem with making the interpretation “probability 0 = impossible”, and this sort of weirdness does not occur.
Countability is not precisely the condition needed to avoid this, but it’s certainly a sufficient condition.
Uh, what sort of weirdness does not occur?
Measure theory tends to be a lot simpler with countable sets.
No, although, if you answer that the space of propositions is countable, then I would argue that all propositions in that space should have a real probability between 0 and 1.
I would like you to answer the question, rather than speculating on hidden meaning in the question, so that I can know what kind of probability distributions we should be talking about.
Ok, I don’t know the cardinality of the space we are talking about, but since I have trouble imagining a language permitting an uncountable number of sentences, lets assume that the space is countable. What are the consequences of that?
If the space is countable, then as long as you can order the propositions in some way, say by complexity, you can assign non-zero probability to every proposition so the total adds up to 1, so you don’t have the same excuse you have in the case of predicting which point in a continuous space is special for using infinitesimal probabilities.
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.
The two are related. Most relevantly, if my set is countable then I must have some singletons with non-zero measure. Moreover, the subset of points who have zero measure itself has zero measure, so they don’t matter at all. It is only in higher cardinality sets that you can have a collection of points each with zero measure that still have positive measure.
Ok, I can see that this tends to rule out my use of the unit-square analogy to justify my suggestion that the probability of the God Hypothesis is infinitesimal. I’m going to have to look more closely at the math, and in particular at my references for non-standard analysis to see whether any of my intuitions can be saved.