On Tue, May 24, 2005 at 10:10:19PM +0100, Patrick Leahy wrote:
> This is very reminiscent of Lewis' argument. Have you read his book? IIRC 
> he claims that you can't actually put a measure (he probably said: you 
> can't define probabilities) on a countably infinite set, precisely because 
> of Cantor's pairing arguments. Which seems plausible to me.

It makes a very big difference whether he said probability or
measure. One can easily attach a measure to a countable set. Give each
element the same value (eg 1). That is a positive measure. However, it is not a
probability, as it cannot be normalised.

One can also sample from a measure without mean - however the rules
for computing expected outcomes differs somewhat from just taking the
mean as the expectation.

For example with a uniform measure, the expected outcome is any point
in the set. Assume some property is distributed over those points -
for example the property is identical (the delta distribution). The
the expected value of that property is the constant value. and so on.

> Lewis also distinguishes between inductive failure and rubbish universes 
> as two different objections to his model. I notice that in your articles 
> both you and Russell Standish more or less run these together.

I'm interested in this. Could you elaborate please? I haven't had the
advantage of reading Lewis.

If what you mean by by the first is why rubbish universes are not
selected for, it is because properties of the selected universe follow
a distribution with well defined probability, the universal prior like
measure. This is dealt in section 2 of my paper.

If you mean by failure of induction, why an observer (under TIME)
continues to experience non-rubbish, then that is the white rabbit
problem I deal with in section 3. It comes down to a "robustness"
property of an observer, which is hypothesised for evolutionary
reasons (it is not, evolutionarily speaking, a good idea to be
confused by hunters wearing camouflage!)

In that case, how am I conflating the two issues? If I'm barking up
the wrong tree, I'd like to know.


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A/Prof Russell Standish                  Phone 8308 3119 (mobile)
Mathematics                                    0425 253119 (")
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