Bruno Marchal writes:
> Le 26-mai-05, à 18:03, Hal Finney a écrit :
>
> > One problem with the UD is that the probability that an integer is even
> > is not 1/2, and that it is prime is not zero. Probabilities in general
> > will not equal those defined based on limits as in the earlier
> > paragraph.
> > It's not clear which is the correct one to use.
>
> It seems to me that the UDA showed that the (relative) measure on a
> computational state is determined by the (absolute?) measure on the
> infinite computational histories going through that states. There is a
> continuum of such histories, from the first person person point of view
> which can not be aware of any delay in the computations emulated by the
> DU (= all with Church's thesis), the first persons must bet on the
> infinite union of infinite histories.

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Sorry, I was not clear: by UD I meant the Universal Distribution, aka
the Universal Prior, not the Universal Dovetailer, which I think you
are talking about. The UDA is the Universal Dovetailer Argument, your
thesis about the nature of first and third person experience.
I simply meant to use the Universal Distribution as an example probability
measure over the integers, where, given a particular Universal Turing
Machine, the measure for integer k equals the sum of 1/2^n where n is
the length of each program that outputs integer k. Of course this is an
uncomputable measure. A much simpler measure is 1/2^k for all positive
integers k.
I don't know whether there would be any probability measures over the
integers such that the probability of every event E equals the limit as n
approaches infinity of the probability of E for all integers less than n.
Actually on further thought, it's clear that the answer is no. Consider
the set Ex = all integers less than x. Clearly the probability of Ex
being true for integers less than n goes to zero as n goes to infinity.
But the only way a probability measure can give 0 for a set is to give
0 for every element of the set. That means that the measure for all
elements less than x must be zero, for any x. And that implies that
the measure must be 0 for all finite x, which rules out any meaningful
measure.
Hal Finney