Waclaw (01)
Any interval of real numbers has cardinality Aleph-1 (not
Aleph-0) - that is rather the point of real numbers. In the usual
axiomatizations of Real numbers (and numbers in general) division is the
inverse of multiplication and 0 is not equal to 1 (the multiplicative
identity). The use of the Alephs is not part of the real number system. (02)
Sean Barker
0117 302 8184 (03)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
> [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of
> Waclaw Kusnierczyk
> Sent: 19 June 2007 07:43
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Probabilistic Ontologies
>
>
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> Kathryn Blackmond Laskey wrote:
> > To do probability right, you need real numbers, and you need to be
> > able to do continuous distributions. For example,
> consider a number
> > picked at random between zero and one. (Imagine a spinner that can
> > land anywhere on the disk with equal likelihood; measure
> the distance
> > around the edge to where it landed and divide by the
> circumference --
> > the number you get is uniformly distributed between zero
> and one.) The
> > probability of getting a number in any sub-interval of the unit
> > interval is equal to the length of the sub-interval. But the
> > probability of getting any specific number is zero.
> >
> > This raises some tricky mathematical issues. In finite
> domains, we can
> > equate probability zero with unsatisfiability and
> probability 1 with
> > validity. But in the example I gave above, UniformRand = X is
> > satisfiable for any X between zero and one, yet (Prob (UniformRand =
> > X)) is zero for all X, and moreover, Prob(Exists X
> UniformRand=X)) is
> > equal to 1.
>
> Isn't the issue due to approximation? In P(X = x) = 0 for
> any x in the selected interval of real numbers, what is the
> zero? The probability that a random variable will take on a
> specific value from a particular set of values is per
> definitionem equal to the inverse of the cardinality of the
> set -- right? An interval -- a dense set of real numbers,
> e.g., the unit interval (0.0, 1.0) -- has the cardinality
> aleph_0 (I guess). The zero in P(X = x) = 0 is in fact the
> inverse of aleph_0; it is not exactly the usual 0, it is an
> approximation. For every x in the interval, P(X = x) is
> infinitesimal, but is not exactly 0. If infinitesimals were
> equal to 0, would integration make sense?
> It may be practically zero, but when discussing paradoxes of
> probability, it may be worth mentioning.
>
> (Ignore if I am getting this wrong.)
>
> vQ
>
>
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> (04)
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