Virgil
Posts:
4,479
Registered:
1/6/11
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Re: Continuous and discrete uniform distributions of N
Posted:
Dec 25, 2012 1:48 AM
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In article
<14be47c2-c95a-45b7-b9e9-686ec5c1b79f@c16g2000yqi.googlegroups.com>,
Butch Malahide <fred.galvin@gmail.com> wrote:
> On Dec 24, 11:22 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <6e340766-2e08-4966-b497-0c4d83497...@10g2000yqo.googlegroups.com>,
> > Butch Malahide <fred.gal...@gmail.com> wrote:
> >
> > > On Dec 21, 8:41 pm, Bill Taylor <wfc.tay...@gmail.com> wrote:
> >
> > > > On Dec 22, 5:23 am, FredJeffries <fredjeffr...@gmail.com> wrote:
> >
> > > > > No one has ever anywhere actually used the concept of a uniform
> > > > > distributions on N to solve any problem.
> >
> > > > Sure they have. You can use it to calculate the probability
> > > > that two randomly chosen naturals will be co-prime, for example.
> > > > And many others of that type.
> >
> > You cannot do it using the standard reals because it would require the
> > existence in the standard real number system of e an infinitesimal
> > non-zero lambda smaller than any positive standard real but itself
> > positive, and an infinite cardinality, card(|N), of the infinite set of
> > naturals such that lambda * card(|N) = 1.
>
> You are tacitly assuming that probability must be a countably additive
> (as opposed to finitely additive) measure. It seems kind of arbitrary
> to say that probability has to be countably additive but does not have
> to be, say, aleph_2-additive.
The standard definition of probability requires that the sum of the
probabilities over ANY set of sets making up a partition of the space be
equal to one.
If you want to impose some non-standard definition, that is your
prerogative, but you must not expect all others to accede to it.
>
> > Which cannot occur within the essentially unique (up to isomorphism of
> > complete ordered Archimedian fields) standard real number system.
>
> "Archimedean" is redundant; a complete ordered field is necessarily
> Archimedean.
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