Virgil
Posts:
8,833
Registered:
1/6/11


Re: Continuous and discrete uniform distributions of N
Posted:
Dec 25, 2012 1:48 AM


In article <14be47c2c95a45b7b9e9686ec5c1b79f@c16g2000yqi.googlegroups.com>, Butch Malahide <fred.galvin@gmail.com> wrote:
> On Dec 24, 11:22 pm, Virgil <vir...@ligriv.com> wrote: > > In article > > <6e3407662e084966b4970c4d83497...@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 coprime, 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 > > nonzero 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_2additive.
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 nonstandard 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. 

