In article <email@example.com>, Butch Malahide <firstname.lastname@example.org> 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. --