
Re: Continuous and discrete uniform distributions of N
Posted:
Dec 27, 2012 12:17 AM


On Dec 26, 9:30 am, Butch Malahide <fred.gal...@gmail.com> wrote: > On Dec 26, 5:50 am, gus gassmann <g...@nospam.com> wrote: > > > On 25/12/2012 5:57 AM, Butch Malahide wrote: > > > > Did anyone say anything about wishing sets of equal cardinality all to > > > have the same probability? > > > The subject line says "uniform distributions". What can that mean OTHER > > than "sets of equal cardinality have equal measure"? > > In the case of the continuous uniform distribution on the interval > [0,1], it means that "intervals of the same length have equal > measure." All intervals have the same cardinality, but their measures > vary. > > I'm not sure what it means for a (finitely additive) measure on N. > Perhaps, that all points have equal measure (which would have to be > zero)? Or that sets of the same natural density should have equal > measures? Maybe it should be translationinvariant? > > > > > > > > > > Is that what the OP was talking about? I > > > didn't try to read his post, as it seemed kind of obscure. Of course > > > it's not possible, in an infinite sample space, for sets of the same > > > cardinality all to have the same probability. > > > Bingo.
It seems the properties of a uniform (or regular) probability distribution of the natural integers would have the properties that the probability of each value of the sample space and here support set would be equal.
Then as these sum to one then this value would be infinitesimal and 1/ omega and yes we all know that's not a standard real number, or rather: not a member of the Archimedean complete ordered field (and 1/ omega is an element of nonArchimedean extracomplete ordered fields like the surreal numbers, which include all the real numbers as described, for example, by Ehrlich or Conway.)
Here then as well I would actually described that instead they are elements of the continuum best represented by the ring of iotavalues, with ratherrestricted transfer principle, in that the elements of continuum as iotavalues and elements of the continuum as complete ordered field are the same thing, but they have different representations and obviously their rules of formation and manipulation are different. Their properties as magnitudes hold.
Then, with regards to the probability of elements from U(n) or U_c(n) and U_d(n) for the "continuous" and "discrete" uniform probability distributions of natural integers being in particular subsets of the natural integers, then as above how the U(r) or U(x) would be represented by measure, here the notion would be as to density, and from du BoisReymond, then to cardinality only for the finite, which is then basically simply of multiplicity and only coincidentally cardinality. P(n even  n e N) = 1/2, "the probability of n being even given n is from the naturals is one half", P(n e {1,2}) = P(n e {3, 4}), "the probability of n being 1 or 2 is the same as n being 3 or 4", where here it is implicit that the probability is regular and uniform.
This is simply founded with treating the probabilities as opaque quantities with additivity, simply underdefined, as to satisfying the general sense of rigor. Of course, that is merely apologetics, where the true inquiry is to the actual nature of these quantities.
So, the probability of any given natural being selected at uniform random from all the natural integers is obviously: the same as that of any other. It is that first, then as to how to divide the unity of the sum of those probabilities equally among them is in then to developments in foundations of real numbers as to support the reasonable expectations thereof. And, there is longstanding tradition in the discussion since antiquity as to ruminations on the nature of these things. Yet, since Cauchy and Weierstrass, and Cantor, many of those lines of inquiry as posed by our greatest thinkers do not have their historical place in the context, of the infinitely divisible and infinitely divided, for the potentially and complete infinity.
And: the only way to line up the points of the line is in a line, and in the line. Draw the line. EF starts that.
In earlier discussions on this as well I described a framework for constructing U(n) in ZFC. As well, as above it's described as modeled by real functions. Again that's to appease the modern, for the avenue to the real, and concrete, and of the continuum, and, in the continuum, as divisible, and as divided, as composed.
Then it's a reasonable notion to consider the infinitesimals as the unity as divisible and the unity as divided in a similar manner as to consideration of the infinity as the unity potentially and the unity completely.
Regards,
Ross Finlayson

