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Re: Continuous and discrete uniform distributions of N
Posted:
Dec 27, 2012 9:18 PM
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On Dec 26, 9:17 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Dec 26, 9:30 am, Butch Malahide <fred.gal...@gmail.com> wrote:
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> > On Dec 26, 5:50 am, gus gassmann <g...@nospam.com> wrote:
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> > > On 25/12/2012 5:57 AM, Butch Malahide wrote:
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> > > > Did anyone say anything about wishing sets of equal cardinality all to
> > > > have the same probability?
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> > > The subject line says "uniform distributions". What can that mean OTHER
> > > than "sets of equal cardinality have equal measure"?
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> > 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.
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> > 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 translation-invariant?
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> > > > 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.
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> > > Bingo.
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> 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 non-Archimedean extra-complete 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 iota-values,
> with rather-restricted transfer principle, in that the elements of
> continuum as iota-values 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 Bois-Reymond, 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.
>
One of the properties of the standard real numbers is that are
iteratively re-divisible infinitely. Basically we know that for any
standard real quantity, it can be halved then halved again ad
infinitum, that multiplied by the series of the negative powers of
two, the limit evaluates to zero as standard real quantity.
Then, another consideration is as to the division or rather partition
of the unit, one, to infinitely many equal parts, that sum back to
one, or that multiplied by the number of partitions, the part is equal
to one. Then, this is the notion that there's a difference between
dividing the continuum, and having divided the continuum.
We're familiar with the fundamental theorems of calculus and
integration, and the definition of the sequence as Cauchy as having a
limit. As well, Leibniz' notation survives: the large S for
summation and d for differentials, that the differential ratio is as
to the free variable, of an infinitesimal rate of change of it, non-
zero and for no finite change. Then, there's at least present the
founder's intent and intuition that the notation reflect the summation
of infinitely many infinitesimal quantities, to retrieve a finite
quantity, and here one that exactly and perfectly corresponds with our
geometrical methods for the computation of area, between real-valued
curves and axes of the coordinate plane or about the origin in general
methods of exhaustion for the computation of area. Then, it's
certainly not unreasonable nor unprecedented to consider that the real
unit of the continuum is composed of points and those of a uniform
partitioning or division _to the points_, besides the inexhaustibility
of division or partition of finite quantities, _to non-points_.
dividing: regions -> regions
divided: regions -> points
Then, where there's the consideration that points have zero width,
that they can't comprise a region with non-zero width, then that's in
as to where only an assemblage of points, and infinitely many,
comprise exactly a region of width one, individua only of the
continuum, where as points they are our familiar points of Euclid and
Hardy, and of Hardy as real numbers.
Then, for this uniform distribution of naturals, or for a putative
mapping from natural integers to the unit interval of reals, then
there is that these infinitesimal quantities, or iota-values, among
real values of the continuum as the continuum includes all magnitudes
and elements of the "linear continuum" of elements that satisfy
trichotomy, density, and continuity between zero and one, are defined
together as a completed thing, that the very definition is as to the
composition of the unit line segment, then via deduction the nature of
the points, instead of definition of the points as elements of a field
from the integers, then ordered field, then complete ordered field.
Plainly there is a consideration that either of these notions
establish trichotomy, density, and continuity of these elements,
between zero and one.
Then, in as to where EF this function is simply representative of
these notions of the continuum as its range, it's of some mathematical
interest. Courtesy its simple properties and resulting available
theorems, given its construction of primary nature in theory of
numbers and geometry, it's in fact a basic building block for a wide
range of theorems, and in particular, an entire class of applicable
theorems, due its properties inaccessible (or otherwise not known to
be accessible) to the standard.
So, then there's a consideration that for the pure or applied, a
putative uniform distribution of the naturals is of interest, and for
that in asymptotics there are established reasonable notions of
density to match those of measure for transfer between the discrete
and continuous. Also, that's applied in number theory as for example
theorems on prime numbers, and as well in systems of for example
random needles dropped on the line and in alternatives to attenuation
of the central limit theorem, such a thing is of more than marginal
utility and placement.
Then for EF, the function, the domain of which is the naturals and
range is [0,1], that in the number-theoretic results on uncountability
of the reals it is as no other function, as well it satisfies being a
CDF, of the naturals. Quite remarkably, it would also be a p.d.f. of
a continuous distribution as its integral evaluates to one, meeting
first the properties of a probability function, before considerations
of its standardness, in its regularity. This complements 1/omega
being a discrete distribution, where the sum of it over the omega-many
terms of the naturals is again: one. Now, where in ZFC it was shown
that there would be a set that would construct a uniform probability
distribution over the elements of the set of naturals, then it remains
that there would be uniform probability distribution over the
naturals.
EF is that.
Regards,
Ross Finlayson
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