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Re: The Distinguishability argument of the Reals.
Posted:
Jan 6, 2013 1:42 PM
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On Jan 5, 10:10 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <694aece8-90bf-400b-8663-dd86266ad...@ah9g2000pbd.googlegroups.com>,
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
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> > On Jan 5, 7:37 pm, fom <fomJ...@nyms.net> wrote:
> > > On 1/5/2013 6:35 PM, Ross A. Finlayson wrote:
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> > > > On Jan 4, 10:20 pm, Virgil <vir...@ligriv.com> wrote:
> > > >> In article
> > > >> <7850ae29-08d9-49ef-8c7b-e8979e037...@m4g2000pbd.googlegroups.com>,
> > > >> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
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> > > >>> Consider the function that is the limit of functions f(n,d) = n/d, n =
> > > >>> 0, ..., d; n, d E N.
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> > > >> You mean the zero function?
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> > > >> For every n, the limit of f(n,d) as d -> oo is 0, so your limit function
> > > >> would have to be the zero function: f(n,oo) = 0 for all n.
> > > >> --
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> > > > No, none of those is the zero function, and each d->oo has it so that
> > > > d/d = 1.
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> > > That is true.
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> > > The problem is that as d -> oo the value at any
> > > given fixed n -> 0.
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> > > 2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9, 2/10, ...
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> > > So, the pointwise limit of the function is zero.
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> > lim_n->d n/d = 1
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> Since the set of values of n is finite for each value of d, no limit
> process is required, or even defined.
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> One has f(d,d) = 1 for all d, but one does not have f(n,d) = 1 for any
> n less than d, and one has the properly defined limit:
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> lim_(d -> oo) f(n,d) = 0 for every n in N
> --
No, d/d = 1, n ranges from 0 to d.
lim_(n->d) n/d = 1
Then, where the limit isn't the only tool to evaluate properties of a
function, here there is a symmetry between the extents. The constant,
monotone, positive progression of values is from zero to one. And,
the sum of their differences is one. And, the complementary or
reverse equivalency function with it, sees a symmetry about 1/2.
Thus, it starts from end to beginning, as beginning to end: toward
the other extent.
Well my, my, my, that looks just like Leibniz' "naive" notion of an
infinitesimal. And, as one of the discovers of the original
infinitesimal analysis, these days well known as the integral
calculus, his notation of the integral bar S for summation and d for
differential matches it quite clearly. So, this kind of notion is at
least perfectly familiar to those familiar with the development, of
the calculus. And, in the very real, as it were, application of the
integral calculus, it is where the differential vanishes but does not
disappear, that Int 1 dx = 1.
Int_0^1 1 dx = 1
Int_0^1 1 dx =/= 0
S_0^1 dx = 1
It is for no finite differential that this is so: and for no zero
differential, in the asymptotic of differences. And, transfinite
cardinals' place in measure theory is that the measure of [0,1] = 1.
There are most certainly infinities in mathematics, and the Universe
would be its own powerset, and the reals standardly are a
construction, with rules, to support analysis, over the continuum.
The linear continuum, then as real numbers, of the trichotomy of all
scalar values, has the real numbers as defined to fulfill being both
the complete ordered field: and the sequential, the points in a line
of the linear continuum. And, from the Infinitarcalcul, there are
more of those than standard reals. Even Archimedes, was not solely,
Archimedean.
So: add (a positive quantity) to zero, it's greater than zero.
Regards,
Ross Finlayson
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