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Re: The Distinguishability argument of the Reals.
Posted:
Jan 6, 2013 6:40 PM
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On Jan 6, 2:59 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <7a163160-c36a-46d0-ab7a-97cf0fa11...@q16g2000pbt.googlegroups.com>,
> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
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> > On Jan 6, 11:55 am, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <1038fe29-f169-4511-bd13-c7ade7fd1...@pd8g2000pbc.googlegroups.com>,
> > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > > > > lim_(d -> oo) f(n,d) = 0 for every n in N
> > > > > --
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> > > > No
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> > > If, as Ross defined it, f(n,d) = n/d for all d in N and all n in
> > > {0,1,2,...,d}, then for any n, lim_(d -> oo) f(n,d) = 0
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> > > And no amount of denial by Ross will alter that fact.
> > > --
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> > Wait, aren't you going to misquote Einstein? Because, you have quite
> > the practice of misquoting me. Now, I'm no Einstein, but, I generally
> > heartily agree with him, of the rather conscientious sort.
>
> > d/d = 1
> > lim_(n->d) n/d = 1
> > lim_(n->d, d->oo) n/d = 1
>
> That last one is false because
> lim_(n -> d) [ lim_(d -> oo) n/d ] =lim_(n -> oo) 0 = 0
> lim_(d -> oo) [ lim_(n -> d) n/d ] =lim_(n -> oo) 1 = 1
> SO
> lim_(d->oo)(lim_(n->oo) n/d =/= lim_(n->oo)(lim_(d->oo) n/d
>
> SO Ross's lim_(n->d, d->oo) n/d does not exist!
>
> At least in stndard mathematics.
>
> But he might try in in WMytheology or RAFeology.
> --
http://en.wikipedia.org/wiki/Order_of_integration_(calculus)
It's not quantifier dyslexia so much as that there's only one free
parameter modeling the function: d.
There d, for denominator, for n, numerator: is free and unbounded, as
is n, as it ranges through elements simply enough in d. For all
values of d, the range is [0,1].
Some years ago you had that the sine wave divided by the parameter had
no standard limit, now it's rather de rigeur that it does: lim_x->oo
sinc(x) = zero. (The standard is not constant, the status quo is
always contemporaneous.) And, half of the integers are even, some
time ago you stoutly averred there was no "standard" way that was so:
it is so, of those numbers from number theory. Just last month I
found another definition for continuity. So, besides that the
function has a limit in the unbounded, your notion of "standard" isn't
necessarily forward. So, I hope you look forward to the next flood
down the steps of the ivory tower.
Here, look to Dirac's delta: modeling it as real function, the width
of the spike is zero. Yet, its area is one. It maintains its
property of having area one, in the asymptotic, as modeled by real
functions. And, it's of quite wide applicability in the solution of
differential equations, with even particular application. So it's at
least not unprecedented to consider functions as so defined this way,
with the potential for application, and indeed wide application.
Then, the range of the function is from zero to one, the constant
difference in elements in the range is exactly correlated with
constant difference in the domain, the range meets definitions of
continuity, and in nested intervals and similar arguments and the
antidiagonal argument and similar arguments, it is unique among
functions, under composition, with different results.
With an infinite alphabet or infinite words, each real number has at
least its own identity, or value.
With the infinite alphabet, the equivalency function's values are 1,
11, 111, ..., the antidiagonal is 111..., the end.
Regards,
Ross Finlayson
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