Re: Cantor's "diagonal argument". My Objection.

David Formosa (aka ? the Platypus) wrote:
On Thu, 04 Dec 2008 21:51:55 +0000, John Jones <jonescardiff@xxxxxxx> wrote:
David Formosa (aka ? the Platypus) wrote:
On Wed, 03 Dec 2008 23:56:46 +0000, John Jones <jonescardiff@xxxxxxx> wrote:
[...]
IF I allow you the possibility of having an infinite placement for the digits of possible reals, then there is no reason why you should not allow me an infinite placement for the digits of possible naturals.
No you can't. The first natural is 0 and the second S0 (or 1 in
binary). There is no natural number between 0 and its successor.
Again, I can say the same for the reals. There is no real between .0 and .1

There exists a real x such that 0<x<0.1 however there does not exist a
natural number y such that 0 < y < S0.


That's why if some nonstandard reals were sequential from zero to one, they wouldn't have representations as expansions.

Where these reals must fulfill the properties of being at once complete ordered field (and perhaps then some, with regards to projectively extended reals) and partially ordered sequential ring (re Schmieden and Laugwitz, and Bishop and Cheng, who investigated these kinds of considerations of the real numbers of the "natural" continuum around the same time as Robinso(h)n's Nonstandard Analysis, which re Nelson's IST is compatible with the standard for non-conclusive acknolwedgment of Cantor's "Cholera bacilli", infinitesimals) of "integral iota-multiples" as I've called them, which go from zero to one, then arises the question of the sufficiency of the Eudoxus/Cauchy/Dedekind representation via iterated fractioning.

The E/C/D, standard, way, to represent reals is as a limit of a convergent sequence of rationals (the sequence of the expansion in a fixed modulus/radix). A nonstandard way to accommodate the reasoning of the unit interval's continuum as naturally well-ordered, which is certainly a familiar concept at least since the invention of the integral calculus (infinitesimal analysis), has that representation being insufficient to represent these nonstandard values, which are defined by their ordering and membership, only from one to the next, instead of in terms of the recursively defined standard definition.

In the standard there is 0.5 to 1, but not half infinity (the semi-infinite as it is acknowledged in analysis) to infinity, infinity standardly isn't a scalar.

In this nonstandard case, based on features of the natural continuum besides rational arithmetic, then, certain arguments related to the standard representation don't hold.

The point I would like to make is that, on a level playing-field, whatever you do with the reals you can do with the naturals. If you object that I am not treating the naturals like the naturals ought to be treated,
No I'm objecting that your not treating the naturals in line with
Peano Axioms. And when you do that the "natural"'s stop being
naturals the same way chess stops being chess if you treat knights the
same way as bishops.
Don't we want the naturals to stop being naturals?

No if one wishes to prove statements with regard to the properties of
tha natural numbers then one has to treat them as natural numbers.

We want an index after all. We just want the digits of the naturals (for it is the digits that are counted) to form an index, in which case there is no other property of the natural number that is of use to us.

The Principal of induction is also usefull in this case.

It's basically a compromise to make "rigorous" analysis via the definition of the standard open topology (which is recursively defined), and is the result of basically the 1800's, where before that there is general treatment of the infinitesimals as comprising a sweep of the unit interval. Today, many still see that as the intuitive "real" structure of these items, individua of the continuum. The limit of the partial sums is the sum.

The standard unit measure is defined in terms of the standard unit measure. Well-order the reals, some would have they already are.

Regards,

Ross F.
.

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