Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/19/05 

Date: 19 Jan 2005 14:43:57 -0800

About 1/n, 1/N, 1/n for n E N, inductively for m > n there exists 1/n >
1/m > 0, for no n does there not exist n+1 thus that 1/n <= 1/(n+1).
In the limit as n diverges 1/n = 0, and for any finite integer n: 1/n >
0.

Where that's agreed upon, there is this notion that the reals are
points on a line, beads on a string. As discrete elements if you
consider the reals to be a contiguous sequence of points between zero
and one, then somewhere between zero and one in the normal progression
a single point is the first element of the reals of that interval to
satisfy 1/n for some finite positive integer n. No particular
"definite" value of the reals can be considered to be that value, but
some "indefinite" element, an element of the contiguous, totally
normally ordered sequence of reals is that value, because the set of
all real numbers includes all subsets of the real numbers.

An obvious problem with that is that in the normal ordering if you did
select any finite integer n then there would be infinitely many values
of the sequence preceeding it, where the sequence is not to be doubly
infinite, that being an opposite of semi-infinite.

So a problem with, or reason to abandon the notion that, the numbers
can be defined in that way is that it implies the infinite set of
finite integers contains an infinite element, which while conceptually
feasible in terms of ubiquitous naturals and the set of all sets
containing itself, is still not a clear issue.

In that sense the normal ordering progression could contain only
indefinite values, with some notion that they were the same numbers as
the set of definite values. That is similar to the consideration of
the open interval only containing indefinite values: (iota, 2iota,
...), except that as "iota" could not be "rational", as one is the
least positive integer the first element of the sequence 1/n would be
the least positive rational instead of iota, the first positive real.

Here my goal is to avoid confusion and crystallize points of dissent,
towards reconciliation and a broader firmament.

In the consideration of dissent on points, the unit interval of reals,
totally, contains everywhere reals. The notion that there exists for
each real number x a real number y greater than x and less than every
other real, except for the upper boundary, that the reals are
contiguous, sequential, as points besides being continuous as the real
number line, is fraught with vagaries on the borderlands of sound, or
rather, practiced, definitions.

So, of the reals, I consider that there is not a "next" after zero, and
also that there is.

Where there is some least positive (infinitesimal transcendental) real
called iota, and integral multiples of iota are representative of the
real numbers and necessarily not translatable to "definite" real
numbers, except in terms of scalar infinities (x/2x = 1/2), then the
set of reals is a contiguous point set on the real number line, besides
being a field.

If the reals were defined in that way, perhaps obviously in not the
standard model, with the multiple definition of rationals and
irrationals and as well contiguous points in a sequence, then the
normal total ordering would be a well-ordering for any finite interval,
where they are not defined in that way, it is not so.

Obviously, in standard practice, it is said that it is not so and I'm
aware of that. I develop some lines of argument as to what
considerations would take place in a redefinition that supports the
reasoning otherwise, which is justified for various deductively
determined reasons, and consequences of such reasoning.
Regards,

Ross Finlayson

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