Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Distinguishability of paths of the Infinite Binary tree???
Replies: 69   Last Post: Jan 4, 2013 11:11 PM

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Distinguishability of paths of the Infinite Binary tree???
Posted: Dec 30, 2012 7:05 PM

On Dec 30, 1:27 pm, forbisga...@gmail.com wrote:
> On Sunday, December 30, 2012 12:57:05 PM UTC-8, Ross A. Finlayson wrote:
> > Draw the line, Cantor shows is it's point-to-point, not the stippling
> > of the stellation, our complete ordered field as structure above the
> > continuum follows from simpler principles (of points that make space).

>
> You've used this language before.  Isn't the assertion of
> countability a claim about stippling rather than drawing a line?
> I would not try to count the reals in numeric order for any interval
> containing more than one entry.

If the points are drawn from the origin to one, over an infinity of
points, that's different than stippling the points, marking instead of
drawing. The notion is that the points can only coalesce into a line
as they are drawn, that they are one-sided, in a sense, contrasted to
the two-sided points of the complete ordered field with limits above
and below or from left and right.

Of course we know that in our "standard" reals the complete ordered
field, there is not a smallest non-zero positive quantity. As well,
we know that it is not alien to the discussion that there be
infinitesimals of a not-necessarily-Archimedean real, linear
continuum. These are regular matters of discussion of, for examples,
the discoverers of the integral calculus, or original infinitesimal
analysis, with infinitely many constant infinitesimals that sum to
one. (And sometimes Archimedes is non-Archimedean.)

Then, in building the equivalency function, so named as that in the
context of the antidiagonal argument that the "antidiagonal" is at the
end, and that in the context of the nested intervals argument that the
two first points are contiguous or side-by-side, thus that the range
is cardinally equivalent to the naturals and the unit interval of real
numbers, building the equivalency function then has a rather simple
construction, with the properties of its range being from zero to one,
and that its values monotonically increase constantly,
infinitesimally, from zero to one.

Then the construction of the reals as complete ordered field and the
construction of the very same reals as ring, see those constructions
are tools, with caveats. There is a transfer of theorems true of one
and the other between them, simply restricted due their structure and
import. This has the reals as complete ordered field, and ring, with
rather-restricted transfer principle.

The "vague fugue" then of the real numbers is as the puzzle, with
different theorems on either side, the pieces all flipped only at
once, either complete, of the continuous elements of the linear
continuum. Put together either way, on the one side, it's simple as
putting the points only in a row, the pieces are only found and even
defined from the pieces before them, then that they complete the
puzzle.

So, draw the line, in the sense of drawing the thread or drawing the
steel. The mark starts the line and where it stops, marks the line.
Those between have a mark, from their being drawn: through.

The complete ordered field is as of infinite simple constructions, in
the geometric sense. (Eh, some would have that only to algebraics,
but pi is courtesy the compass.) The line as points drawn does too,
of purely abstract constructions, of plethora of things.

As a mathematical abstraction, this construction or EF is then a
simple one. Yet, its placement in the theory (of numbers, and
geometry, then sets) yields very rich possibilities: progress, from
the foundations.

Regards,

Ross Finlayson

Date Subject Author
12/23/12 Zaljohar@gmail.com
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/25/12 mueckenh@rz.fh-augsburg.de
12/25/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Virgil
12/28/12 mueckenh@rz.fh-augsburg.de
12/28/12 Virgil
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/30/12 fom
12/30/12 mueckenh@rz.fh-augsburg.de
12/30/12 fom
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
1/4/13 ross.finlayson@gmail.com
12/30/12 forbisgaryg@gmail.com
12/30/12 ross.finlayson@gmail.com
12/30/12 Virgil
12/26/12 Zaljohar@gmail.com
12/26/12 Virgil
12/26/12 Zaljohar@gmail.com
12/26/12 gus gassmann
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Zaljohar@gmail.com
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Zaljohar@gmail.com
12/28/12 mueckenh@rz.fh-augsburg.de
12/28/12 Zaljohar@gmail.com
12/28/12 Virgil
12/29/12 Zaljohar@gmail.com
12/29/12 Virgil
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/28/12 Zaljohar@gmail.com
12/29/12 mueckenh@rz.fh-augsburg.de
12/29/12 Virgil
12/27/12 Virgil
12/26/12 fom
12/26/12 Virgil
12/26/12 fom
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 Virgil
12/26/12 mueckenh@rz.fh-augsburg.de
12/26/12 forbisgaryg@gmail.com
12/26/12 Virgil
12/26/12 fom
12/27/12 gus gassmann
12/27/12 Tanu R.
12/27/12 mueckenh@rz.fh-augsburg.de
12/27/12 Tanu R.
12/27/12 Virgil
12/28/12 Zaljohar@gmail.com
12/28/12 Virgil
12/27/12 fom
12/27/12 Virgil
12/24/12 Ki Song