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.