Post by Jan Post by John Gabriel
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.
In Dedekind's system numbers are defined as certain sets, yes. As long as
they satisfy the desired properties, such things are as good as any other
possible alternate definition.
No Jan, they're not as good as any other definition. Sets don't measure anything in the way that rational numbers do. You can't do anything, even with Dedekind arithmetic which is only possible with rational numbers.
We understand a given magnitude (whether distance, mass, volume, etc) by a number which describes it. Neither a Dedekind Cut nor a Cauchy sequence do this.
Post by Jan
Philosophically many may find this sort of thing unsatisfying and a "kludge"
but mathematically this is sound (or at least as sound as the axioms of
set theory which nobody knows for sure).
How is it sound when it has been proven to be invalid?
The entire concept is absolute hogwash because one can't relate non-numbers to indistinguishable and unreifiable points on a mythical "real" number line.
To reify a point on the number line, all of the following must be possible:
i. It must be possible to mark off the magnitude on the number line.
ii. There must be a number which describes its measure.
You can reify any rational number on the number line, but it is not possible to reify the magnitudes pi, e or sqrt(2) because there is no number describing any of these. These are all symbols for incommensurable magnitudes. They are decidedly not numbers.
Even before one can consider Dedekind Cuts, the real number line must be established. There is no real number line - it's a myth.