On Tuesday, June 11, 2013 11:23:17 AM UTC-7, Julio Di Egidio wrote: > <email@example.com> wrote in message > > news:firstname.lastname@example.org... > > > On Tuesday, 11 June 2013 19:18:34 UTC+2, Julio Di Egidio wrote: > > >> WM wrote: > > > > > >> > Therefore it is not possible to enumerate all rational numbers > > >> > (always infinitely many remain) by all natural numbers (always > > >> > infinitely many remain) or to traverse the lines of a Cantor list > > >> > (always infinitely many remain). > > >> > > >> It is not possible to do so effectively... > > > > > > It is only possible by applying the axiom of infinity. But that axiom > > > will also manage to well-order the rationals. After each one has been > > > attached to a natural, we have the set of naturals that obviously can > > > be re-ordered in any desired way. > > > > Obviously and crucially not in any finite number of steps (you will get the > > rationals well-ordered by magnitude from, say, the rationals > > lexicographically ordered). Anyway, I'm already out of my depths here, so > > I'll leave this to more competent mathematicians. >
Its not that deep really.
A Well Order, is an Order where every subset contains a least element.
In the real line, the rational are Not Well Ordered. The open set (0,1) has no least element. Of couse, [0,1] does, but we only need one counter-example.
They, the rationals, are Ordered under the usual algebraic/geometric "<", however.
> > > (The power set of |N should even include the singleton of the last natural > > > as an element.) > > > > There is no such thing as the last natural number, and not even a next to > > last, and not even a next to next to last, and so on. This was already > > touched in my initial post, the part you have snipped. >
See you got that one.
Keep asking questions. I understand you're learning, not spouting BS.
> Julio > > > > P.S. It's a bit of a pain to have to reformat your posts and quotes every > > time: it would help a lot if you could at least split your paragraphs in > > short lines using carriage returns.