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Topic: Matheology § 285
Replies: 84   Last Post: Jun 15, 2013 6:05 PM

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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: Matheology § 285
Posted: Jun 11, 2013 3:04 PM

On Tuesday, June 11, 2013 11:23:17 AM UTC-7, Julio Di Egidio wrote:
> <mueckenh@rz.fh-augsburg.de> wrote in message
>
>

> > 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.

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.

ZG

Date Subject Author
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 JT
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Tucsondrew@me.com
6/15/13 Virgil
6/15/13 Tanu R.
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Virgil
6/14/13 Virgil
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Scott Berg
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Virgil
6/11/13 Tanu R.