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Topic: fom - 01 - preface
Replies: 35   Last Post: Dec 16, 2012 12:20 AM

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: fom - 01 - preface
Posted: Dec 10, 2012 11:57 PM

On Dec 10, 7:55 pm, fom <fomJ...@nyms.net> wrote:
> On 12/10/2012 2:43 PM, Virgil wrote:
>
>
>
>
>
>
>
>
>

> > In article
> >   WM <mueck...@rz.fh-augsburg.de> wrote:

>
> >> On 10 Dez., 01:02, fom <fomJ...@nyms.net> wrote:
> >>> On 12/9/2012 12:30 PM, WM wrote:
>
> >>>> On 9 Dez., 17:24, fom <fomJ...@nyms.net> wrote:
> >>>>> On 12/9/2012 3:20 AM, WM wrote:
>
> >>> <snip>
>
> >>>>> So, why is there no global axiom of choice?
>
> >>>> As far as I am informed, *the* axiom of choice is global. There is no
> >>>> exception. Zermelo proved: Every set can be well-ordered.

>
> >>> The axiom of choice only applies to sets within
> >>> a given model.

>
> >> Zermelo proved that every set can be well-ordered - without mentioning
> >> any model. My interest is solely the set of real numbers. It is
> >> covered by Zermelo's proof.

>
> > Does WM claim that Zermelo's "proof" must hold in every model, that it
> > is somehow universal?

>
> Well, it certainly holds whenever the theory
> being modeled has the axoim of choice.
>
> I wonder how the claim holds when the axiom
> of determinacy is in force and the axiom of
> choice is inconsistent.
>
> I suppose, that the claim is interpretable along
> the lines of finitism.  Completeness is of no
> issue.  What can be proved using a sound deductive
> system is what is true.  Then the only real numbers
> are the constructive real numbers.

I'm for that they're mutually constructed as the complete ordered
field of Eudoxus/Cauchy/Dedekind and also as a partially ordered ring
a la Bishop and Cheng (strong constructivists), with between the two
forms a rather restricted, but existent, transfer principle.

Besides Archimedes, and, say, Newton's first fluxions or Leibniz'
infinitesimals: where do we find the infinitesimals in natural
theoretical order?

http://en.wikipedia.org/wiki/Infinitesimal

Thinkers since antiquity.

Regards,

Ross Finlayson

Date Subject Author
12/7/12 fom
12/8/12 Zaljohar@gmail.com
12/8/12 mueckenh@rz.fh-augsburg.de
12/8/12 fom
12/8/12 mueckenh@rz.fh-augsburg.de
12/8/12 Virgil
12/9/12 mueckenh@rz.fh-augsburg.de
12/9/12 Virgil
12/8/12 fom
12/8/12 ross.finlayson@gmail.com
12/9/12 fom
12/9/12 mueckenh@rz.fh-augsburg.de
12/9/12 fom
12/9/12 mueckenh@rz.fh-augsburg.de
12/9/12 fom
12/10/12 mueckenh@rz.fh-augsburg.de
12/10/12 Virgil
12/10/12 fom
12/10/12 ross.finlayson@gmail.com
12/11/12 fom
12/12/12 ross.finlayson@gmail.com
12/12/12 Virgil
12/16/12
12/9/12 ross.finlayson@gmail.com
12/9/12 fom
12/9/12 ross.finlayson@gmail.com
12/9/12
12/10/12 fom
12/9/12 mueckenh@rz.fh-augsburg.de
12/8/12 Virgil
12/9/12 mueckenh@rz.fh-augsburg.de
12/9/12 Virgil
12/9/12 Zaljohar@gmail.com
12/9/12 mueckenh@rz.fh-augsburg.de
12/10/12 Dan Christensen
12/11/12 fom