Date: Dec 10, 2012 11:57 PM
Author: ross.finlayson@gmail.com
Subject: Re: fom - 01 - preface
On Dec 10, 7:55 pm, fom <fomJ...@nyms.net> wrote:

> On 12/10/2012 2:43 PM, Virgil wrote:

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> > In article

> > <e88797b7-7c0e-456a-9cf7-87f0a5247...@gu9g2000vbb.googlegroups.com>,

> > 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