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

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

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> >> 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.

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> > 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.
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> 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