```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:>>>>>>>>>> > 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 orderedfield of Eudoxus/Cauchy/Dedekind and also as a partially ordered ringa la Bishop and Cheng (strong constructivists), with between the twoforms 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 naturaltheoretical order?http://en.wikipedia.org/wiki/InfinitesimalThinkers since antiquity.Regards,Ross Finlayson
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