```Date: Dec 11, 2012 12:46 AM
Author: fom
Subject: Re: fom - 01 - preface

On 12/10/2012 10:57 PM, Ross A. Finlayson wrote:> 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 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?Abraham RobinsonNon-Standard Analysis provides a model wherein thereare infinitesimals so that the physicists can feelgood about what they do.>> http://en.wikipedia.org/wiki/Infinitesimal>> Thinkers since antiquity.>> Regards,>> Ross Finlayson>
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