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:
>>
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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?


Abraham Robinson

Non-Standard Analysis provides a model wherein there
are infinitesimals so that the physicists can feel
good about what they do.

>
> http://en.wikipedia.org/wiki/Infinitesimal
>
> Thinkers since antiquity.
>
> Regards,
>
> Ross Finlayson
>