Date: Dec 10, 2012 10:55 PM
Author: fom
Subject: Re: fom - 01 - preface

On 12/10/2012 2:43 PM, Virgil wrote:
> In article
> <e88797b7-7c0e-456a-9cf7-87f0a5247cbb@gu9g2000vbb.googlegroups.com>,
> WM <mueckenh@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.