Date: Dec 10, 2012 10:55 PM
Subject: Re: fom - 01 - preface
On 12/10/2012 2:43 PM, Virgil wrote:
> In article
> WM <email@example.com> 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:
>>>>> 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.