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.