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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>>

>>

>>

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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

>