On Dec 10, 7:55 pm, fom <fomJ...@nyms.net> wrote: > On 12/10/2012 2:43 PM, Virgil wrote: > > > > > > > > > > > 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?