fom
Posts:
1,969
Registered:
12/4/12


Re: fom  01  preface
Posted:
Dec 11, 2012 12:46 AM


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: >> >> >> >> >> >> >> >> >> >>> In article >>> <e88797b77c0e456a9cf787f0a5247...@gu9g2000vbb.googlegroups.com>, >>> WM <mueck...@rz.fhaugsburg.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 wellordered. >> >>>>> The axiom of choice only applies to sets within >>>>> a given model. >> >>>> Zermelo proved that every set can be wellordered  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
NonStandard 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 >

