Posts:
822
Registered:
9/1/10


Re: fom  01  preface
Posted:
Dec 16, 2012 12:20 AM


On Dec 11, 9:09 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Dec 10, 9:46 pm, fom <fomJ...@nyms.net> wrote: > > > > > > > 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. > > Two, three, and four are clearly integers, and clearly rationals, and > clearly reals, and clearly ordinals, and clearly hypercomplex numbers > from each Cartesian product of a spacelike and spacelike, timelike, > and lightlike dimensions. > > Multiplication of reals with those except transfinite ordinals is > quite welldefined, with their products in the reals. > > Robinso(h)n's hyperreals don't much add to the analytical character of > the real numbers. The halos of infinitesimals about a point are > simply dense halos about them without defining, for example, the > intuitive fluxions of Newton or for that matter iotavalues, which is > a term I use to describe the infinitesimals, as reals or elements of > the continuum of reals, as sequence here from zero to one, dense and > continuous and contiguous. Then the hyperintegers are not much > different than transfinite ordinals, for distinct infinite sequences > of elements of a finite alphabet, and the cumulative limit hierarchy > of ordinals. > > Basically the iotavalues have various characters from how many > dimensions they are in or perspective. On the line, the 1D line, the > elements we construct as rational approximations to be elements of the > complete ordered field, are twosided points on the line. The > elements drawn as iotavalues from one of the twosided points as > endpoint to the next are onesided points on the line. Then the iota > values as elements of the continuum see R as R^bar and R^dots, or > R^crown (R^bar^dots, was R^bar^umlaut: Re^bar). > > Then the reals of the real continuum satisfy at once being the > complete ordered field and the ring of infinitesimals about the > origin, that in their infinite extent are the continuum, as the > infinite rational approximations are the continuum, of real numbers, > pure and applied. > > That's from first principles. > > Regards, > > Ross Finlayson > > 2y.time.Musatov

