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 > >>> <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.
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 space-like and space-like, time-like, and light-like dimensions.
Multiplication of reals with those except transfinite ordinals is quite well-defined, 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 iota-values, 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 iota-values have various characters from how many dimensions they are in or perspective. On the line, the 1-D line, the elements we construct as rational approximations to be elements of the complete ordered field, are two-sided points on the line. The elements drawn as iota-values from one of the two-sided points as endpoint to the next are one-sided 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.