Date: Dec 16, 2012 12:20 AM
Subject: Re: fom - 01 - preface
On Dec 11, 9:09 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> 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.
> That's from first principles.
> Ross Finlayson