Date: Dec 16, 2012 12:20 AM Author: Subject: Re: fom - 01 - preface 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

> > >>> <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.

>

> Regards,

>

> Ross Finlayson

>

>

-2y.time.Musatov