Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity

From: ken quirici (kquirici_at_yahoo.com)
Date: 01/20/05 

Date: 20 Jan 2005 05:47:53 -0800

Ross A. Finlayson wrote:
> Perhaps one reason the well-ordering of the reals by their normal
> ordering seems a worthwhile theoretical excursion is that a
> well-ordering of the reals is proven to exist, but no examples are
> provided.
>
> Thus there is consensus that a well-ordering of the reals exists, so
a
> notion is to look directly under the the collective nose.
>

Hi Ross,

I'm not sure what you mean above. My first take on it is that you mean
you're really just POSTULATING that the normal ordering (3< 3.1,
4.57483 < 4.58483, etc.) is a well-ordering of the reals simply to
allow the POSSIBILITY of a well-ordering of the reals to be
entertained.

If that's what you mean, sounds good to me.

> One possible construction of a well-ordering of the reals might come
> from the discussions of surreal numbers. John Horton Conway, a
> prolific and quite excellent mathematician who I think of most for
his
> sphere packing book written with N.J.A. Sloane, derives a system of
> what he calls the surreal numbers in his book "On Numbers and Games."
>
> The surreals are constructed in a similar manner as naive
enumerations
> of the rational numbers as a closure of their field for inductively
> increasing members of the natural integers.

I didn't understand the naive enumerations of the rational numbers you
mention above. Is it something like the enumeration of the rational
numbers n1,d1,n2,d2,n3,d3,.... where you 'interleave' numerators and
denominators? (I didn't follow 'as a closure of their field for
inductively increasing members of the natural integers').

>There is the notion of a
> time component, with new sets of surreal numbers generated each
"day",
> and on day omega new rationals are yet being generated and as well
> irrationals numbers are first, which leads to a conundrum where omega
> is not a finite natural integer, regardless of whether it is a member
> or element of the set of all finite natural integers.
>
> Could thus the surreal numbers be totally ordered by their
generation?
> Yes, they can, although it is unclear if all real numbers are
> generated.
>

What do you mean by 'omega' above? Is it the number of elements in
the natural numbers? If so, 'day omega' is troublesome, at least for
me. Also, how do you generate the numbers? Also, there seems to be
a switch from surreal to rational to irrational which left me
hopelessly
befuddled.

> As that would be at least a total ordering, then there is the
> consideration whether it could be a well-ordering, as any
well-ordering
> is necessarily a total ordering. If it's not a total ordering, it's
> obviously not a well-ordering, being a total ordering is a necessary,
> but not sufficient, condition to be a well-ordering.
>
> It seems obvious that the total ordering of the surreals by their
> generation day and index is not a well-ordering.
>

It seemed a well-ordering to me (with a caveat about day omega and my
general befuddlement)!

> Then, in consideration of the reals and a well-ordering of the reals,
> besides facile rationalizations of the normal, linear, total ordering
> as a candidate for a well-ordering, another line or direction of
> enquiry is which properties a well-ordering specifically of the real
> numbers would have to have besides just the characteristics of a
> generic well-ordering because of the properties of the real numbers
as
> being comprising each point on the line and being at once an infinite
> field with multiplicative identity.
>
> So, besides that I want to show some characteristics of the normal
> ordering of the reals to help explain EF, the natural/unit
equivalency
> function, where sets dense in the reals are Megill's inadvertent
> hostages, I hope to see that a redefinition of the reals that is at
> once that of the field of the reals as basically the closure of the
> hyperintegers or p-adics is as well a coincident, non-conflicting,
> definition as a contiguous sequence of points, and geometrically,
> towards new mathematical subject categories to do with relations
among
> the continuous and discrete.
>

Way over my head - many terms I am completely unfamiliar with -
obviously not your problem.

> So, in the consideration that a mapping between the naturals and
reals
> that is non-constant and either strictly monotonic or everywhere
> non-monotonic and divergent is not precluded from being a bijection
by
> Cantor's first proof of the unaccountability of the reals, such a
> function would as well be, as necessarily qualified, a well-ordering
of
> the reals.
>

ditto (to my above remark)

> It thus does not seem unjustified to consider the real numbers for
what
> they are in terms of search for a well-ordering of the real numbers,
as
> a set, that is their normal ordering.
>
> Ken, I've read some of your posts to sci.math and sci.logic over the
> past month or so.
>

and?

> I promote my theory with no non-logical axioms, where excluded middle
> applied to the minimal (and dual and maximal) ur-element enables a
set
> theory with a set of all sets and resolution of Cantor, Russell, and
> Burali-Forti, towards a set theory that can be consistent and
complete
> and compatible with a concrete T.O.E., "Theory of Everything."
>

OK, more over my head.

> There are only, and everywhere, real numbers between zero and one.

Don't follow this.

> Regards,
>
> Ross Finlayson

same to you.

Basically interesting stuff but again much of it in realms of math
and/or logic that I am completely ignorant of.

However the notion of ordering the reals by a DIFFERENT but even more
clever ordering than the usual one is very interesting.

Thanks.

Ken

Relevant Pages

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