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

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/23/05 

Date: 22 Jan 2005 16:27:29 -0800

Timothy Little wrote:
> Ross A. Finlayson wrote:
> > What do you think about well-ordering the non-negative reals and
the
> > normal ordering as a candidate well-ordering?
>
> The normal ordering on the non-negative reals is a failed candidate.
> What is so difficult for you to understand about this?
>
> Is a well-ordering even what you want?
>
>
> - Tim

Hi Tim,

If the reals are well-orderable, then I have some considerations of how
to make use of those well-orderings.

Consider Cantor's first proof of the uncountability of the reals, can
you think of any well-ordering that would not suffer its conclusion of
uncountability, ie, not containing each element of the reals, and thus
being empty for some non-empty subset, and not containing a least
element? I know you say to just use the "uncountable ordinals". Is
that a corollary, that there can be no well-ordering of a dense set of
reals?

Where the normal ordering is a well-ordering, both of those problems
are resolved.

That's not to say that you can use the indefinite values for anything
besides enumerating the elements towards a choice function leading to a
well-ordering.

Similar arguments hold for the rationals, and other sets dense in the
reals, as do for the reals, as a set dense in the reals.

I'm interested in this Caicedo's discussion about the projectively
extended reals, latching onto an ephemeral quote that they're
"well-ordered", where that generally implicitly means "by their normal
total ordering", where it is generally accepted that the reals are
"well-orderable". I'm not claiming he says that without further
agreement, only that it was written in a casual sense. So I'm
wondering about just the _statement_ that the reals, in some way,
shape, or form, are well-ordered.

http://www.logic.univie.ac.at/~caicedo/papers.html

There there is some discussion of the "super-real" fields that were
passingly mentioned earlier here. I am unfamiliar with them.

What's 1/0? From http://mathworld.wolfram.com/FieldAxioms.html:

"a * a^-1 = 1 = a^-1 * a IF a =/= 0" (emphasis mine)

So, as long as you don't attempt to divide by zero, the field axioms
appear to be preserved. Does x/x = 1?

Tim, I'm looking for a well-ordering of the reals. If the definition
of the real numbers is amended thus that the normal properties hold
true for "definite" values, yet for these "indefinite" values they are
a contiguous sequence of points, then the normal ordering is a
well-ordering.

Perhaps instead I should only concern myself with the sequence of
points. That reduces into the same problem of showing the real numbers
to be a sequence of points. In a way, that's consideration that the
multiples of n, ..., 2, 1, 1/2, ..., 1/n for finite n are each
sequences of points. For each finite n it is a set not dense in the
reals, as a union for all of the integers it is.

The contiguous sequence that would be dense does not contain
identifiable elements that obey the general properties of the reals,
except for possibly one of those elements, the definite element. Yet,
where they exist, they would be real numbers because there are only
real numbers between zero and one.

It's like a jigsaw puzzle where each piece is numbered on the back,
with perhaps a direction. Where you can only flip them all over at the
same time, either way it's possible to reconstruct the puzzle, but you
can only see the picture on one side. If you want to draw a continuous
line between point a and point b on a piece of paper, generally that's
accomplished by drawing a continuous line from point a to point b.
Continuity is beyond the limits of precision, just barely.

I guess I need to learn more about generally accepted results about
well-ordering the reals, so I can utilize those results. Where V = L,
it may be feasible to do that.

About the or "my" theory, the ur-element is at once empty and
universal, it has multiple aspects, and infinite sets are equivalent.

Do you use transfinite cardinals for anything? I'm aware that standard
measure theory has that any countable domain has measure zero,
including sets dense in the reals.

In 2's complement computer logic, the 32 bits 0xffffffff equals
negative one, or (int) UINT_MAX == -1. Why would anybody ever think
that zero equals infinity? Perhaps it's because they use mathematical
logic to prove it to themselves, as I do.

That aside, this is a discussion of well-orderings of the reals and
other sets dense in the reals. It's shown in ZFC that there is at
least one well-ordering, yet no one wants to provide one because then
you could index that well-ordering by the natural integers.

Look directly to the normal ordering, and assume that it is not
contradictory to do so for reasons yet to be developed, or fiat, and
that it is not allowed to conflict with the "usual arithmetic
properties" because they're useful to solve real-world problems.

While you're at it, found a non-standard measure theory based on that.

If you don't care to consider that, there is some ordering of the reals
for which there is a least element of any subset of those numbers. If
you do consider it, then that's what it is.

Do infinitesimals exist among the real numbers? Short answer: no. Do
the hyperreals contain any element that is not an element of the reals?
Short answer, there are everywhere reals between zero and one.

Does the generic extension of N contain any elements not in N? No.
Regards,

Ross Finlayson

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