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

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/19/05 

Date: 18 Jan 2005 22:44:56 -0800

I think we agree here that the reals are well-orderable.

Any well-ordering is a total ordering, so any well-ordering of the
reals comes from the set of total orderings of the reals.

One obvious total ordering of the reals is the normal ordering.

Here, in the consideration of the reals, it is appropriate to start
with the unit interval of reals, in determining total and
well-orderings on the unit interval of reals, and then from those,
where there is a well-ordering on all of the reals, then the
well-ordering of the reals is obviously a well-ordering of the (set of
elements of the) unit interval of the reals.

Besides the obvious normal ordering of the reals, other total orderings
of the reals imply the definition of a comparator, eg "<[". If there
is a function f between the naturals and reals of the unit interval
then the comparator is obvious the total ordering on the naturals of
f', in this context the inverse of f instead of its first derivative or
other meaning.

Now, I don't agree with that there doesn't exist a function between the
naturals and unit interval of reals, I say that infinite sets are
equivalent, in acceptance that many do not accept that fact. So
anyways the function f I use is the Equivalency Function, where "<["
has the same meaning of "<", that is to say, if the Equivalency
Function is a bijection between the naturals and unit interval of
reals, then the normal ordering of the reals is a well-ordering of the
unit interval of reals, as is the reverse EF and sufficient
compositions of EF.

Putting EF aside, as it is not accepted by everybody, then I hope you
could help me define total orderings on the unit interval of reals that
at no point are compositions of the normal ordering. What's a total
ordering, not well-ordering, on the reals besides the normal ordering
or anything constructed from the normal ordering?

For the unit interval of rationals, any bijection from the naturals to
the rationals is an obvious total ordering of the unit interval of
rationals. If you do not allow bijections between the natural integers
and reals of the unit interval, then there is not an obvious total
ordering beside the normal ordering and "compositions" of the normal
ordering. That is to say, if you define an invertible function with
the domain being the naturals that explicitly gives for each element of
the unit interval of reals a natural to indicate their total ordering,
then there exists a bijection between the naturals and unit interval of
reals.

Before, you indicated that the "sequence" or "function" would instead
have as an index or domain the infinite ordinals, or perhaps
hyperintegers. This is where the ordinals are well-ordered, there is a
minimal element in the set theory, the empty set. Is that the way it
is?

My idea is that if you define a well-ordering of the reals, then I
would challenge myself to present real numbers that were either
indistinguishable under your definitions or illustrative of the lack of
a total ordering and thus a well-ordering.

When the normal ordering is a well-ordering, then that would not be
possible because the normal ordering of the reals is a well-ordering,
the reals are totally ordered under the usual definition of "less than"
or "greater than".

Now to go back to the previous post, about the discussion of the
well-ordering of the unit interval by its normal, total ordering in as
necessary a non-standard model of the real numbers: if the infinite
sequence (0, iota, ...) represents the least two elements of the
non-negative reals, and each successive real in its normal ordering,
then the non-negative reals are well-ordered by their normal ordering.
Regards,

Ross Finlayson

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