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

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/21/05 

Date: 20 Jan 2005 19:33:05 -0800

Hi Ken,

It's a theorem of ZFC, in this case I guess it was Zermelo himself who
proved this, that every set has a well-ordering.

So, it is known the set of reals, where the collection of all real
numbers is a set, has a well-ordering.

The idea of considering the normal ordering and why or why not and
under what circumstances it could be correctly called a well-ordering
partially comes about from wanting an _example_ of a well-ordering of
the reals.

The "Axiom of Choice", AC, basically says that a "choice function"
exists for every non-empty set, and the choice function returns an
element of the set, where the domain is the entire set and the range is
one element of the set. Then, if you subtract that element of the set
to get some proper subset, then it has a choice function, and you can
get an element from it. Repeated ad infinitum, that constructs a list
of those elements, return values from the choice function, that
sequence represents a well-ordering of the initial set, and thus
obviously each of its subsets. There exists x E X, x E X, , not x E
X-x, choice reduces to the quantifier.

Basically then, I'm looking for the choice function that for any set of
non-negative real numbers returns an element less than any other in the
set. For the set of positive real numbers I say there is one and that
the value returned is iota. There's that, and that for any set X of
non-negative reals numbers there is some least real number. That
contradicts with the true notion that for any positive real x, that
that number can be halved or otherwise divided by y > 1 and that x/y is
less than x. So, I say that the value returned, for example iota,
which while necessarily being a real number, is indefinite in that it
represents the necessarily existent point, less than any other, on a
sequential progression of the contiguous reals between zero and one.

That leads to the requirement of proof that the unit interval comprises
some sequence of reals with the normal ordering. That's exactly
equivalent to the statement that the non-negative reals are
well-ordered, by their normal ordering. It is necessary to not
presuppose that which is to be proven. However, it would be
acceptible, to begin, by presupposing the opposite and showing it
always contradictory.

That the normal ordering is a well-ordering (for the non-negative reals
or the unit interval) is the same thing as that there exists a choice
function to select not just an element but the least element, in the
normal-ordering, that there is a least element in the normal ordering.

As we agree, with the standard definition of the reals or set dense in
the reals there is _not_ a least element of the reals. That's because
the standard definition of the set of non-negative reals is not the
sequence of points from zero onwards that is continuous on a straight
line or ray. Where the set of reals is a contiguous sequence of
points, there is a least element by the normal ordering of each
subsequence and the normal ordering is a well-ordering.

One possible notion to address is that the non-negative reals are a
contiguous sequence of points or they would not be complete.

At one point I described that a set might be "ordering-sensitive", that
the sets' elements vary based upon the sequence of choice functions
used. I consider why the reals might be such a set as they are
continuous. That has to do with the two-sidedness of discrete points
or endpoints on the lines, and the one-sidedness of interior or rather
non-endpoints. Ken, that's that kind of conjecture, or, yes, perhaps
even postulate. Dedekind/Cauchy provides _examples_ of real numbers.

That the normal ordering of the reals would be a well-ordering is
conceptually similar in some ways to accepted theory about the
well-ordering of the reals: there is not a way to tell the "value" of
the next real in the ordering except that it is that. If the normal
ordering was a well-ordering, there is not a way to tell the "value" of
the next real number, only that it is a real number and less than each
other remaining.

With the normal ordering coinciding with a well-ordering, it is easier
to show infinite sets equivalent. Indeed, the range of EF is basically
the normal ordering (of the unit interval), but EF can use limit. EF
is a putative building block of bijections between the naturals and
reals.

This is where I think infinite sets are equivalent, which is _not_
standard, although I have proven it to myself. "Set of all sets, class
of all classes", and "no classes in set theory", those are little
mantras. So, I promote a non-standard set theory, and neither en- nor
disourage its use by you. I use it.

I guess the conclusion here is that if the unit interval of reals is a
contiguous sequence of points, then the normal ordering is a
well-ordering.

Regards,

Ross Finlayson

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