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

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/30/05 

Date: 29 Jan 2005 20:36:35 -0800

We're arguing about definitions here. It appears that the definition
of "computable" is somewhat more pathological than that of, say,
"equipollent", please disregard use of that term, re Torkel's "which
definition of computable". In examining the "PlanetMath" entry for
"computable number", you can currently note a minor discrepancy between
the current entry: empty, and the cached entry via Google, with an
"alternate definition" about a recursive function to generate a value
to any given precision, with some notion of a single free variable to
combinatorially enumerate the numbers.

http://www.google.com/search?q=cache:O-pLJ5ALioYJ:planetmath.org/encyclopedia/ComputableNumber.html

As I said, I'm not very familiar with the definition of "computable".
Is Chaitin's constant useful for anything? I'd like to know an
application of it.

With computable versus non-computable numbers there seems to be a
similar inability as Dedekind/Cauchy to describe each real number.

I figured you would note the "apples and oranges" of reals vis-a-vis
hyperreals, but I think it's possible that the transfer principle still
applies because the reals are _complete_. I'm glad you did. I don't
grasp why the positive hyperintegers are not well-ordered, as the
positive integers are, and those are the ones under consideration. As
well, with regards to indexing the elements of the well-ordering of the
reals by the ordinals, it still seems unclear why "Cantor's first"
wouldn't apply, due to the completeness of the reals. Does Cantor's
first not apply, or do two successive elements of the well-ordering
contain no real between them in the normal ordering of the reals?

How does any well-ordering of the real numbers avoid the contradictory
consequences of Cantor's first?

About a working (agreed upon) definition of "well-ordering", it is
obvious that that is one currently available to this discussion and
that is in use for some time. The definition and its usage are quite
clear.

About proving the existence of a least element in each subset of the
reals, that's part of the rationale of addending to the definition of
real numbers iota-value or atomic infinitesimals towards what you have
labelled the "Finlayson numbers" or "pseudo-reals", which I call the
"reals" or real numbers. I derive enjoyment from calling them
"Finlayson numbers" to the extent that I think they're useful, is that
selfish or wrong? I'm reminded of Glass' pronouncements in joviality.
I just call them the reals or real numbers with atomic iota-values,
perhaps R-bar or R-umlaut. This is where re-reading the entire thread
helps to refresh understanding.

Here's another analogy, it's like a Pachinko machine, a method to
randomly generate a real. That's where there is a triangular (or
lattice) grid of pegs at an incline and a marble is dropped on the top
of the triangle (from the center or evenly distributed side-to-side),
bouncing somewhat randomly at each peg to the left or right, to
eventually fall within one of many separate bins, labelled left to
right by ascending or descending ordinals. The average value of all of
the reals of the unit interval is 1/2, as discussed above about the
fair coin and random reals. That adds a notion of "computability" to
"beads on a string" or points on a line.

About the notion that 1/n has no least element, the idea is that for
some ordinal x that is indefinite and unknown except to be greater than
one and less than for any other value of n that x*iota is the least
element of the range of 1/n. _If_ you accept that iota is the least
value of the interval (0,1) by the normal ordering, _then_
correspondingly every subset of (0,1) has a least element.

About the equivalency of infinite sets, I've argued that infinite sets
are equivalent, where that means there is a bijection between them, for
some years. "Equivalent" is here seeing acceptible usage, and it is in
regular usage in that way, and in the context it is quite clear.

I'm glad that you consider these notions of the extension of the real
numbers with atomic iota-values, indefinite and ubiquitous. While that
is so, the _completeness_ of the reals implies that the elements of
these nonstandard extensions can only be real numbers.

So, what insight you give towards helping to establish the definition
of these "extensions" to the real numbers is appreciated.

Thanks for briefly re-reading the entire thread.
Regards,

Ross Finlayson

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