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: 23 Jan 2005 00:13:51 -0800

Hi,

"Generic extension" normally applies to a model, I talk about the
generic extension of a set as just the set in the generic extension.
It was recently discussed with regards to IZF and perhaps even CZF.

There are only reals between zero and one, and everywhere reals, real
numbers, where are the infinitesimals?

An infinitesimal is sometimes defined as being less than 1/n for any
finite n. In the hyperreals infinitesimals can be greater than zero in
terms of their scalar magnitude. Internal Set Theory has notions of
infinitesimals, and is conconsistent with ZFC. Cantor was against
infinitesimals. I don't figure to add any elements to the real
numbers, as they are complete or on the line continuous.

Paraconsistency is a notion of a logic where an inconsistency doesn't
invalidate every other true statement. Burali-Forti is difficult to
avoid. While that is so, it is not so bad for "not void."

If the normal ordering of the positive reals was a well-ordering, then
it would be.

(0, iota, 2iota, ...)

For positive and non-positive reals that kind of shorthand of a
well-ordering would be along the lines of

(0, iota, -iota, 2iota, -2iota, ...)

I wonder about the statement that the reals, in some way, shape, or
form, are well-ordered.

About the jigsaw puzzle analogy, it captures some of the notions of the
normal ordering as sequenceable but not all. It's still kind of
useful. That's about the sequence of contiguous, next to each other,
points on the real number line, from having a point being the
intersection of two lines on a plane, as it has been for several
thousands of years.

Do you use transfinite cardinals for normal, day-to-day things? If so,
is it to talk about transfinite cardinals?

Here's an idea: compare Zeno's paradox to uncountable sets. Consider
a straight line there and back.

About constructing the well-orderings of the reals, there is probably a
body of work describing exactly their necessary structure. It appears
to have to do with this Goedelian Kolmogorov complexity in terms of the
randomness. Yet, Enderton says that from L, the constructible
universe, is "definable" a choice function. What are Feferman's
"natural well-orderings", particularly of the reals?

Anyways, in my theory with no non-logical axioms, ur-element as
Thing-in-Itself and Being and Nothing, and ubiquitous ordinals and
naturals, the normal ordering of the positive reals is a well-ordering,
and infinite sets are equivalent.

Regards,

Ross Finlayson 

--
"Have you seen my pen?"
"Say hello to my little pen."

Relevant Pages

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