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

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/18/05 

Date: 17 Jan 2005 22:38:53 -0800

Hi Piotr,

That's interesting, for you to divide the subsets into more than
aleph_1 many. I hadn't thought of that. How is that done?

It's an interesting notion there that there would be aleph_2 ways to
divide the reals into sets with that characteristic. I'm not sure
about how to proceed showing that.

I partially agree with you about that infinite sets are equivalent. A
key concern is to be able to provide to others the rationale or logic
that they can use themselves or share with others so they can feel
rational and socially acceptible in terms of mathematical discourse in
their belief that infinite sets are equivalent. I proffer a set theory
with logical axioms, ubiquitous ordinals, and ur-element.

I think one term I have used that was unclear was "dense", referring to
the density property. It's about saying that the sets are dense "in
each other" and the reals, instead of just saying they are dense in the
reals and nowhere continuous which is the condition. The rationals are
not dense in the irrationals, for example, in the irrationals there are
no rationals to be in place in the normal ordering between any two
irrationals. The rationals are dense in themselves. In the reals,
each of the rationals and irrationals is dense and as well the
rationals and irrationals are disjoint and their union is the reals.
When it's written that the sets in question are dense in each other,
that means that any pair of irrationals that is defined as a sequence
of rationals is having infinitely many rationals between them.

That's about an inductive impasse between that on the "macro-" scale,
viewing the reals as rationals and sequences of rationals, between any
pair of rationals there are infinitely many rationals, that on the
"micro-" scale, it's difficult to say into which sets the reals are
divided thus that in terms of their propensity or, heh, "perspacity",
it is possible to describe or specify which real number is "next" after
zero or greater than zero and less than all other positive real
numbers, the least real number.

It might be intellectually easy to say something along the lines of
that: the well-ordering exists, it's the same as the normal ordering,
after zero is iota, and because zero is rational iota is irrational
(because it's a real number and all real numbers are rational or
irrational and _no two elements of the same NCD subset of the reals may
be neighbors_). Then again it's trivial to lump zero in with the
irrationals and say that iota is not an element of that set. A more
troubling progression is that iota is not an element of any of those
sets. The hyperreals are the reals, because the reals are continuous.
By the same token, lumping zero with the irrationals would make the
function defined on that set continuous at zero instead of everywhere
discontinuous, thus, the "trivial" set can be outlawed refining the NCD
condition to include everywhere discontinuous, nowhere continuous, or
NC2D.

So, you shouldn't have to go to the hyperreals or non-standard models,
because the real number line is continuous.

Thank you for your input. If you would please further develop the
notion of how many sets can be generated that would be useful for
proving your point.

Regards,

Ross Finlayson

Relevant Pages

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