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 22:31:31 -0800

Hi Tim,

I'm talking about the null axiom or axiom-free set theory with no
non-logical axioms, ubiquitous ordinals and an ur-element that is the
unique proper class and a set. In it, the powerset is successor is
order type.

By "no non-logical axioms", that means it revolves around tautology and
the excluded middle. Besides regularity, the axioms (non-logical
axioms) of ZFC are theorems. I guess it's Hilbertian.

Then, I think that's strong enough to model the integers, and
concurrently everything relevant about them.

A partial summary to whit is noted in the listing of "Claims" last year
on sci.logic.

I use ZFC in these discussions because as long as I don't make use of
regularity then it's largely the same thing, and in general people who
read sci.logic have been introduced to ZFC, the Zermelo-Fraenkel axioms
of set theory with as well the axiom of Choice: Z, F, C.

You might not be aware of this, I claim infinite sets are equivalent
and a variety of participants here have discussed that for some time,
with some agreement that what I say is acceptible when I'm not trying
to apply it to all foundations of mathematics, which I do.

I don't say "infinite sets are equivalent" in ignorance of, say,
Cantor, although I did originally. There are hundreds of pages of
discussions of arguments about the equivalency of infinite sets.
Antidiagonal is discussed vis-a-vis dual representation, and
uncertainty, as is the powerset result. Nested intervals are
confronted with monotonic mapping and sets dense in the reals, and to
some extent this discussion. Measure theory is claimed to require
retrofit. The dearth of results in terms of transfinite cardinals with
regards to anything besides transfinite cardinals is noted.

I don't find paradoxes acceptible (acceptable, acceptible). A paradox
is a _contradiction_. Infinite sets are equivalent. If I accept that,
then by reason I would have to not accept that contradictory to it,
basically throwing away infinite cardinals. I think what is _gained_
from that is a foothold of sorts for the consideration of analytic
results of functions with the domain being the naturals and the range
being the unit interval of real numbers.

As well, model expansion and higher order logic is shown to be the same
problem as Burali-Forti, which is the same problem as Cantor, which is
a similar problem to Russell. Skolemize: your model is countable.

Where everything is an ordinal, and the ur-element represents at once
null and U, the universal set, then the powerset is, in a way, the
order type and successor, or it has the same ordinal value. Besides
ubiquitous ordinals, it might be restrictable to ubiquitous naturals.
Then with f(x)=x+1 with the domain being the naturals, the result is
the naturals, or empty set. That has to do with dual representation.
About uncertainty, the ur-element is one or both of empty set and
universal set, and it doesn't matter which because the conclusion is
the same.

I realize that might not seem clear. It's not meant to be otherwise.

So, I've put forth little bits and pieces of logical progression, some
assembly is required.

V = L.

Anyways, in terms of "Well-Ordering(s) of Sets Dense in the Reals", I'm
trying to figure out a way to define the reals as not only the field
with multiplicative inverse that they are but also the sequence of
points that they may be. Unless otherwise noted, assume I use ZFC.

It would be coincidental that the normal ordering would satisfy the
Goedelian requirements for a well-ordering, I rather expect it would,
with a perhaps brusque interpretation.
What's iota?

Regards,

Ross Finlayson

Relevant Pages

  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... non-logical axioms, ubiquitous ordinals and an ur-element that is the ... You might not be aware of this, I claim infinite sets are equivalent ... and the ur-element represents at once ... ubiquitous ordinals, it might be restrictable to ubiquitous naturals. ...
    (sci.math)
  • Re: Why an inconsistent ZF may be desirable, and should be welcome.
    ... Chris Menzel wrote: ... In a theory with no non-logical axioms, how can I say what is or isn't ... discourse and infinite sets are equivalent, ... burn, Hollywood, burn ...
    (sci.logic)
  • Re: Two results of set geometry
    ... if each is a proper subset of some other in the set. ... we PROVE from the axioms that w is a counterexample. ... Moreover, you say you do endorse that there exist infinite sets, so ... and the union of the set is infinite. ...
    (sci.math)
  • Re: Cantors circular "proof" that evens = integers
    ... In order to understand why this "proof" is actually just a circular argument, i.e., an argument that merely ADDS the belief that N and E are equinumerous as an axiom to the already existing axioms of set theory, we have to make use of some basic facts concerning proofs. ... We will refer to the possibility that sets N and E are equinumerous as property "P." The possibility that they are not equinumerous is "not-P." ... not-P means that we can have infinite sets of the natural numbers of different sizes. ...
    (sci.logic)
  • Null-Axiom Set Theory
    ... The idea of having no axioms is for several reasons. ... empty, or an ur-element or minimal element, exist, that from anything ... Goedel sentence G_0 and the infinite chain of ghost axioms that each ... and applied mathematics. ...
    (sci.logic)
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