Re: Rational numbers, irrational numbers: each dense in real numbers

On Oct 8, 6:38 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Oct 8, 5:58 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

On Oct 6, 9:14 pm, lwal...@xxxxxxxxx wrote:

On Oct 5, 10:41 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

Good call. Though, it would be interesting to me any progress you make
toward a systemm that upholds "infinite-case pigeonhole" and is
adequate for proving an alternative real analysis

That's still my goal, but of course it won't be easy (otherwise TO,
RF,
etc., would already have accomplished it).

That's an interesting goal, one held by many, who aren't necessarily
cranks,

Right. To work toward a particular alternative mathematical goal is
not itself to be a crank. What is crank is in the manner of operation
and communication regarding such manners.

and I'm not a crank.

You're quintessentially a crank.

Here's quintessentially crank writing:



I think that in the infinite the universal quantifier can be
specialized into forms that reflect whether the transfer principle
applies, and, that might have some meaning.

Consider for example, if there was a set of only all the finite
ordinals, that for each n-set of finite ordinals ({0, 1, ..., n}),
it's a finite ordinal, but for all n-sets of finite ordinals, it's not
a finite ordinal.

As another example, for each finite integer n there exists a value
between 1/n and zero, but for all finite integers n, there doesn't
exist a value between 1/n and 0.

Obviously, sometimes the transfer principle applies.

I think an adequate theory is the null axiom theory because it can be
both complete and consistent. Things exist no matter what we say, so
why demand to proscribe them? I think that the null axiom theory,
which corresponds to a variety of long-held creation stories, has
objects in its universe, and that they're not inscrutable.

Then, in terms of discovering meaningful interactions among these soi-
disant mathematical objects, largely in the sense of what could be
done with them, instead of generation of primary/primitive elements
along the lines of 0, 1, ..., instead it is 0, universe, ....
(Interactions among mathematical objects are discovered moreso than
invented, yet of course methods of expression might be moreso invented
than discovered, in terms of novel mathematics.)

So, then there's the empty set (as a set, null, void, nothing) and the
universal set (everything, being). As sets, they contain other
things, or not, with variously the empty set containing nothing and
universal set containing everything, in a set theory.

The empty set is the universe, then there's everything else,
everything other, everything, in the universe. Or, the universe is
empty, then ....

With some principles along the lines of sufficiency and necessity, the
infinitely many elements uniquify themselves, differentiate, only to
the point yielding a continuum of them as they wouldn't more. The
continuum of elements is recognized as having the properties of a
number-theoretic continuum, and much analysis derives from it.

As mathematicians there are many perfectly reasonable theories like
geometry and number theory. They're great. Since Goedel and the 50's
many would believe that there is no set of all true statements about,
say, the natural integers,

Let me interject your nonsense at this particular point to say: WRONG.
Godel said no such thing.

or, points and lines, yet via a simple
counterargument, true statements about those objects can only be
expressions of the theory, those that aren't inconsistent, otherwise
they would be provable. That's not necessarily to say that according
to Goedel, or rather others as many refer to Goedel, there is no set
of only true statements about, say, the natural integers or points and
lines. Then there wouldn't be a collection of false statements
either. Given all the statements, true and false, Goedel can't
separate them.

Anyways, in terms of infinite sets being irregular, consider the
casual notion that infinity + 1 = infinity. It is presumed in
transfinite ordinal arithmetic that omega + 1 = omega,

I do hate to interrupt again when you've got such a great rhythm
going, but WRONG. In ordinal arithmetic it is NOT the case that w+1 =
w.



while in
transfinite cardinal arithmetic aleph_0 + 1 = aleph_0. So, consider
infinity as an ordinal, and adding another means to include the set
itself, when infinity + 1 = infinity, the infinity already included
itself: infinity is irregular. Consider some countable universe, if
the universe contains itself because it contains everything, then
according to Russell's paradox some least infinite set is irregular,
so all of them are.

In terms of an alternative analysis, it has much to do with geometry,
and geometrical mutations of a sort in the large and small, and
polydimensional points and the space containing them.

So, do you think the above argument holds, about the "transfinite
recursion schema" describing an uncountable set and for each element
of it a distinct rational? If not, why is not the "transfinite
recursion scheme" defined for more than countably many irrationals
unles there are not more than countably many?

Half of the integers are even.

I think I already have indicated the theory, because it's the only one
that could possibly be, the question then is how to make theorems of
it.

The question is how you got so mixed up. Or rather, why in the world
you think that rambling inchorently about mathematics should move
anyone to think anything other than that you're a huge nutcase or a
huge phoney.

MoeBlee



No, in denial, when Goedel claims that no consistent theory can be
complete, and that thus there are "true" statements about the objects
of the theory that are not theorems of the theory, I disagree with
that, because I think "true" means "provable."

Then, consider a collection of all theorems, consistent and
inconsistent, about the objects of a theory strong enough to represent
the, say, natural integers. That's all the theorems about those
objects. If Goedel could separate that collection into consistent and
inconsistent theorems, then the collection of consistent theorems
would represent all facts about the objects. There couldn't be more
facts about those objects because they are completely and only defined
in terms of the theory already. Any new fact would represent new
objects, or objects augmented with new properties, and a different
theory, not some truism about the objects of the initial theory. So,
Goedel can't generally separate consistent from inconsistent theorems
of a theory.

I was in the process of explaining to Larry here some of my positions
on various statements having to do with philosophical foundations of
mathematics and so on. So, Moe, while you might be an expert in what
it means to be a crank, the above is not an inaccurate representation
of consequences of inexistence of complete and consistent theories
accorded to Goedel's rresults. Larry, I hope you don't mind if I call
you Larry, you don't have a very accurate understanding of what I say
about these things, although it is agreeably in the right vein.
Generally I don't disagree, in various specific technicalities there
is very much material in terms of foundations and the infinite but in
your initial analyses not so much to directly contradict. For the
most part, I think the majority of people here have little idea, or
care, of the contents of discussion. I appreciate your comments,
because they illustrate a refreshing open-mindedness as opposed to
knee-jerk jingoism. I think that you recognize that there are
perceived and addressable deficiencies in standard theories is a sign
of philosophical maturity, Larry. Larry, I'd appreciate having a name
to call you.

About omega + 1 > omega, that was a typographical error, excuse me.
As is hopefully obvious from the context, in illustrating a difference
between regular ordinal arithmetic and an ordinal arithmetic with
irregular ordinals, of a sort, where infinity = infinity + 1 for a
mechanistic reason, the standardly correct statement (fact) was meant.

I don't ramble incoherently about mathematics, I ramble in a very
concise and specific manner. I'm direct, dammit. That I saw reason
to inspect the status quo's foundation of mathematics and find them
lacking, is, I would agree, not a conventional perspective.

Moe, you would deny in a set theory where everything is a set that
everything is a set. Here's another one, consider for example G(zero)
= G(alpha) = G(lambda), G(i) defined up to On, the "proper class" of
ordinals, to be "the union restricted to i and i is an ordinal",
there's Burali-Forti again, the paradox. You can't have the simple
predicate x=x to define a set in ZFC. That is to say, one of the most
direct truisms: tautology, identity, can't be used to apply to sets in
ZFC, because there is no universe in ZFC. Yet, in theories containing
the universe of ZFC, ZFC contains itself. When you read the above,
does it not make sense?

The above defined transfinite recursion schema about sets dense in the
reals shows ZFC inconsistent.

Ross

--
Finlayson Consulting

.

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