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

On Oct 8, 11:16 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
On Oct 8, 6:38 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Oct 8, 5:58 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
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.

No, in denial, when Goedel claims that no consistent theory can be
complete,

But Godel did not say that.

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."

Aside from the fact that you're incorrectly summarizing Godel, now you
go on to say that your point is based on taking a different
definition. That's no interesting. Anyone can get different results
just by changing definitions.

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.

Except (1) a union of consistent theories is not necessarily a
consistent theory and (2) there is no effective procedure for
determining whether a theory is consistent, so there'd be no effective
procedure for determining the axioms of such a union theory.

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,

I'm not an expert on the subject of cranks, but I do know enough about
it that I can see that certain people are indeed cranks through and
through.

[...]

About omega + 1 > omega, that was a typographical error, excuse me.

Thanks. That's a bit of progress. Though, relatively, not much more
than closing the curtains in face of a hurricaine.

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.

Now you're not just rambling, but also lying about me.

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?

No.

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

Sure it does, Ross, sure it does. Now time for the rec room, Ross, to
meet all your friends there.

MoeBlee

.

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