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

On Oct 9, 9:51 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
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

Goedel has a theory about theories that no consistent theory strong
enough to represent, say, the natural integers would be complete. So,
is Goedel's theory incomplete? Then, where that's the only statement
of the theorem, for there to be any different property of his theory,
it is that there is a complete theory. Otherwise, according to
himself, it's inconsistent.

How else would you describe truth, in terms of the objects of a
theory, except in terms of facts proven by theorems of the theory? Is
it not true that 1+1 = 2? Is it not a fact? What do you see as the
difference between a truth and a fact?

Perhaps it's similar to the difference between a proper class, and a
set, where each are defined by their elements.

If everything (each thing) in a set theory is a set, then why not
everything (all things)? That the universe in ZFC is called a proper
class in ZFC with classes instead of a set in ZFC, a set theory,
doesn't help quantifying over sets in the universe (domain of
discourse, all sets) of ZFC. I certainly am not misattributing you,
I'm not a liar, instead simply illustrating via simple plain language
statements the inconsistency of a variety of your assumptions about
(all) the objects of a theory to which you claim adherence (ZFC).

Where Goedel claims that no consistent theory strong enough to
represent the natural integers could be complete, where a variety of
trivial theories are obviously consistent and complete (eg x = x), the
above is not an incorrect summarization of Goedel, just partial and in
context.

Then, for a given (consistent) theory and all of its theorems, i.e.,
sentences finite and infinite in the language of the theory both
consistent and inconsistent, then they can't be separated, else there
would be a collection of one or the other. Now, the consistent ones
are theorems of the theory. They're facts. Being all the theorems
about the objects of the theory, there are not more theorems of the
objects of the theory. Any "new" theorems would be via the addition
of "new" properties of the objects. Then, given a collection of all
theorems, there aren't any "new" theorems without "new" properties of
those objects. If there are undecideable statements about those
objects, that means they have properties unaccounted for in the
original specification of the objects, "hidden properties."

So, there is thus an implicit non-logical axiom for any theory Goedel
analyzes: some anonymous axiom that purports to have a fact about the
objects. Otherwise the objects are fully specified. As there is thus
one for that and another ad infinitum, theories analyzed by Goedel are
no longer finitely axiomatized. Then, Goedel's results don't apply.

Then, as I was trying to explain to Larry before you decided to re-
enter the conversation, there are some reasonable justifications of
the rejection of Goedel's results, that there can be no complete and
consistent finitely-axiomatized theory strong enough to represent the
natural integers. Here, and nobody's ever told me this before, I
introduce the notion that for objects of a theory to have anonymous
properties that there are anonymous non-logical/proper axioms, and
when those are thus witnessed, then the finitely axiomatized theory
analyzed by Goedel has fleas ad infinitum.

Ah, that's a novel concept, in this discussion. Enjoy.

If there are uncountably many irrationals then the class function of
the transfinite recursion schema is defined up to some uncountable
ordinal, and then due the denseness properties of the rationals in the
reals ZFC is inconsistent.

Ross

--
Finlayson Consulting

.

Relevant Pages

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