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

On Oct 9, 11:41 am, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

Goedel has a theory about theories that no consistent theory strong
enough to represent, say, the natural integers would be complete.

No, you're leaving out that the theory must be formally axiomatizable.
We'd say recursively axiomatizable these days. The theory that is the
set of true sentences of arithmetic is complete and consistent; but
it's not recursively axiomatizable. And Godel doesn't have a "theory
about theories" so much as it would be more reasonable to say he
proved certain theorems about theories.

So,
is Goedel's theory incomplete?

What "Godel's theory"?

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.

I have no idea what you're trying to say that makes sense.

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?

Truth of sentences in the language of a theory is defined by the
method of models.

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?

As a formal sentence, it's true in any model of that sentence. And
it's true in the standard model of the language of arithmetic. Also,
it would seem to me that the sentence expresses a proposition that is
a basic finitary fact.

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

Perhaps someday, before or after Doomsday, who knows, you'll actually
read a book on mathematical logic.

If everything (each thing) in a set theory is a set, then why not
everything (all things)?

You'd have to ask someone who thinks not.

That the universe in ZFC is called a proper
class in ZFC with classes instead of a set in ZFC,

I have no idea what you mean by "the universe 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).

When you said, "[Moe] would deny in a set theory where everything is a
set that everything is a set", you said something untrue of me. But
i'll give you the benefit of the doubt now that you didn't intend to
lie.

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.

Your argument after that point went on by missing the crucial fact
that what are incomplete are recursively axiomatized theories. Godel
never said that a theory that is not recursively axiomatized can't be
sufficient for arithmetic, consistent and complete.

Then, for a given (consistent) theory and all of its theorems, i.e.,
sentences finite and infinite in the language of the theory

We're not dealing with infinite sentences in the context of Godel's
incompleteness.

both
consistent and inconsistent, then they can't be separated, else there
would be a collection of one or the other.

I don't know what you're trying to say. A sentence on itself is
inconsistent iff it's a self-contradiction. Anyway, it's a corollary
of Church's theorem that the set of self-contradictions is not
recursive. Maybe that's what you have in mind.

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."

None of what you said generally holds for first order theories as
discussed in ordinary mathematical logic. So I really don't know what
theories you think you're talking about.

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.

You really need to read a book on mathematical logic so that you can
write a post using the words you use but in a meaningful way.

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.

I enjoy various things. Your babbling is not one of them.

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.

Breathtaking.

MoeBlee

.

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