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

(RAF is getting as notorious as AP and JSH lately, sigh.)
From: "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx>
when Goedel claims that no consistent theory can be complete

No, he says no consistent *AND* arithmetic-containing *AND*
finitely expressible theory can be complete in the sense of
deciding all well-formed sentences.

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."

You are wrong: If you can't prove a sentence is true, and you
can't prove it's false, then what the ***, is it true or not?
What do you say? Are such sentences true or false? Remember that if
S is true then notS is false, so you can't simply make them all
true or all false. Or do you look at the first quantifier, if it
starts "FOR ALL" then it's true but if it starts "THERE EXISTS"
then it's false?

Goedel proved there exists at least one WTF sentence, in fact
probably an infinite number of independent WTF sentences. What do
you say? How do you assign truth to sentences that can be neither
proven nor disproven? You can't just guess, because you'd need an
infinite number of guesses to get all of them decided by fiat.

If Goedel could separate that collection into consistent and
inconsistent theorems,

There's no such thing as an inconsistent theorem.

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

That's a lie. There's just no way you could move your fingers to
type omega + 1 and have 1 + omega appear, or vice versa.
It was a *mental* lapse you suffered, a gaffe not a typo.

I don't ramble incoherently about mathematics,

I disagree: "inconsistent theorem" is incoherent.

You can't have the simple predicate x=x to define a set in ZFC.

I believe you may have said something correct there.
Is that a strawman you denied there?
I never saw anyone attempt to use x=x to define a set. Have you?

in theories containing the universe of ZFC, ZFC contains itself.

It depends on what you mean by "contains", whether you mean
subset or element. It makes a difference, an essential difference.
{1, 2, {5}} contains {5} as an element, and {1} as a subset,
but not vice versa.
{1, 2, {1}} contains {1} as both subset and element, but not for
the same reason.
Should I believe you have no idea what the difference is?

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

I'm guessing you're wrong about this claim.
Can you get ahold of a ZFC interactive theorem-proving program
(you enter your premises, and the rules of inference, and it
checks each step is correct)
and try to make it accept your alleged derivation of an inconsistency?
If you succeed, please post the ID of the program you've used and a
transcript of the session. (Post on Web page, with summary and URL here.)
.

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