Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/02/2024 07:23 PM, Ross Finlayson wrote:

On 12/02/2024 05:22 PM, Jim Burns wrote:

On 12/2/2024 7:43 PM, Ross Finlayson wrote:

On 12/02/2024 04:32 PM, Jim Burns wrote:

On 12/2/2024 9:28 AM, WM wrote:

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Infinite endsegments contain an infinite set each,
infinitely many elements of which
 are in the intersection.

>
Yes to:
⎛ regarding finite.cardinals,
⎜ for  each  end.segment   E(k)
⎜ there is a subset S such that
⎝ for each finite cardinal j, j < |S| ≤ |E(k)|
>
No to:
⛔⎛ regarding finite.cardinals,
⛔⎜ ⮣ there is a subset S such that ⮧
⛔⎜ ⮤ for  each  end.segment   E(k) ⮠
⛔⎝ for each finite cardinal j, j < |S| ≤ |E(k)|
>
A quantifier shift tells you (WM) what you (WM) _expect_
  but a quantifier shift is untrustworthy.
>

An empty intersection cannot come before
an empty endsegment has been produced by
losing one element at every step.

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No.
Because see below. [redacted by JB]

>

The usual idea of wrestling with a pig is
that you both get dirty, and the pig likes it.

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However it seems to you,
I'm not really hostile to the pig,
even if it does piss me off sometimes.
Let it like the wrestling.
Am I hurt by that?
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I get dirty.
Oh! I get dirty! Oh my! Oh my! Oh my!
Have you never gotten dirty, Ross?
One gets dirty,and then one gets clean again.
And then, one gets dirty.
Welcome to life.
>

Quit letting that pig dirty things.

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Refresh my memory, Ross.
Was it you, Ross, who told me that,
even though I tell you
I'm talking about standard integers,
you will take me to be talking about _everything_
?
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In what way is talking with you (RF)
different from wrestling with a pig
?
>
Oh, wait. I forgot.
You (RF) don't answer questions.
Never mind.
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You mean like "do you pick?".
>
Remember "do you pick?".
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See, in mathematics, all of which are mathematical
objects, in one theory called mathematics, there's
the anti-diagonal argument, which here used to be
called the diagonal argument which is the wrong name,
has after an only-diagonal argument, what results
that you either get both or none, though that the
only-diagonal itself is constructive for itself
while the anti-diagonal is a non-constructive argument
when you look at it that way.
>
So, "do you pick?".
>
I think us long-term readers can generously,
generously, aver the "Burse's memories", are,
at best, regularly erased.
>
Many of which elicited a spark of thought
then out-went-the-lights. Most of which
went starkers bat-shit.
>
>
So, do you even remember? Or did you just get told again?
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See, the results here, for example about the
extra-ordinary after Mirimanoff, are about
making "bridge" results in mathematics -
those results that "bridge" and make the
"analytical bridges", "analytische Bruecken",
that _resolve_ paradoxes in logic, not make
more contradictions and make "work" contradicting,
the ideal goal of the "bridge" results" are what
make for "a mathematics", and not just whatever
pet theory.
Then when you got "Borel versus Combinatorics",
essentially, and _neither_ win, then _neither_ does
mathematics, which means it's not doing work.
It ain't workin'.
So, Mirimanoff, is considered clarion and truthful,
so, add up _all_ the theories of mathematics,
and arrive at one.
Of course this is about the most usually totally
dogmatic academic tradition since antiquity.
Which is strikingly successful, and, very much
for the standard linear curriculum - and then some.
And then some - and the remainder, not a corner.
Some will have that a constructivist mathematics,
which to constructivists is the only mathematics
of any sort - has no contradictions at all, and
not even any "proofs by contradiction", except
matters of the symmetric, reciprocal, and closed.
Which provide proofs of elimination, ..., more than whim.
So anyways I most certainly enjoy sometimes writing
with reasonably demonstrably well-meaning yet
either wrongly guided or partial and incomplete
mathematical reasoners about foundations and all
the objects of the mathematics - it usually enough
just results a balk and silence, with all these
great bridge results connecting the promontory ponts
of the features: of mathematics.
The in-teg-ral.