Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/11/2024 3:41 AM, WM wrote:

On 10.11.2024 18:49, Jim Burns wrote:

On 11/10/2024 4:35 AM, WM wrote:

On 10.11.2024 00:27, Jim Burns wrote:

In the first case, with the not.changing sets,
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 which
  has only true.or.not.first.false 𝗰𝗹𝗮𝗶𝗺𝘀
has only true 𝗰𝗹𝗮𝗶𝗺𝘀.

>
But it

>
"It" refers to who or what?

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To that finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲.
>

But it will never complete
an infinite set of claims.

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We do not need an infinite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 completed.
We do not want an infinite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 completed.

>
But you claim that
_all_ fractions are in bijection with
all natural numbers,
don't you?

Yes, I claim that.
This is one 𝗰𝗹𝗮𝗶𝗺:
⎛ All fractions are in bijection with
⎝ all natural numbers.
We can peel that claim apart,
truthfully and finitely saying what it means, and
we can situate 𝗶𝘁 in a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀,
  each 𝗰𝗹𝗮𝗶𝗺 of which is true.or.not.first.false.
Such a 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 has only true 𝗰𝗹𝗮𝗶𝗺𝘀.
That 𝗰𝗹𝗮𝗶𝗺, being in that 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲, is a true 𝗰𝗹𝗮𝗶𝗺.
⎛ I have been 𝗳𝗹𝗶𝗽𝗽𝗶𝗻𝗴 back 𝗮𝗻𝗱 𝗳𝗼𝗿𝘁𝗵 between 𝗳𝗼𝗻𝘁𝘀,
⎜ in an attempt to visibly mark a distinction between
⎜ claims as bearers of meaning and
⎜ 𝗰𝗹𝗮𝗶𝗺𝘀 as objects, as vibrations in the air,
⎜ smears of colored bear fat on a cave wall, bits, or pixels.
⎜
⎜ Finite sequences of smears of bear fat have
⎜ certain properties which we ("we" very broadly)
⎜ have learned to use in our exploration of infinity.
⎜
⎜ Note: that's _finite_ sequences.
⎜ Issues involving Scrooge McDuck and Disappearing Bob
⎝ do not come into play for these properties.

It will forever remain in the status nascendi.
Therefore
irrelevant for actual or completed infinity.

>
A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.or.not.first.false,
will forever remain
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.or.not.first.false.

>
Therefore
such a sequence does not entitle you
to claim infinite mappings.

No.
Such a 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 does entitle me.
A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.or.not.first.false,
is
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀, each of which
is true.

Infinite sets can correspond to
other infinite sets which,
without much thought about infinity,
would seem to be a different "size".

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But they cannot become such sets.

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Our sets do not change.

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My intervals I(n) = [n - 1/10, n + 1/10]
must be translated to all the midpoints
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5,
2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ...
if you want to contradict my claim.

Your 𝗰𝗹𝗮𝗶𝗺𝘀 start with "Sets change".
If we are in the same discussion,
they are already contradicted at that point,
If we aren't in same discussion,
I don't see how to respond sensibly.
End.of.debate, I guess? If there ever was a debate?