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

On 11/14/2024 5:20 AM, WM wrote:

On 14.11.2024 00:16, Jim Burns wrote:

A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
each claim is true.or.not.first.false
is
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
each claim is true.
>
Some claims are true and we know it
because
they claim that
when we say this, we mean that,
and we, conscious of our own minds, know that
when we say this, we mean that.
>
Some 𝗰𝗹𝗮𝗶𝗺𝘀 are not.first.false and we know it
because
we can see that
no assignment of truth.values exists
in which 𝘁𝗵𝗲𝘆 are first.false.
𝗾 is not first.false in ⟨ 𝗽 𝗽⇒𝗾 𝗾 ⟩.
>
Some finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲𝘀 of 𝗰𝗹𝗮𝗶𝗺𝘀 are
each true.or.not.first.false
and we know it.
>
When we know that,
we know each claim is true.
>
We know each claim is true, even if
it is a claim physically impossible to check,
like it would be physically impossible
to check each one of infinitely.many.

>
Here is a single claim which is true:

You don't say what reason you (WM) have
for knowing that that single claim is true.
Is that single claim in some finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲𝘀 of 𝗰𝗹𝗮𝗶𝗺𝘀
which are each true.or.not.first.false?
Highly unlikely that that's your reason.
You (WM) sneer at such sequences.
More likely, you assign what.you.mean so that
part of what.you.mean is that what.you.say is true.
AKA the Axiom of Because.You.Say.So.
Something along the lines of this:

Some claims are true and we know it
because
they claim that
when we say this, we mean that,
and we, conscious of our own minds, know that
when we say this, we mean that.

However,
that doesn't work for anything beyond
saying what you are conscious of in your own mind,
for anything beyond axioms and definitions.
There was a demagogue who "changed"
the projected path of a hurricane by
drawing on a map with a felt.tip.pen.
It didn't work.
You (WM) seem to be trying to do
the kind of same thing.
It doesn't work.

The covering of a geometric figure by
a set of similar smaller intervals is independent of the order of the intervals.
That holds for every finite figure and,

You are drawing on the word 'finite'
with a felt.tip.pen,
trying to make 'finite' mean
"WM is correct, because WM says so"
⎛ A finite set A can be ordered so that,
⎜ for each subset B of A,
⎜ either B holds first.in.B and last.in.B
⎜ or B is empty.
⎜
⎜ An infinite set is not finite.
⎜ An infinite set C can _only_ be ordered so that
⎜ there is a non.empty subset D of C, such that
⎜ either first.in.D or last.in.D or both don't exist,
⎜ not visibly and not darkly.
⎜
⎝ And our sets do not change.
You cannot change what.we.mean
with a felt.tip.pen.

by applying the analytical limit,
also for infinite figures like

You are calling sets which are finite
_according to what those words mean_
infinite sets
in order to prove (WM.style) that
infinite sets behave finitely.