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

On 11/28/2024 12:34 AM, Jim Burns wrote:

On 11/27/2024 2:59 PM, WM wrote:

On 27.11.2024 16:57, Jim Burns wrote:

On 11/27/2024 6:04 AM, WM wrote:

>

However,
one makes a quantifier shift, unreliable,
to go from that to
⛔⎛ there is an end segment such that
⛔⎜ for each number (finite cardinal)
⛔⎝ the number isn't in the end segment.

>
Don't blather nonsense.

>
Yes:
⎛ for each j in {1,2,3}
⎜ there is k in {4,5,6} such that
⎝ j+3 = k
>
No:
⛔⎛ there is k in {4,5,6} such that
⛔⎜ for each j in {1,2,3}
⛔⎝ j+3 = k
>

If all endsegments are infinite
then infinitely many natbumbers
remain in all endsegments.

>
Yes:
⎛ for each end.segment
⎜ there is an infinite set such that
⎝ the infinite set subsets the end.segment
>
No:
⛔⎛ there is an infinite set such that
⛔⎜ for each end.segment
⛔⎝ the infinite set subsets the end.segment
>
Swapping quantifiers _in that direction_
is not VISIBLY not.first.false.
>
⎛ Many claims are known because
⎜ a finite sequence of claims is known
⎜ in which each claim is true.or.not.first.false.
⎜
⎜ In such a sequence of claims, each is true.
⎜ Thus,
⎝ we have a stake in not.first.false.ness.
>
>
Consider the sequence of claims.
⎛⎛ [∀∃] for each end.segment
⎜⎜ there is an infinite set such that
⎜⎝ the infinite set subsets the end.segment
⎜
⎜⎛ [∃∀] there is an infinite set such that
⎜⎜ for each end.segment
⎝⎝ the infinite set subsets the end.segment
>
We cannot SEE,
just by looking at the claims,
that, after [∀∃], [∃∀] is not.first.false.
>
>
On the other hand,
swapping quantifiers _in the other direction_
is VISIBLY not.first.false.
>
⎛⎛ [∃∀′] there is a Grand Poobah such that
⎜⎜ for each Water Buffalo
⎜⎝ the Grand Poobah is over the Water Buffalo.
⎜
⎜⎛ [∀∃′] for each Water Buffalo
⎜⎜ there is a Grand Poobah such that
⎝⎝ the Grand Poobah is over the Water Buffalo.
>
However little we know about
the Loyal Order of Water Buffaloes,
we can SEE,
just by looking at the claims,
that, after [∃∀′], [∀∃′] is not.first.false.
>
>
What we want is truth,
but visible not.first.false.ness is not truth.
>
In a certain sense,
visible not.first.false.ness may be better,
more accessible, more visible than truth,
and, sometimes,
when the stars align just right,
visible not.first.false.ness is also truth.
>
>

It's at least not false on its own account.
It's rather so that inductive accounts may not
vouch their own infinite completions, as being
"partial", yet that deductive inference may
arrive at that they're yet "whole", "full".
This though I read as more reasonable than
your previous accounts, about not.ultimately.untrue.