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

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.