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

On 11/19/2024 5:38 PM, FromTheRafters wrote:

Jim Burns was thinking very hard :

On 11/19/2024 4:38 PM, Ross Finlayson wrote:

On 11/19/2024 11:56 AM, Jim Burns wrote:

On 11/19/2024 12:52 PM, Ross Finlayson wrote:

The "bait-and-switch" and "back-slide"
don't go well together.
>
In either order, ....

>
⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ 3. (Paul Stäckel)
⎜ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has both
⎝ a least and a greatest element in the subset.
>
https://en.wikipedia.org/wiki/Finite_set

>
Yeah we looked at that before also,
and I wrote another, different, definition of finite.

>
Thank you for admitting that.
>
However, you (RF) might NOT be bait.and.switch.ing
if the definitions are equivalent.
>
What was the definition you wrote before?
I didn't see it in the rest of your post.

>
Didn't he use not.ultimately.untrue
instead of not.first.false?
Is that just an inversion?
It seems to imply an ultimate or last
instead of a first or least.
Just like WM
when he inverted the naturals
to unit fractions.

That sounds to me like a reasonable possibility.
I don't myself right now see how to make it work.
All I will ever have are guesses, though,
unless Ross changes his tradition of
not.answering questions like "What do you mean?"
My guess -- and it is only a guess --
is that not.ultimately.untrue is
part of the innards of what I'll name
The Discontinuous Function Paradox
(note: not a paradox).
In The Discontinuous Function Paradox,
each of an infinite sequence is one way
(let us say, each a true claim)
_but the limit is something else_
(a false claim)!!!
_Obviously_ skullduggery is afoot!
Quod erat demonstrandum about something something infinity.