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

On 11/19/2024 02:38 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.
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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.

Bourbaki: is a French panel of "algebraic geometers".
Now, you should know that algebra and geometry are
two _different_ theories, besides for what all and
where they may agree, in part, in parts.
So, some for example Lefschetz make for being much
more algebraic GEOMETERs, than, ALGEBRAIC, geometers.
One of the concepts out of Bourbaki is, "strictly",
with regards to their vacillations about "positive",
whether "positive always means non-zero", "strictly".
No, not like "WM". We're algebraic GEOMETERS.
Then, besides models of potential, practical, effective,
and/or actual infinity, with regards what is "finite"
and what is "infinite", is here for what would be
matters of "the infinite limit", that results finites.
(Finite quantities.)
Otherwise it's gratifying you might recall that
remark in passing, because, it's a thing.
In "Replacement of Cardinality (infinite middle)", 8/19 2024, this was:

>

I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering
and that all the orderings of the set are well-orderings.
That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?
The entire idea here that there's a particular _regularity_
due dispersion and modularity only courtesy division down
from a fixed-point, that "Peano's axioms" don't give integers,
they only give increments, i.e. not necessarily constant increments,
that there's more than one _regularity_, REQUIRED, is another
little fact of mathematics missing from your neat little hedgerow.