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

On 11/19/2024 07:59 PM, Ross Finlayson wrote:

On 11/19/2024 07:45 PM, Ross Finlayson wrote:

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.
>
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.
>
>
>

>
>
This went on for some time, ....
>
>
>
 >>>> Why is it you think that Stackel's definition of finite
 >>>> and "not Dedekind's definition of countably infinite"
 >>>> don't agree?
 >>
 >> I don't think they disagree, normally.
 >>
 >> Note: If you mean Dedekind's definition of infinite,
 >> it isn't limited to countably.infinite.
 >>
 >>>> 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.
 >>>
 >>> ..., REQUIRED, ....
 >>
 >> Things missing from my neat little hedgerow are
 >> missing because I intend for them to be missing.
 >> My neat little hedgerow has no weeds.
 >> It has not had and will not have weeds.
 >> And weeds would not be an improvement.
 >>
 >> My neat little hedgerow is well.ordered;
 >> each non.empty subset holds a minimum.
 >>
 >> In my neat little hedgerow,
 >> each Little Bunny Foo Foo has a successor,
 >> scooping up the field mice and bopping them on the head,
 >> and is a successor, except the first, named 0.
 >>
 >> Successors are non.0 non.doppelgänger non.final.
 >>
 >> You are welcome to talk about something else, Ross.
 >> Note, though, that,
 >> if you are talking about something else,
 >> then you are talking about something else.
 >> Non.triangles are not counter.examples to triangles.
 >> Non.Bunny.Foo.Foos are not counter.examples to Bunny.Foo.Foos.
 >>
 >> Have a nice day.
 >>
 >>
 >
 > The other day I read or leafed through and enjoyed
 > this pretty good little book called "Us & Them: The
 > Science of Identity", by a D. Berreby. Now, I don't
 > necessarily adhere to any same opinions, yet it's
 > rather didactic and establishes a sort of discourse
 > about what is so and considered so and what's not
 > and considered not.
 >
 > Then, the idea that that sort of reflexivity is or isn't
 > symmetrical, about the usual notions of conservation
 > and symmetry in this sort of world, is explored as
 > for matters of Berreby's opinion and lens about
 > how science that isn't physics or "mathematical",
 > i.e. that it's "non-logical", at all, isn't science.
 >
 > So, for nominalist fictionalists of the formalist
 > sort, while there may be strong mathematical
 > platonists who are also formalist constructivists,
 > it's suggested that a reading of Berreby might
 > result them being non-logical and fundamentally
 > as of matters of mere opinion and not of relevance,
 > here as with respect to the Relephant, since at least
 > times when flying rainbow sparkle ponies were
 > putative models of continuous domains or "sets
 > of reals", and various ones at that.
 >
 >
 >
 > Huntington's postulates are mentioned again,
 > quite all about universals. (A president of the MAA.)
 >
 > Peter of Spain's appositve and suppositive and
 > about use/mention distinction making it so that
 > "terms" in some "universal particulars" are
 > REQUIRED their context, helps explain why
 > theories like universal ordinals for any model
 > of an integer continuum and the duBois-Reymond
 > long-line of all real expressions, which has a larger
 > cardinal than c and is on the same line already,
 > all make one milieu, and it's logical.
 >
 > No-one's trying to take away your triangles,
 > nor anything else that's mathematical for
 > that matter, it is though pointed out that
 > this wider world of a strong mathematical
 > platonist's universal criteria _always exists_,
 > basically pointing out that you can't wish that away.
 >
 > Often this is mentioned, "that is like the pot,
 > one of the implements in the fire along with
 > the kettle, who are both blackened by the fire,
 > that is like the pot, calling the kettle black,
 > when indeed the pot and the kettle are both
 > quite black", yet it's not relevant here, because,
 > the issue is that for all your reasonable and correct
 > criticisms of perceived and demonstrated formal
 > incorrectness according to formal constructions,
 > then you claim ignorance of "theories with universes",
 > for example, without which there isn't one, or,
 > this simple "only diagonal" after you've spent
 > an entire course establishing why the non-constructive
 > "anti-diagonal" makes your system of inequalities
 > giving measure after least-upper-bound (axiomatized)
 > and measure 1.0 (axiomatized), why the one is so yet
 > the other with pretty much the exact same form
 > is not: it demonstrates that a hedgerow without
 > it would be a mathematical absurdity, and thus
 > not mathematical.
 >
 > Or, you know, trivial, which is acceptable for itself,
 > a "fragment", of a, "the mathematics", this though
 > is about a "the mathematics" for _all_ the objects
 > of the universe of mathematical objects, including
 > itself.
 >
 > For example, at one point it was brought out that
 > in the theories about relations of triangles, that
 > sine and cosine and the Pythagorean has another
 > way to make it, where the Pythagoarean triples
 > are basically the end result or the completions
 > instead of the other way around, demonstrating
 > that lines don't make points and points don't make
 > lines, according to induction, yet they do, according
 > to deduction, hence/whence/thence they do.
 > Then for example the Phythian, more or less
 > does the same for uniqueness of Fourier-style series,
 > liberating what are some "uniqueness" results to
 > "distinctness" results, and making more "repleteness"
 > of this "completeness", "re-pletion".
 >
 > 2500 years later, ....
 >
 > So anyways, there's basically the "only diagonal" bit
 > that sets up there's a non-Cartesian function so that
 > there's a model of a countable continuous domain,
 > with least-upper-bound and measure according to
 > there being sigma algebras established, then you
 > get both or none.
 >
 > Of course you needn't _apply_ such definitions,
 > that for whatever reasons you don't use, there's
 > though for the mathematically conscientious,
 > that one established is an _enduring effect_.
 >
 >
>
Sort of like you don't apply the inductive cases
that each stay "nope" and instead only affirm
that each one "goes", where, it goes.
>
Where it _goes_.
>
Deductively, that would be a wrong case for induction,
it's not justified, or in the usual sense of the word
it's not juxtaposed, as to where it _goes_, where it _goes_.
>
>
Then a claim like "I don't pick wrong" eventually
results there are "right" and "wrong", and "why",
and answering all the question words.
>
This is then that once established the enduring effect
of the mathematical proof or the existence of a model
its structure, those being equi-interpretable or same,
has that a _restriction_ of comprehension, like defining
away the character of an inductive set, is _not_ the
unrestricted comprehension of the naive and natural,
and afterward it is _not_ unique thusly.
>
>
Mein Hut hat Drei Ecken ....
>
>
So, well-order the reals. Ha, they already are (well-ordered).
Pick one of "only" or "anti" diagonal. Get both or none.
>
>
Good sir
>
>

You see, if you have a mathematical hedge-row,
in it are infinitely many bunnies foo-foo, and as
you go thrashing through the row bagging each
you find, eventually, one will say, "there's another
bunny in this hedge-row:  BUNNY FOO FOO".
"And he's going to bop you in the head."