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

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