Re: Intéressant...

On 03/03/2024 01:36 PM, Olivier Miakinen wrote:

[crosspost supprimé]
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Le 03/03/2024 21:48, Richard Hachel a écrit :

Le 03/03/2024 à 21:44, Ross Finlayson a écrit :
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Je ne le connais pas, mais j'ae etudie "l'INFINI",
l'infinit mathematique, apres j'ae entendu que
c'est un nombre plus grand que lui-meme, "+1".

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  Il faut en parler aux mathématiciens.
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  Ils auront peut-être ta réponse.

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Il y a tellement de sens différents du mot infini en mathématiques qu'il
est impossible de répondre à une déclaration aussi vague.
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Par exemple, si on considère le cardinal (en gros le nombre d'éléments) des
ensembles infinis, alors l'infini des entiers est le même que celui des
entiers pairs, et le même que celui des rationnels, mais il est strictement
plus petit que celui des nombres réels.
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Il existe des infinis qui ne sont pas des cardinaux d'ensembles, par exemple
la limite d'une suite de nombres réels, suite qui pourrait tendre vers plus
l'infini ou vers moins l'infini.
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Et je n'ai pas parlé des ordinaux infinis. Bref.
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Bonjour, voici on ecrit un essai, bref, mais en anglais,
sur les concettes mathematique de les "infinitesimals".
Excuse moi, or pardonnez-moi si vousd voudriez, mais,
mais, nous pouvons establir un peu des preliminaires,
qui suivents ces concettes.
Interessant, ...
https://fr.wikipedia.org/wiki/Concept_essentiellement_contest%C3%A9
https://fr.wikipedia.org/wiki/Concept_(philosophie)
On peut savoir "Kant a decrive le Sublime, il presente
la notion d'un sens-objet qui peut comprendre l'infini".
https://fr.wikipedia.org/wiki/Recherche_philosophique_sur_l%27origine_de_nos_id%C3%A9es_du_sublime_et_du_beau
https://fr.wikipedia.org/wiki/Sublime
Jean Duns Scotus et Baruch Spinoza, ils sont philosophes,
qui sonts attribue un idee ";'infinite c'est verite",
comme, "l'infinite il verifierait", "'infinite c'est
ce que qui a fait".
Si Kronecker a dit "ils ne sonts pas mais les Anzahlen",
il y a sept siecles Scotus a dit "mais bien sur, Es muss sein".
Dot dot dot, ....
This is that: the first infinitesimal,
_standardly_, is zero.
It's "not.ultimately.untrue".
Now, there are others, kinds of infinitesimals,
it's called "nonstandard analysis".
Of course, that's only since, before Weierstrass,
the integral calculus was called "infinitesimal
analysis", because, it involves quantities that
are _not_ standard and finite, in the infinite limit.
So, while MacLaurin well detailed this, with that
MacLaurin's was the best rigorous formalization,
then there was Weierstrass with the best rigorous
formalization, then Cauchy defined some things,
making Cauchy-Weierstrass, then around Cantor,
Dedekind stepped in, what results that needing
to combine set theory and point-sets, with
analysis, resulted that being best rigorous
formalization.
So, for modern mathematics, where it all arises
as "descriptive set theory: a model theory",
after axiomatic set theory, then there was
Robinson, Robinsohn, with "hyper-reals", yet
only a conservative extension of ZF, infinitesimals,
and Nelson with "Internal Set Theory: co-consistent
with ZF", that there are infinitesimals, these
are others, kinds of infinitesimals, in
what's called "nonstandard analysis",
in modern mathematics today.
Then, the idea of infinitesimals and a continuum
of them, has that there are many models of infinitesimals,
for example Peano, Dodgson, Veronese, Stolz, you
can point to Bishop and Cheng with "a rather
restricted transfer principle", Paris and Kirby
or Boucher with "an infinity", even something like
Sergeyev's "infinity computer", and of course
Conway's, "sur-real numbers", a non-Archimedean
field, while yet, as of a sort of, conservative
expression usually according to ZF.
Especially duBois-Reymond and the Infinitarcalcul
and orders of infinity, and infinitesimals, much
anticipates the usual cumulative hierarchy or
ordinals in set theory.
Yet, most people's intuitive notion of a clock
arithmetic, that from zero to one goes empty
to full, "standard infinitesimals", is for a
sort of "continuum infinitesimal analysis",
after formalizing the infinite limit and
continuum limit, to make for a nonstandard
analysis, with its own, and even deeper,
contribution to real analytical character.
So, it's a very stunning and beautiful fact
of mathematics, that the continuous domain
has at least three different models:
line-reals,
field-reals the standard,
signal-reals,
and their "rather restricted transfer principle"
as Bishop and Cheng put the idea when building
in "constructivist mathematics" this kind of thing,
though resulting yet a "conservative extension"
because otherwise it would introduce a "contraction",
that here it's actually constructivist that it's
just a result that what bridges the _discrete_
and the _continuous_ is so unique that it's
as a function not a Cartesian function, and
that of course via inspection, is not contradicted,
otherwise about the fact of mathematics, that is,
uncountability.
So, yeah, usually it's just called "zero".