Re: Replacement of Cardinality

Le 28/08/2024 à 08:13, Jim Burns a écrit :

On 8/27/2024 3:11 PM, WM wrote:

The function exists if
actual infinity exists.
The function does not exist if
only potential infinity exists.
>

¬∃ᴿx>0: NUF(x) = 1

>
Then NUF(x) does not exist
and infinity is not actual
and sets are not complete.

 A potentially.infiniteᵂᴹ set is
an infiniteⁿᵒᵗᐧᵂᴹ set.

A collection.

 An actually.infiniteᵂᴹ set is
a not.potentially.infiniteᵂᴹ set with
a potentially.infiniteᵂᴹ subset.

Subcollection.

¬∃ᴿx>0: NUF(x) = 1

>
Then NUF(x) does not exist

 What exists?
 I propose a very conservative answer:
that we accept at least
the empty set existsᴲ,

Does it?
Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German the word "Menge" has the meaning of many or great quantity. Often we find in German texts the expression "große (great or large) Menge", rarely the expression "kleine (small) Menge". Therefore Bolzano apologizes for using this word in case of sets having only two elements: "Allow me to call also a collection containing only two parts a set." [B. Bolzano: "Einleitung zur Grössenlehre", J. Berg (ed.), Friedrich Frommann Verlag, Stuttgart (1975) p. 152]
Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non-empty set of less than two elements: "For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is element of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element, shall be excluded completely for certain reasons, although it may be convenient for other investigations to fabricate such." [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig (1887), 2nd ed. (1893) p. 2]
Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [B. Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]
Gottlob Frege shared his opinion: "If, according to our previous use of the word, a class consists of things, is a collection, a collective union of them, then it must disappear when these things disappear. If we burn down all the trees of a forest, then we burn down the forest. Thus an empty class cannot exist." [G. Frege: "Kleine Schriften", I. Agelelli (ed.), 2nd ed., Olms, Hildesheim (1990) p. 195]
Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O; P  O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146] And even Ernst Zermelo who made the "Axiom II. There is an (improper) set, the 'null-set' 0 which does not contain any elements" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same author himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)] So it is all the more courageous that Zermelo based his number system completely on the empty set: { } = 0, {{ }} = 1, {{{ }}} = 2, and so on. He knew that there is only one empty set. But many ways to create the empty set can be devised, like the empty set of numbers, the empty set of bananas, the uncountably many empty sets of all real singletons, and the empty set of all these empty sets. Is it the emptiest set? Anyhow, "zero things" means "no things". So we can safely say (pun intended): Nothing is named the empty set.
Regards, WM