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Subject: Re: Replacement of Cardinality
Date: Wed, 28 Aug 2024 15:30:31 +0200
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Le 28/08/2024 à 15:25, WM a écrit :
> 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.

"pun intended"