http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv1i1p17bwm Fundamenta Mathematicae 1920 | 1 | 1 | 129-131 Sur la notion d'ensemble fini Kazimierz KuratowskiJ?zyki publikacji FR Abstrakty FR Le but de cette note est d'introduire une definition d'un ensemble fini et de demontrer son equivalence avec la definition donnee par Wac?aw Sierpi?ski.
(PDFからOCRして手直し引用) Sur la notion d'ensemble fini. Par Casimir Kuratowski (Warszawa).
M. W.Sierpinski a donne dans son ouvrage L'axiome de M. Zermelo et son role dans la Theorie des Ensembles et l'Analyse 1) une nouvelle definition de l'ensemble fini. Cette definition se distingue essentiellement par ce fait qu'elle ne depend ni de la notion de nombre naturel ni de la notion generale de fonction, qui entre d'habitude dans les definitions faisant usaged de la notion de correspondance. La definition en question est la suivante:
"Considerons des classes K d'ensembles dont chacune satisfait aux ,conditions suivante: 1° tout ensemble contenant un seul element appartient a la classe K, 2° si.A. et B sont deux ensembles appartenant a la classe K, leur ensemble-somme A + B appartient aussi a K. Appelons fini tout ensemble qui appartient a chacune des classes K satisfaisant aux conditions 1°et 2°".
Comme on sait, l'ensemble de tous les objets (s'il existe) jouit des proprietes paradoxales : contrairement a un theoreme connu de G. Cantor, la puissance, de cet ensemble ne serait point inferieure a celle de la classe de tous ses sous-ensembles. Il en est de meme de la classe composee de tous les ensembles contenant un seul element; donc, les classes K ne verifient pas, le theoreme de Cantor. En tenant compte de ce fait, on pourrait mettre en doute l'existence meme des classes K.
En modifiant la definition de M. Sierpinski de facon a en supprimer cet inconvenient, j'obtiens la definition suivante:
L'ensemble M est fini, lorsque la classe de tous ses sousensembles (non vides) est l'unique classe satisfaisant aux conditions: 1. ses elements sont des sous-ensembles (non vides) de M; 2. tout ensemble contenant un seul element de M appartient a cette classe; 3. si A et B sont deux ensembles appartenant a cette classe, leur ensemble-sornme A+B lui appartient aussi.
Nous allors demontrer qu'un ensemble fini d'apres cette definition l'est aussi au sens ordinaire et reciproquement. En d'autres termes: pour qu'un ensemble soit fini d'apres la definition proposee, il faut et il suffit que le nombre de ses elements puisse etre exprime par un nombre naturel (la notion de nombre naturel etant supposee connue). En effet,soit M un ensemble dont le nombre d'elements peut etre exprime par un nombre naturel; soit Z une classe quelconque satisfaisant aux conditions 1-3. Nous allons montrer que tout sous-ensemble de M appartient a Z. Il en est ainsi - en vertu de la condition 2 - des sous-ensembles composes d'un seul element; en meme temps, s'il en est ainsi des sous-ensembles contenant n elements, il en est de meme - d'apres 3 - de ceux qui en contiennent n+l. Comme le nombre d'elements de chaque sous-ensemble de M se laisse exprimer par un nombre naturel, il en resulte par induction que Z contient tous les sous-ensembles de M. Donc, la classe Z etant necessairement identique a celle de tous les sous-ensembles de M, elle est l'unique classe satisfaisant aux conditions 1-3. Ainsi, tout ensemble dont le nombre d'elements peut etre exprime par un nombre naturel est un ensemble fini dans notre sens. Supposons, d'autre part, que le nombre d'elements d'un ensemble donne M ne se laisse pas exprimer par un nombre naturel. Designons par Z la classe de tous les sous-ensembles de M dont le nombre d'elenlents peut etre exprime par un nombre naturel. Cette classe satisfait evidemment aux conditions 1-3; en meme temps, d'apres l'hypothese, M n'appartient pas a Z et, par suite, Z n'est pas identique a la classe de tous les sous-ensembles de M; donc, la classe de tous les sous-ensembles de M n'est pas l'unique classe satisfaisant aux conditions 1-3 et M n'est pas fini dans notre sens, c. q. f. d.
(Google 仏→英訳) On the notion of finite set. Through Casimir Kuratowski (Warszawa).
Mr. W.Sierpinski gave in his book The axiom of Mr. Zermelo and his role in the Theory of Ensembles and Analysis 1) a new definition of the finite set. This definition is essentially distinguished by the fact that it does not depend either on the notion of natural number or on the general notion of function, which usually enters into the definitions that make use of the notion of correspondence. The definition in question is as follows:
"Consider classes K sets each of which satisfies the following conditions: 1 ° any set containing a single element belongs to class K, 2 ° si.A. and B are two sets belonging to the class K, their set-sum A + B also belongs to K. Let's call finite everything that belongs to each of classes K satisfying conditions 1 ° and 2 ° ".
?As we know, the set of all objects (if it exists) enjoys paradoxical properties: unlike a theorem known to G. Cantor, the power of this set would not be inferior to that of the class of all its subassemblies. It is the same of the class composed of all the sets containing a single element; therefore, K classes do not check, Cantor's theorem. ?Taking this fact into account, one could question the very existence of classes K.
By modifying Mr. Sierpinski's definition so as to remove that drawback, I get the following definition:
The set M is finite, when the class of all its subsets (not empty) is the only class satisfying the conditions: 1. its elements are subsets (not empty) of M; 2. any set containing a single element of M belongs to this class; 3. if A and B are two sets belonging to this class, their set -sorn A + B also belongs to it.
?We can show that a finite set according to this definition is also in the ordinary sense and reciprocally. In other words: for a set to be finite according to the proposed definition, it is necessary and sufficient that the number of its elements can be expressed by a natural number (the notion of natural number being assumed to be known). ?Indeed, let M be a set whose number of elements can be expressed by a natural number; let Z be any class satisfying the conditions 1-3. We will show that every subset of M belongs to Z. This is - under condition 2 - subsets composed of a single element; at the same time, if this is so subsets containing n elements, it is the same - according to 3 - of those which contain n + 1. Since the number of elements of each subset of M is expressed by a natural number, it follows by induction that Z contains all the subsets of M. Therefore, since the class Z is necessarily identical to that of all the subsets of M, it is the only class satisfying the conditions 1-3. Thus, any set whose number of elements can be expressed by a natural number is a finite set in our sense. ?Suppose, on the other hand, that the number of elements of a set gives M does not let itself be expressed by a natural number. Let Z be the class of all the subsets of M whose number of elements can be expressed by a natural number. This class obviously satisfies conditions 1-3; at the same time, according to the hypothesis, M does not belong to Z and, consequently, Z is not identical to the class of all the subsets of M; therefore, the class of all subsets of M is not the only class satisfying the conditions 1-3 and M is not finite in our sense, c. q. f. d. (引用終り) 以上 0543現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/30(土) 21:00:02.37ID:4Ujjq2jv>>542 補足
?We can show that a finite set according to this definition is also in the ordinary sense and reciprocally. ↑ ?は、先頭のブランクが、文字化けしているんだ Google翻訳の仕様なのでしょうね 目で見ると、ブランクで通常と変わりないが、5CH板に貼ると化けるんだ(^^; 0544現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/30(土) 21:01:00.40ID:4Ujjq2jv>>540
というなら、それは誤り 0612現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/06(金) 07:56:31.62ID:eTcHIROk 参考 https://en.wikipedia.org/wiki/Axiom_of_infinity Axiom of infinity (抜粋) It was first published by Ernst Zermelo as part of his set theory in 1908.[1]
References [1] Zermelo: Untersuchungen uber die Grundlagen der Mengenlehre, 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f.
https://glossar.hs-augsburg.de/Zermelo,_E._(1908):_Untersuchungen_%C3%BCber_die_Grundlagen_der_Mengenlehre Datenschutz Uber GlossarWiki Lizenzbestimmungen (抜粋) Zermelo, E. (1908): Untersuchungen uber die Grundlagen der Mengenlehre Zermelo (1908b): Ernst Zermelo; Untersuchungen uber die Grundlagen der Mengenlehre; in: Mathematische Annalen; Band: 65; Nummer: 2; Seite(n): 261?281; https://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065 (このサイトからPDFが落とせる) Untersuchungen uber die Grundlagen der Mengenlehre. I. Von E. ZERMELO in Gottingen. P261
PDFをOCRして、表題だけGoogle翻訳すると Untersuchungen uber die Grundlagen der Mengenlehre. I. Von E. ZERMELO in Gottingen. ↓ Studies on the basics of set theory. I. From E. ZERMELO in Gottingen.
(>>612より) https://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065 (このサイトからPDFが落とせる) Untersuchungen uber die Grundlagen der Mengenlehre. I. Von E. ZERMELO in Gottingen. P261 (抜粋英訳) P263 Axiom I. If every element of a set M is simultaneously an element of N and vice versa, that is, if M = E N and N = E M at the same time, then M = N is always M or shorter: every set is determined by its elements.
P266 But in order to secure the existence of "infinite" sets, we still need the following axiom, which derives from its essential content by Mr. R. Dedekind. Axiom VII. The domain contains at least a set Z which contains the null set as an element and is such that each of its elements a is another element of the form {a}, or which with each of its elements a is also the corresponding set {a } as an element. (Axiom of the infinite.) 14 VII. *) If Z is an arbitrary set of the properties required in VII, then for each of its subsets Z1 it is definite whether it possesses the same property. For if a is any element of Z1 ', it is definite whether {a} ∈ Z1, and all the elements a of Z1 thus constituted form the elements of a subset Z1' for which it is definite whether Z1 '= Z1 or Not. Thus, all subsets Z1 of the considered property form the elements of a subset T = E UZ, and the average corresponding to them (# 9) Z0 = DT is an amount of the same nature. つづく 0618現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土) 08:43:56.92ID:H2e5WMAT>>617 つづき
For once 0 is a common element of all elements Z1 of T, and on the other hand, if a is a common element of all these Z1, then also {a} is common to all and therefore also an element of Z0. If Z 'is any other quantity of the nature required in the axiom, then in the same way as Z0 it corresponds to Z for a smallest subset Z0' of the property under consideration. Now, however, the average [Z0, Z0 '], which is a common subset of Z and Z', must have the same properties as Z and Z and, as a subset of Z, the constituent Z0 and, as a subset of Z ', the constituent Z0 ' contain. After I it follows that [Z0, Z0 '] = Z0 = Z0', and that Z0 is therefore the common component of all possible quantities, such as Z, although these do not need to form the elements of a set. The set Z0 contains the elements 0, {0}, {{0}}, and so on, and may be called a "series of numbers" because their elements can represent the location of the numerals. It is the simplest example of a "countless infinite" set (Nos. 36).
(ドイツ語原文) P263 Axiom I. Ist jedes Element einer Menge M gleichzeitig Element von N und umgekehrt, ist also gleichzeitig M =E N und N =E M, so ist immer M = N. Oder kurzer: jede Menge ist durch ihre Elemente bestimmt.
P266 Um aber die Existenz "unendlicher" Mengen zu sichern, bedurfen wir noch des folgenden, seinem wesentlichen Inhalte von Herrn R. Dedekind**) herruhrenden Axiomes. Axiom VII. Der Bereich enthalt mindestens eine Menge Z, welche die Nullmenge als Element enthalt und so beschaffen ist, das jedem ihrer Elemente a ein weiteres Element der Form {a} entspricht, oder welche mit jedem ihrer Elemente a auch die entsprechende Menge {a} als Element enthalt. (Axiom des Unendlichen.) 14 VII. *) Ist Z eine beliebige Menge von der in VII geforderten Beschaffenheit, so ist fur jede ihrer Untermengen Z1 definit, ob sie die gleiche Eigenschaft besitzt. Denn ist a irgend ein Element von Z1' so ist definit, ob auch {a} ε Z1 ist, und alle so beschaffenen Elemente a von Z1 bilden die Elemente einer Untermenge Z1', fur welche definit ist, ob Z1' = Z1 ist oder nicht. Somit bilden alle Untermengen Z1 von der betrachteten Eigenschaft die Elemente einer Untermenge T =E UZ, und der ihnen entsprechende Durchschnitt (Nr. 9) Z0 = DT ist eine Menge von der gleichen Beschaffenheit.
Denn einmal ist 0 gemeinsames Element aller Elemente Z1 von T, und andererseits, wenn a gemeinsames Element aller dieser Z1 ist, so ist auch {a} allen gemeinsam und somit gleichfalls Element von Z0. Ist nun Z' irgend eine andere Menge von der im Axiom gefordertenN Beschaffenheit, so entspricht ihr in gen au derselben Weise wie Z0 dem Z eine kleinste Untermenge Z0' von der betrachteten Eigenschaft. Nun mus aber auch der Durchschnitt [Z0, Z0'] , welcher eine gemeinsame Untermenge von Z und Z' ist, die gleiche Beschaffenheit wie Z und Z haben und als Untermenge von Z den Bestandteil Z0, sowie als Untermenge von Z' den Bestandteil Z0' enthalten. Nach I folgt also, das [Z0, Z0'] = Z0 = Z0' sein mus, und das somit Z0 der gemeinsame Bestandteil aller moglichen wie Z beschaff (men Mengen ist, obwohl diese nicht die Elemente einer Menge zu bilden brauchen. Die Menge Z0 enthalt die Elemente 0, {0}, { {0} } usw. und moge als "Zahlenreihe" bezeichnet werden, weil ihre Elemente die Stelle der Zahlzeichen vertreten konnen. Sie bildet das einfachste Beispiel einer "abzahl bar unendlichen" Menge (N r. 36). (引用終り) 以上 0621現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土) 08:49:51.11ID:H2e5WMAT>>618 補足
(引用開始) The set Z0 contains the elements 0, {0}, {{0}}, and so on, and may be called a "series of numbers" because their elements can represent the location of the numerals. It is the simplest example of a "countless infinite" set (Nos. 36). 注:36節(Nos. 36 P280)で、ZERMELOは無限("unendliche")について論じている。 (引用終り)
https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem Lowenheim?Skolem theorem (抜粋) The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem.
Many consequences of the Lowenheim?Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.
Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute. 0628現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/12/07(土) 14:54:57.80ID:H2e5WMAT>>626-627
(引用開始) レーヴェンハイム−スコーレムの定理 定理の上方部分の証明は、いくらでも大きな有限のモデルを持つ理論は無限のモデルを持たねばならないことをも示す。 The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. (引用終り)