https://en.wikipedia.org/wiki/Finite_set Finite set (抜粋) Contents 1 Definition and terminology 2 Basic properties 3 Necessary and sufficient conditions for finiteness 4 Foundational issues 5 Set-theoretic definitions of finiteness 5.1 Other concepts of finiteness
Foundational issues Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory. Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets. Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo?Fraenkel set theory with the axiom of infinity replaced by its negation.
Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. The difficulty stems from Godel's incompleteness theorems. One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular, there exists a plethora of so-called non-standard models of both theories. A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model.
(This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately.
More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann?Bernays?Godel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.
A formalist might see the meaning[citation needed] of set varying from system to system. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.
Set-theoretic definitions of finiteness In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form {\displaystyle \{x\,|\,x<n\}}{\displaystyle \{x\,|\,x<n\}}. Mathematicians more typically choose to ground notions of number in set theory, for example they might model natural numbers by the order types of finite well-ordered sets. Such an approach requires a structural definition of finiteness that does not depend on natural numbers.
Various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. (Kuratowski's is the definition used above.)
A set S is called Dedekind infinite if there exists an injective, non-surjective function {\displaystyle f:S\rightarrow S}f:S\rightarrow S. Such a function exhibits a bijection between S and a proper subset of S, namely the image of f. Given a Dedekind infinite set S, a function f, and an element x that is not in the image of f, we can form an infinite sequence of distinct elements of S, namely {\displaystyle x,f(x),f(f(x)),...}x,f(x),f(f(x)),.... Conversely, given a sequence in S consisting of distinct elements {\displaystyle x_{1},x_{2},x_{3},...}x_{1},x_{2},x_{3},..., we can define a function f such that on elements in the sequence {\displaystyle f(x_{i})=x_{i+1}}{\displaystyle f(x_{i})=x_{i+1}} and f behaves like the identity function otherwise. Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers. Dedekind finite naturally means that every injective self-map is also surjective.
Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice. Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S).[8] Intuitively, K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons.
Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:
・X contains the empty set; ・For every set T in P(S), if X contains T then X also contains the union of T with any singleton.
Then K(S) may be defined as the intersection of M.
In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs. That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks. (引用終り) 以上 0508現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木) 00:22:27.05ID:QdpmOFrx>>504 追加
Necessary and sufficient conditions for finiteness In Zermelo?Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:[citation needed]
2.(Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.)
Set-theoretic definitions of finiteness
Various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. (Kuratowski's is the definition used above.)
Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice. Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S).[8] Intuitively, K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons.
Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:
X contains the empty set; For every set T in P(S), if X contains T then X also contains the union of T with any singleton. Then K(S) may be defined as the intersection of M.
In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs. That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks. 0510現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木) 00:30:28.71ID:QdpmOFrx>>508-509 ＞ 2.(Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.) ＞Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that: ＞X contains the empty set; ＞For every set T in P(S), if X contains T then X also contains the union of T with any singleton. ＞Then K(S) may be defined as the intersection of M.
なるほど ”Kuratowski finiteness”の定義では、 CやRやQやNのシングルトン {C}や{R}や{Q}や{N} 達は 有限集合にはならんな！ 思った通りだったな！ｗｗ（＾＾； 0511現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木) 00:37:22.52ID:QdpmOFrx>>509 >Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S).[8] Intuitively, >K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons.
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
