[過去ﾛｸﾞ] 現代数学の系譜 カントル 超限集合論 (1002ﾚｽ)
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508(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木)00:22 ID:QdpmOFrx(1/7)調 AAS
>>504 追加
https://en.wikipedia.org/wiki/Finite_set
Finite set
(抜粋)
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.)
つづく
509(2): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木)00:24 ID:QdpmOFrx(2/7)調 AAS
>>508
つづき
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.
510(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木)00:30 ID:QdpmOFrx(3/7)調 AAS
>>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（＾＾；
536(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/30(土)20:49 ID:4Ujjq2jv(1/17)調 AAS
>>508 追加
Kuratowsk有限(1920),iは、仏文らしいね（＾＾；
https://en.wikipedia.org/wiki/Finite_set
Finite set
(抜粋)
References
・Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini" (PDF), Fundamenta Mathematicae, 1: 129?131
http://matwbn.icm.edu.pl/ksiazki/fm/fm1/fm1117.pdf
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.
つづく
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