[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
上下前次1-新
抽出解除 ﾚｽ栞

---

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索 歴削→次ｽﾚ 栞削→次ｽﾚ 過去ﾛｸﾞﾒﾆｭｰ

---

ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

---

506(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/27(水)22:12 ID:qnEhNItW(11/12) AAS
>>505

つづき

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.
省4

---

507: 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/27(水)22:13 ID:qnEhNItW(12/12) AAS
>>506
つづき

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.
省3

---

上下前次1-新書関写板覧索設栞歴

---

ｽﾚ情報 赤ﾚｽ抽出 画像ﾚｽ抽出 歴の未読ｽﾚ AAｻﾑﾈｲﾙ

---

ぬこの手 ぬこTOP 0.031s