現代数学の系譜　カントル　超限集合論

[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
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
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｡

505: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:11:18.53 ID:qnEhNItW

>>504

つづき

(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.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/505

506: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:12:50.56 ID:qnEhNItW

>>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.

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.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/506

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 0.044s