現代数学の系譜工学物理雑談古典ガロア理論も読む62

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152: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/10(日) 11:10:57.56 ID:rk/29Zdt

>>151 これ失敗でボツな（＾＾；

貼り直し
>>150
つづき
（参考引用）
https://en.wikipedia.org/wiki/Well-founded_relation
(抜粋)
Well-founded relation
"Noetherian induction" redirects here. For the use in topology, see Noetherian topological space.

In mathematics, a binary relation, R, is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if
(∀ S ⊆ X)[S ≠ Φ → (∃ m ∈ S)(∀ s ∈ S) ￢ (sRm)].
Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

Equivalently, assuming the axiom of dependent choice, a relation is well-founded if it contains no countable infinite descending chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn+1 R xn for every natural number n.[1][2]

In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order.

In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x.
The axiom of regularity, which is one of the axioms of Zermelo?Fraenkel set theory, asserts that all sets are well-founded.

A relation R is converse well-founded, upwards well-founded or Noetherian on X, if

つづく

http://rio2016.5ch.net/test/read.cgi/math/1551963737/152

153: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/10(日) 11:12:05.10 ID:rk/29Zdt

>>152

つづき

Contents
1 Induction and recursion
2 Examples
3 Other properties
4 Reflexivity

Induction and recursion
On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X, R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ X and a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion.
If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.

There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.
When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1551963737/153

171: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/10(日) 19:37:27.48 ID:rk/29Zdt

>>170

つづき

６．さらに、正則性公理の意味の補足
　　「>=, we have 1 >= 1 >= 1 >= ・・・」の例類似で、”∈を使った順序”で、∋は >=では無く、>（等号=含まず）(>>150と>>152)」だとか、
　　正則性公理の意味の別の側面で、それは極小元の存在保証（無限降下列禁止）の意味があるとか、そういう蘊蓄を、付け加えておけば、新歓としては良いだろうね（＾＾

（参考）
http://fuchino.ddo.jp/misc/goedel-universe.pdf
渕野 昌，連続体仮説とゲーデルの集合論的宇宙（ユニヴァース）， 現代思想，2007年2月臨時増刊号 (2007), 94-116
https://www.jstage.jst.go.jp/article/sugaku/65/4/65_0654411/_pdf/-char/ja
特別企画 これから学ぶ人のために 公理的集合論 渕 野 昌 - J-Stage 渕野昌 著 数学 ?2013
https://ja.wikipedia.org/wiki/%E3%83%9A%E3%82%A2%E3%83%8E%E3%81%AE%E5%85%AC%E7%90%86
ペアノの公理 （ここにフォン・ノイマンの構成法がある）
http://evariste.jp/kagami/index.html
かがみのホームページ プロフィール 学生時代の専攻は数学。今の趣味も数学。
http://evariste.jp/kagami/diary/0000/200401.html#20040103-1
2004年1月3日 自然数の構成と ω
http://evariste.jp/kagami/diary/0000/200402.html#20040201-2
2004年2月1日 自然数と数学的帰納法
http://evariste.jp/kagami/diary/0000/200402.html#20040207-2
2004年2月7日 順序
http://evariste.jp/kagami/diary/0000/200403.html#20040320-1
2004年3月20日 順序数の定義
http://evariste.jp/kagami/diary/0000/200403.html#20040322-1
2004年3月22日(月) 整列順序
http://fuchino.ddo.jp/foundation.html
基礎の公理の成り立たない集合論 (non well-founded set theory) について
渕野 昌(Sakae Fuchino) Last modified: Sat Aug 13 14
以上

http://rio2016.5ch.net/test/read.cgi/math/1551963737/171

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