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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151: 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) 11:09:21.34 ID:rk/29Zdt

or 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/1551963737/151

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.

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.

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

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