[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む62 (1002ﾚｽ)
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153(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/03/10(日)11:12 ID:rk/29Zdt(10/28) AAS
>>152

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Contents
1 Induction and recursion
2 Examples
3 Other properties
4 Reflexivity
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154: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/03/10(日)11:13 ID:rk/29Zdt(11/28) AAS
>>153

つづき

Other properties
If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer.
Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n ? 1, n ? 2, ..., 2, 1 has length n for any n.

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R) is isomorphic to (C,∈).

Reflexivity
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