21/05/08 19:39:01.83 X3IvmoGN.net
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つづき
URLﾘﾝｸ(en.wikipedia.org)
Epsilon-induction
In mathematics, {\displaystyle \in }\in -induction (epsilon-induction or set-induction) is a variant of transfinite induction.
Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction.
It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction.
Contents
1 Statement
1.1 Comparison with natural number induction
2 Independence
Comparison with natural number induction
The above can be compared with {\displaystyle \omega }\omega -induction over the natural numbers {\displaystyle n\in \{0,1,2,\dots \}}{\displaystyle n\in \{0,1,2,\dots \}} for number properties Q.
Independence
In the context of the constructive set theory CZF, adopting the Axiom of regularity would imply the law of excluded middle and also set-induction. But then the resulti