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URLﾘﾝｸ(en.wikipedia.org)
Epsilon-induction
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In mathematics, ∈ -induction is a variant of transfinite induction that can be used in set theory to prove that all sets satisfy a given property P[x].
If the truth of the property for x follows from its truth for all elements of x, for every set x, then the property is true of all sets.
This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity given the other ZF axioms. ∈ -induction is a special case of well-founded induction.
The Axiom of Foundation (regularity) implies epsilon-induction.
URLﾘﾝｸ(en.wikipedia.org)
Axiom of regularity
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However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on {(n,α)｜ n∈ ω ∧ α is an ordinal }.
Given the other axioms of Zermelo?Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction.
The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.
In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.
Contents
1.1 No set is an element of itself
1.2 No infinite descending sequence of sets exists
1.3 Simpler set-theoretic definition of the ordered pair
2 The axiom of dependent choice and no infinite descending sequence of sets implies regularity
3 Regularity and the rest of ZF(C) axioms
4 Regularity and Russell's paradox
(引用終り)
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