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einductin
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In mathematics, * * * * ∈ * * * {\displaystyle \in } * -induction (Epsilon-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. In symbols:* * * * ∀ * x * * * ( * * * ∀ * y * ( * y * ∈ * x * → * P * [ * y * ] * ) * → * P * [ * x * ] * * * ) * * * → * ∀ * x * * P * [ * x * ] * * * {\displaystyle \forall x{\Big (}\forall y(y\in x\rightarrow P[y])\rightarrow P[x]{\Big )}\rightarrow \forall x\,P[x]} * This principle, sometimes called the axiom of induction (in set theory), is equivalent to the axiom of regularity given the other ZF axioms. * * * * ∈ * * * {\displaystyle \in } * -induction is a special case of well-founded induction. The Axiom of Foundation (regularity) implies epsilon-induction. * The name is most often pronounced "epsilon-induction", because the set membership symbol * * * * ∈ * * * {\displaystyle \in } * historically developed from the Greek letter * * * * ϵ * * * {\displaystyle \epsilon } * . |