Epsilon-induction

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:

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 ${\displaystyle \epsilon }$.

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