Epsilon-induction

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In mathematics, $\in$ -induction (epsilon-induction) is a variant of transfinite induction, which 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:

$\forall x (\forall y (y \in x \rightarrow P(y)) \rightarrow P(x)) \rightarrow \forall x P(x)$

This principle is equivalent to the axiom of regularity. It can be converted into a transfinite induction on the rank of the set x.

The name is most often pronounced "epsilon-induction", because the set membership symbol $\in$ historically developed from the Greek letter $\varepsilon$ .