Epsilon-induction

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In mathematics, ε-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:

This principle, sometimes called the axiom of induction (in set theory), 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 historically developed from the Greek letter .