In mathematics, ∈ -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 ] 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 name is most often pronounced "epsilon-induction", because the set membership symbol ∈ historically developed from the Greek letter ϵ .
