∈-induction

From Freepedia

In mathematics, ∈-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). The induction hypothesis is

for every set x, if ∀y (y∈x → P(y)), then P(x),

from which we can infer that P(x) holds for all sets x. The principle is equivalent to the axiom of regularity.

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