# Epsilon-induction

In mathematics, ${\displaystyle \in }$-induction (epsilon-induction) is a variant of transfinite induction.

It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction.

## Statement

It states that 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:

${\displaystyle \forall x{\Big (}\forall y\,(y\in x\rightarrow P(y))\rightarrow P(x){\Big )}\rightarrow \forall z\,P(z)}$

## Independence

Considered as alternative set theory axiom, it's called axiom of induction.

Given the other ZF axioms, the statement is equivalent to the axiom of regularity or foundation.

In the context of CZF however, foundation implies the law of excluded middle while the induction sheme does not. In other words, with constructive logic, set induction is strictly weaker than foundation.