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
.
