Epsilon-induction

In mathematics, $in$-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:

: "$forall\; x\; (forall\; y\; (y\; in\; x\; ightarrow\; P\; [y]\; )\; ightarrow\; P\; [x]\; )\; ightarrow\; forall\; x\; P\; [x]$"

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 $in$ historically developed from the Greek letter $epsilon$.