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(Redirected from ∈-induction)
In mathematics, 'ε-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:
: ''
ightarrow P(y))
ightarrow P(x))
ightarrow orall x P(x)''
This principle 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 .