Epsilon-induction

Under Epsilon - induction ( also ∈ - induction) is understood in mathematics, a special proof methods of set theory. Does it prove that a statement is true for all sets, it is sufficient, according to Epsilon - induction to show that it applies to the quantities for the elements of which it applies. Precise words stating the Epsilon - induction so

The validity of the Epsilon - induction can be used in IF ( the axiom of choice is not necessary for this ) prove. Decisive is in the proof of regularity that one. This allows even show that the epsilon induction of regularity is equivalent to. That means you exchanged in the ZF of regularity against the epsilon induction would create an equivalent system of axioms.

Your name, the Epsilon - induction due to the small Greek letter ε, from which developed the current element characters.

Sketch of proof

Usually one proves the Epsilon induction by contradiction. So If it is false, there would be a lot which does not satisfy condition is met. Now we consider the quantity

The transitive closure of is, ie a transitive set that contains as a subset. Per condition can not be empty, so let the regularity provides an epsilon minimal element. Every element of can not be because of the epsilon- minimality of. The elements of are but due to the transitivity of in. So the statement is true for all. The condition provides us now, but this implies the desired contradiction.

Application

The Epsilon - induction is used, for example, used to show that each set is contained in the von Neumann hierarchy. Thus, any set can be found with an ordinal number. In the corresponding proof of the statement is so by

Defined.