A note on transfinite iteration

Extract

For some of the well-known set-theoretic operations a natural definition of transfinite iterations (powers) seems impossible. If, however, an operator D has the property that, for every X, D(X) ⊃ X, we may write the formula Dⁿ(X) = D(Dⁿ⁻¹(X)) (which defines as usual the finite powers of D) in the form . Assuming this formula to be valid when n is any ordinal, we obtain a natural definition of transfinite powers of those operators. Obviously the dual definition would work in the case D(X) ⊂ X.

As an example of the use of this definition, we show how the sets used by von Neumann to represent ordinals can be constructed by means of the operator N(X) = X⋃{X}. Denoting the union of the elements of X by S(X), we have for the set E(ξ), which represents the ordinal ξ, E(ξ) = S(N^ξ({ϕ})), as can easily be verified by transfinite induction. As the sets E(ξ) are different for different values of ξ, the powers of N are all different as well.

This is not always the case. The powers of certain operators start repeating themselves from a certain ordinal onward. If an operator has the property that, for every set A of sets a, , then the sequence of its powers becomes constant already from ω on. This is a special case of the following theorem.