Non-well-foundedness of well-orderable power sets

Abstract

Tarski [Tarski] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w(X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs’ ℵ-function easily establishes that there can be no infinite descending sequence under the relation |𝒫(X)| = |Y|.