Set-theoretic constructions of two-point sets

A two-point set is a subset of the plane which meets every Une in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice. © Instytut Matematyczny PAN, 2009.