## Set-theoretic constructions of two-point sets

### Volume 203 / 2009

Fundamenta Mathematicae 203 (2009), 179-189
MSC: 03E35, 54G99.
DOI: 10.4064/fm203-2-4

#### Abstract

A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than $ \hbox {ZFC}$, we demonstrate two new constructions of two-point sets. Our first construction shows that in $ \hbox {ZFC}+ \hbox {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 $ \hbox {ZF}$, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice.