Set-theoretic constructions of two-point sets

Volume 203 / 2009

Ben Chad, Robin Knight, Rolf Suabedissen Fundamenta Mathematicae 203 (2009), 179-189 MSC: 03E35, 54G99. DOI: 10.4064/fm203-2-4


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.


  • Ben ChadSt Edmund Hall
    Oxford, OX1 4AR, UK
  • Robin KnightWorcester College
    Oxford OX1 2HB, UK
  • Rolf SuabedissenLady Margaret Hall
    Oxford, OX2 6QA, UK

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