Geodesic
Set-theoretic constructions of two-point sets
Ben Chad 1   ; Robin Knight 2   ; Rolf Suabedissen 3
1 St Edmund Hall Oxford, OX1 4AR, UK
2 Worcester College Oxford OX1 2HB, UK
3 Lady Margaret Hall Oxford, OX2 6QA, UK
Fundamenta Mathematicae, Tome 203 (2009) no. 2, pp. 179-189 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/fm203-2-4
Keywords: two point set subset plane which meets every line exactly points working models set theory other hbox zfc demonstrate constructions two point sets first construction shows hbox zfc hbox there exist two point sets which contained within union countable collection concentric circles second construction shows certain models hbox existence two point sets without explicitly invoking axiom choice

Ben Chad  1   ; Robin Knight  2   ; Rolf Suabedissen  3

1 St Edmund Hall Oxford, OX1 4AR, UK
2 Worcester College Oxford OX1 2HB, UK
3 Lady Margaret Hall Oxford, OX2 6QA, UK 
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Ben Chad; Robin Knight; Rolf Suabedissen. Set-theoretic constructions of two-point sets. Fundamenta Mathematicae, Tome 203 (2009) no. 2, pp. 179-189. doi: 10.4064/fm203-2-4

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