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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-2-4,
author = {Ben Chad and Robin Knight and Rolf Suabedissen},
title = {Set-theoretic constructions of two-point sets},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {179-189},
zbl = {1175.03029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-2-4}
}
Ben Chad; Robin Knight; Rolf Suabedissen. Set-theoretic constructions of two-point sets. Fundamenta Mathematicae, Tome 205 (2009) pp. 179-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-2-4/