The axiom of choice and two-point sets in the plane

In this paper we prove that it consistent to have a two-point set in a model of ZF in which the real line cannot be well-ordered. We prove two results related to a construction of Chad of a two-point set inside the countable union of concentric circles. We show that if the reals are the countable union of countable sets, then every well-orderable set of reals is countable. However, it is consistent to have a model of ZF in which the reals are the ω1 increasing union of sets of size ω1 and ω2 can be embedded into the reals. A two-point set is a subset of the plane which meets every line in exactly two points. It is an open question whether a two-point set can be Borel. In work on the two-point problem Chad [1] came up with the question of whether it might be possible to have a model of ZF in which the reals are the countable union of countable sets and there is an uncountable well-orderable set of reals. If this were possible, then Chad’s method of construction of two-point sets inside concentric circles could be used to construct a Borel example of a two-point set. However, it is impossible: Theorem 1 Suppose the reals are the countable union of countable sets. Then every well-orderable set of reals is countable. Proof Let p : ω × ω → ω be a fixed bijection. For each n ∈ ω define the map πn : 2 ω → 2 by: πn(x) = y iff ∀m ∈ ω y(m) = x(p(n,m)). 1 Mathematics Subject Classification 2000: 03E25 03E15