Title: The axiom of choice and two-point sets in the plane
Author: Arnold W. Miller
Abstract:
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
omega1 increasing union of sets of size omega1 and omega2
can be embedded into the reals.
Latex2e: 8 pages
Latest version at: www.math.wisc.edu/~miller