TITLE: Axiom of choice and a set in the plane that intersects every line in two points
QUESTION [9 upvotes]: In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects each line exactly twice'' (called $2$-point sets [I think]).
I was talking with an students and we were wondering if, in the absence of Choice, this result is still true.
To make my question more precise, I know that we only need Choice to well order the reals and, I believe, you can make the set to be non-measureble but I'm not sure if it is non-measurable itself (I believe that using CH you can make a $2$-point set of measure zero [but, to be honest, I haven't work out the details]).
So, being more concrete, how much choice is need to create such a set? Are there models of ZF without them?

REPLY [3 votes]: I wasn't aware of A. Miller's paper http://www.math.wisc.edu/~miller/res/two-pt.pdf up until recently. In Miller's model, DC (dependent choice) is false (there is a Dedekind finite infinite set). In a paper with M. Beriashvili I show that ZF plus DC plus the existence of a 2-point set does not imply that there is a well-ordering of the reals, see https://ivv5hpp.uni-muenster.de/u/rds/mazurkiewicz_sets.pdf or https://ivv5hpp.uni-muenster.de/u/rds/mazurkiewicz.pdf .