Timeline for Axiom of choice and a set in the plane that intersects every line in two points

7 events

when toggle format	what		by	license	comment
Feb 21, 2019 at 13:45	comment	added	Will Brian		Great question! This one is closely related: mathoverflow.net/questions/272527/….
Feb 21, 2019 at 9:59	answer	added	Ralf Schindler		timeline score: 3
Apr 13, 2017 at 12:58	history	edited	CommunityBot		replaced http://mathoverflow.net/ with https://mathoverflow.net/
Dec 4, 2015 at 0:56	comment	added	Ashutosh		On measure and category: There is a 2-point set within $\{z \in \mathbb{R}^2 : \|z\| \in C\})$ where $C$ is any set of reals which, say, meets every interval on a perfect set . So there is always a meager null 2-point set.
Dec 4, 2015 at 0:13	comment	added	Ashutosh		Arnie Miller showed that you don't need a well ordering of reals in the sense that there are ZF models with 2-point sets where the set of reals cannot be well ordered: math.wisc.edu/~miller/res/two-pt.pdf He also showed that in $L$, there are conanalytic 2-point sets which is the best known upper bound - An analytic 2-point set is necessarily Borel. Mauldin has results connecting this to geometric measure theory: math.unt.edu/~mauldin/papers/no100.pdf
Dec 3, 2015 at 23:14	comment	added	Asaf Karagila♦		It seems that this is open whether or not such set can even be Borel, let alone $G_\delta$. In that case, I think you can probably construct one without appealing to the axiom of choice.
Dec 3, 2015 at 23:04	history	asked	Iván Ongay Valverde	CC BY-SA 3.0