This will be a talk I’ll give at the CUNY Graduate Center Graduate Student Colloquium on Monday, April 11 (new date!), 2016, 4-4:45 pm. The talk will be aimed at a general audience of mathematics graduate students.
Abstract. I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if $x\mapsto A_x$ is a function mapping every real $x\in[0,1]$ in the unit interval to a countable set of such reals $A_x\subset[0,1]$, then there are two reals $x$ and $y$ for which $x\notin A_y$ and $y\notin A_x$. To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis. I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of $\neg$CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps $(x,y)\mapsto A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.
Freiling axiom of symmetry on MathOverflow | On Wikipedia | Graduate Student Colloquium