Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

Abstract

We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $\Pi^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $\Sigma^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $\Sigma^0_4$-complete and that the set of ordinary differential equation which have a global solution for every initial condition is $\Pi^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $\Pi^0_2$-complete.