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 Π20-complete and that the set of Cauchy problems which
locally have a unique solution is Σ30-complete. We prove that the set
of Cauchy problems which have a global solution is Σ40-complete and
that the set of ordinary differential equation which have a global solution for
every initial condition is Π30-complete. We prove that the set of Cauchy
problems for which both uniqueness and globality hold is Π20-complete