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

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 Π2-complete and that the set of Cauchy problems which locally have a unique solution is Σ3-complete. We prove that the set of Cauchy problems which have a global solution is Σ4-complete and that the set of ordinary differential equations which have a global solution for every initial condition is Π3-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is Π2-complete. This paper deals with descriptive set-theoretic questions in the theory of ordinary differential equations (ODEs). Descriptive set theory (DST, from now on) is, roughly, the study of definable sets in Polish (i.e. separable completely metrizable) spaces. Definable means here: being Fσ, Gδ, Borel, analytic, or, more generally, belonging to a “well behaved” collection of sets. The roots of this subject go back to the work of the analysts of the turn of the century: Borel, Lebesgue, and Baire in France and Lusin, Suslin, and Novikov in Russia. After the ’50s DST was revolutionized by the techniques of mathematical logic: these allowed to solve long standing problems and changed the perspective of the subject (see [7] and [4] for more on the history and development of DST). One of the main trends of current research (see [1] and Sections 23, 27, 33 and 37 of [5]) is the classification of natural sets arising in various parts of analysis, topology, etc. A couple of clarifications are in order. To classify here means to pin down the exact complexity of a given set: e.g. to show 1991 Mathematics Subject Classification: Primary 04A15; Secondary 34A12. We thank our colleagues Anna Capietto and Camillo Costantini for helpful conversations.