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 $\Pi^0_2$-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

The iterability hierarchy above I3

In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below

Lebesgue density and exceptional points

Work in the measure algebra of the Lebesgue measure on N2 : for comeagre many [A] the set of points x such that the density of x in A is not defined is \u3a30 3-complete; for some compact K the set of points x such that the density of x in K exists and it is different from 0 or 1 is \u3a00 3-complete; the set of all [K] with K compact is \u3a00 3-complete. There is a set (which can be taken to be open or closed) in \u211d such that the density of any point is either 0 or 1, or else undefined. Conversely, if a subset of \u211dn is such that the density exists at every point, then the value 1/2 is always attained on comeagre many points of the measurable frontier. On the route to these results we show that the Cantor space can be embedded in a measured Polish space in a measure-preserving fashio