Definable partitions and reflection properties for regular cardinals
Let \(\kappa\) \(\to^{\Sigma_ n}(\kappa)^ 2_ 2\) mean that every partition of \([\kappa]^ 2\) into two sets defined by a \(\Sigma_ n\)- formula has a homogeneous set of order type \(\kappa\). Let for every n \(\kappa_ n\) be the first \(\kappa\) for which \(\kappa\) \(\to^{\Sigma_ n}(\kappa)^ 2_ 2\). The author relates the above partition properties to certain \(\Pi^ 1_ 1-reflection\) properties. It follows that assuming \(V=L\) the \(\kappa_ n\) lie between the first inaccessible and the first Mahlo cardinal. In particular \(\kappa_ 2\) lies strictly between these cardinals. It is left open whether \(\kappa_ 1\) is the first inaccessible and whether in general \(\kappa_ n<\kappa_{n+1}\).
scientific article; zbMATH DE number 3937169