ZFC proves that the class of ordinals is not weakly compact for definable classes

Abstract

We prove that the class of all ordinals Ord is not weakly compact with respect to definable classes. Specifically, in any model of ZFC, the definable tree property fails for Ord, in that there is a definable Ord tree with no definable cofinal branch; the definable partition property fails, in that there is a definable 2-coloring of pairs from a certain definable proper class, with no definable homogeneous proper class; and the definable compactness property fails for $\mathcal{L}_{\infty,\omega}$, in that there is a definable theory in this logic all of whose set-sized subtheories are satisfiable, but which has no definable proper class model. In addition, we prove that the definable diamond principle $\Diamond_{\rm Ord}$ holds if and only if there is a definable well-ordering of the universe. And we prove that the common theory of all spartan models of G\"odel-Bernays set theory, those having only definable classes, is $\Pi^1_1$-complete.