Ali Enayat and I have proved that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,
- there is a definable Ord-tree with no definable cofinal branch.
- there is a definable 2-coloring of a definable proper class, with no definable homogeneous proper class.
- there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.
This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal property that reasonable people might have expected to hold true.
The article is now available:
- A. Enayat and J. D. Hamkins, ZFC proves that the class of ordinals is not weakly compact for definable classes, manuscript under review. (arχiv, blog post)