  (3.7) If Constr holds, then there are no cardinals κ for which L(ω1,ω) is compact. Accordingly, it is consistent with the usual axioms of set theory to suppose that there is no cardinal κ such that all languages  L(ω1,ω) are κ-complete. This result is to be contrasted with the fact that all first-order languages are ω-complete.