  Turning to the Löwenheim-Skolem theorem, we find that the downward version has adequate generalizations to  L(ω1,ω) (and, indeed, to all infinitary languages). In fact, one can show in much the same way as for sets of first-order sentences that if Δ ⊆ Sent(ω1,ω) has an infinite model of cardinality ≥ |Δ|, it has a model of cardinality the larger of  ℵ0, |Δ|. In particular, any  L(ω1,ω)-sentence with an infinite model has a countable model.