  It follows that (5.1) and (5.2) fail when  “L” is replaced by  “L(ω1,ω)” and  “H(ω)” by  “H(ω1)”. Moreover, it can be shown that the set Γ ⊆ Sent(ω1,ω) in (5.3 bis) and (5.4 bis) may be taken to be Σ1 on H(ω1). Thus the compactness and generalized completeness theorems fail even for Σ1-sets of  L(ω1, ω)-sentences. 