  We see from (5.4 bis) that the reason why the generalized completeness theorem fails for Σ1-sets in  L(ω1,ω) is that, roughly speaking, H(ω1) is not “closed” under the formation of deductions from Σ1-sets of sentences in H(ω1). So in order to remedy this it would seem natural to replace H(ω1) by sets A which are, in some sense, closed under the formation of such deductions, and then to consider just those formulas whose codes are in A.