  The completeness theorem for  L implies  that Val(L) — regarded as a subset of H(ω) — is recursively enumerable, and hence Σ1 on H(ω). Similarly, the completeness theorem for  L(ω1,ω) (see §2) implies that Val(L(ω1,ω)) — regarded as a subset of H(ω1) — is Σ1 on H(ω1). However, this pleasant state of affairs collapses completely as soon as  L(ω1,ω1) is reached. For one can prove