  If just deductions of countable length are admitted, then no deductive apparatus for  L(ω1,ω) can be set up which is adequate for deductions from arbitrary sets of premises, that is, for which (2.1) would hold for every set Δ ⊆ Sent(ω1,ω), regardless of cardinality. This follows from the simple observation that there is a first-order language  L  and an uncountable set Γ of  L(ω1,ω)-sentences such that Γ has no model but every countable subset of Γ does. To see this, let  L be  the language of arithmetic augmented by ω1 new constant symbols {cξ : ξ &lt; ω1} and let Γ be the set of  L(ω1,ω)-sentences {σ} ∪ {cξ ≠ cη : ξ ≠ η}, where σ is the  L(ω1,ω)-sentence characterizing the standard model of arithmetic. This example also shows that the compactness theorem fails for  L(ω1,ω) and so also for any  L(κ,λ)  with κ ≥ ω1.