  Let A be a countable transitive set such that  LA is a sublanguage of  L(ω1, ω) and let Δ be a set of sentences of  LA. We say that A (or, by abuse of terminology,  LA) is Δ-closed if, for any formula φ of  LA such that Δ  ⊢A φ, there is a deduction D of φ from Δ such that D ∈ A. It can be shown that the only countable language which is Δ-closed for arbitrary Δ is the first-order language  L,  i.e., when A = H(ω). However J. Barwise discovered that there are countable sets A ⊆ H(ω1) whose corresponding languages  LA differ  from  L and yet are Δ-closed for all Σ1-sets of sentences Δ. Such sets A are called admissible sets; roughly speaking, they are extensions of the hereditarily finite sets in which recursion theory—and hence proof theory—are still  possible (for the full definition, see Section 5.1 below).