Let U be an admissible structure. A cPCd(U) class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$ , where K̄ is an U-r.e. set of relation symbols and φ is an U-r.e. set of formulas of L∞ω that are in U. The main theorem is a generalization of the following: Let U be a pure countable resolvable admissible structure such that U is not Σ-elementarily embedded in HYP(U). Then a class K of countable (...) structures whose universes are sets of urelements is a cPCd(U) class if and only if for some Σ formula σ (with parameters from U), M is in K if and only if M is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$ , where HYPU(M), the smallest admissible set above M relative to U, is a generalization of HYP to structures with similarity type Σ over U that is defined in this article. Here we just note that when Lα is admissible, HYPLα(M) is Lβ(M) for the least β ≥ α such that Lβ(M) is admissible, and so, in particular, that HYPHF(M) is just HYP(M) in the usual sense when M has a finite similarity type. The definition of HYPU(M) is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

§0. Introduction. In [16], Barwise described his graduate study at Stanford. He told of his interactions with Kreisel and Scott, and said how he chose Feferman as his advisor. He began working on admissible fragments of infinitary logic after reading and giving seminar talks on two Ph.D. theses which had recently been completed: that of Lopez-Escobar, at Berkeley, on infinitary logic [46], and that of Platek [58], at Stanford, on admissible sets.Barwise's work on infinitary logic and admissible sets is described (...) in his thesis [4], the book [13], and papers [5]—[16]. We do not try to give a systematic review of these papers. Instead, our goal is to give a coherent introduction to infinitary logic and admissible sets. We describe results of Barwise, of course, because he did so much. In addition, we mention some more recent work, to indicate the current importance of Barwise's ideas. Many of the central results are stated without proof, but occasionally we sketch a proof, to indicate how the ideas fit together.Chapters 1 and 2 describe infinitary logic and admissible sets at the time Barwise began his work, circa 1965. From Chapter 3 on, we survey the developments that took place after Barwise appeared on the scene.§1. Background on infinitary logic. In this chapter, we describe the situation in infinitary logic at the time that Barwise began his work. We need some terminology. By a vocabulary, we mean a set L of constant symbols, and relation and operation symbols with finitely many argument places. As usual, by an L-structureM, we mean a universe set M with an interpretation for each symbol of L. (shrink)