A Spector-Gandy theorem for cPCd() classes
Shaughan Lavine · Journal of Symbolic Logic · 1992
Abstract Let be an admissible structure. A cPCd( ) class is the class of all models of a sentence of the form , where is an -r.e. set of relation symbols and Φ is an -r.e. set of formulas of ℒ∞,ω that are in . The main theorem is a generalization of the following: Let be a pure countable resolvable admissible structure such that is not Σ-elementarily embedded in HYP( ). Then a class K of countable structures whose universes are sets of urelements is a cPCd( ) class if and only if for some Σ formula σ (with parameters from ), is in K if and only if is a countable structure with universe a set of urelements and σ, where , the smallest admissible set above relative to , is a generalization of HYP to structures with similarity type Σ over that is defined in this article. Here we just note that when Lα is admissible, HYPLα( ) is Lβ( ) for the least β ≥ α such that Lβ( ) is admissible, and so, in particular, that is just HYP( ) in the usual sense when has a finite similarity type. The definition of 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.