Sequences of n-diagrams

Abstract

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.