Construction of sentences with specific interpretability properties

A construction is presented for generating sentences that satisfy a recursively enumerable set of interpretability properties. This construction is then used to prove three previously announced results concerning the lattice of local interpretability types of theories (also known as the Lattice of Chapters). The purpose of this paper is to prove three theorems originally announced without proof in [MPS]. The theorems are presented as Corollaries 10, 11, and 12 below; they state that no interval in the Lattice of Chapters bounded by r.e. theories is complemented, that the chapter of any r.e. theory is the meet of the chapters of finitely axiomatizable theories above it, and that the meet of the chapters of an r.e. set of finitely axiomatizable theories that all lie strictly above a given r.e. chapter is itself strictly above the given chapter. (As an interesting consequence of the second result it follows that there is a finitely axiomatizable theory T such that |ZF| < |T | < |GB|, and as a consequence of the third it follows that for any r.e. set A of sentences none of which has a finite model, there is a sentence τ , also having no finite models, interpretable in every member of A.) The proofs of the corollaries are all based on a general technique for constructing sentences having very specific interpretability strengths. This general technique is given in Theorem 4, which is the main technical result of the paper. Although this paper can be read independently of [MPS], the reader is encouraged to consult that work for further information and background on the Lattice of Chapters. For present purposes, the following remarks (up through Lemma 3) will serve as a brief introduction. The collection of all first-order sentences can be partially ordered by the relation of interpretability. This yields a naturally induced equivalence relation: two sentences are equivalent if each is interpretable in the other. The equivalence classes are called chapters; the chapter containing a sentence