A computably stable structure with no Scott family of finitary formulas

One of the goals of computability theory is to find syntactic equivalences for computational properties. The Limit Lemma is a classic example of this type of equivalence: X ⊆ ω is computable from 0′ if and only if it is arithmetically definable by a ∆2 formula. A more relevant example for this paper was proved independently by Ash, Knight, Manasse and Slaman [1] and by Chishom [2]: a computable structure is relatively computably categorical if and only if it has a computably enumerable Scott family of finitary existential formulas. (These terms are defined below.) Our main theorem is a negative result which states that there is a computably stable rigid graph which does not have a Scott family of finitary formulas. Therefore, any attempt to modify the syntactic characterization of relative computably categoricity to capture computable stability using notions such as a Scott family must involve an infinitary language. In this section, we give the background definitions and motivation for this theorem. In the second section, we construct a countable family of sets A with a specific list of enumeration properties. In the last section, we code A into a rigid graph G, prove a partial quantifier elimination theorem for G, and use the enumeration properties of A and the quantifier elimination result for G to show that G does not have a Scott family of finitary formulas. Let M be a countable structure in a computable language whose domain |M| is a subset of ω. The degree of M (denoted deg(M)) is the Turing degree of the atomic diagram of