Vaught's theorem on axiomatizability by a scheme

Abstract

In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS₂ of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS₂ has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.