Vaught's theorem on axiomatizability by a scheme
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.
Permanent link to this document: http://projecteuclid.org/euclid.bsl/1344861888
Digital Object Identifier: doi:10.2178/bsl/1344861888
Zentralblatt MATH identifier: 06083933
Mathematical Reviews number (MathSciNet): MR2987522