Bulletin of Symbolic Logic

Vaught's theorem on axiomatizability by a scheme

Albert Visser

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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.

Article information

Bull. Symbolic Logic Volume 18, Issue 3 (2012), 382-402.

First available in Project Euclid: 13 August 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B10, 03B30, 03F25

predicate logic axiom scheme


Visser, Albert. Vaught's theorem on axiomatizability by a scheme. Bull. Symbolic Logic 18 (2012), no. 3, 382--402. doi:10.2178/bsl/1344861888. https://projecteuclid.org/euclid.bsl/1344861888

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