Łoś–Vaught test

In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence, the theory contains either the sentence or its negation but not both.

Statement

A theory ${\displaystyle T}$ with signature σ is ${\displaystyle \kappa }$-categorical for an infinite cardinal ${\displaystyle \kappa }$ if ${\displaystyle T}$ has exactly one model (up to isomorphism) of cardinality ${\displaystyle \kappa .}$

The Łoś–Vaught test states that if a satisfiable theory is ${\displaystyle \kappa }$-categorical for some ${\displaystyle \kappa \geq |\sigma |}$ and has no finite model, then it is complete.

This theorem was proved independently by Jerzy Łoś (1954) and Robert L. Vaught (1954), after whom it is named.

See also

• Robinson's joint consistency theorem – Theorem of mathematical logic

References

• Enderton, Herbert B. (1972), A mathematical introduction to logic, Academic Press, New York-London, p. 147, MR 0337470.
• Łoś, Jerzy (1954), "On the categoricity in power of elementary deductive systems and some related problems", Colloquium Mathematicum, 3: 58–62, MR 0061561.
• Vaught, Robert L. (1954), "Applications to the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability", Indagationes Mathematicae, 16: 467–472, MR 0063993.