In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a
theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
to be
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, 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 ''T'' is
''κ''-categorical for an infinite cardinal ''κ'' if ''T'' has exactly one model (up to isomorphism) of cardinality ''κ''.
The Łoś–Vaught test states that if a satisfiable theory is ''κ''-categorical for some κ ≥ ℵ
0 and has no finite model, then it is complete.
This theorem was proved independently by and , after whom it is named.
References
* .
* .
* .
{{DEFAULTSORT:Los-Vaught test
Mathematical logic
Model theory
Theorems in the foundations of mathematics