View source for Łoś–Vaught test

In [[Model theory|model theory]], a branch of [[Mathematical logic|mathematical logic]], the '''Łoś–Vaught test''' is a criterion for a [[Theory (mathematical logic)|theory]] to be [[Complete theory|complete]], unable to be augmented without becoming inconsistent. For theories in [[Philosophy:Classical logic|classical logic]], this means that for every [[Philosophy:Sentence (mathematical logic)|sentence]], the theory contains either the sentence or its negation but not both.
==Statement==

A theory <math>T</math> with [[Philosophy:Signature (logic)|signature]] ''σ'' is <math>\kappa</math>-categorical for an infinite cardinal <math>\kappa</math> if <math>T</math> has exactly one model (up to isomorphism) of cardinality <math>\kappa.</math>

The '''Łoś–Vaught test''' states that if a satisfiable theory is <math>\kappa</math>-categorical for some <math>\kappa \geq |\sigma|</math> and has no finite model, then it is complete.

This theorem was proved independently by {{harvs|first=Jerzy|last=Łoś|year=1954|txt}} and {{harvs|first= Robert  L.|last= Vaught |authorlink=Robert Lawson Vaught |year=1954|txt}}, after whom it is named.

==See also==

* {{annotated link|Robinson's joint consistency theorem}}

==References==

{{reflist}}

* {{citation
 | mr = 0337470
 | page = 147
 | publisher = Academic Press, New York-London
 | title = A mathematical introduction to logic
 | url = https://books.google.com/books?id=LHjNCgAAQBAJ&pg=PA147
 | year = 1972}}.
* {{citation
 | journal = Colloquium Mathematicum
 | mr = 0061561
 | pages = 58–62
 | title = On the categoricity in power of elementary deductive systems and some related problems
 | volume = 3
 | year = 1954}}.
* {{citation
 | journal = [[Indagationes Mathematicae]]
 | mr = 0063993
 | pages = 467–472
 | title = Applications to the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability
 | volume = 16
 | year = 1954}}.

{{Mathematical logic}}
{{DEFAULTSORT:Los-Vaught test}}

[[Category:Mathematical logic]]
[[Category:Model theory]]
[[Category:Theorems in the foundations of mathematics]]

{{Sourceattribution|Łoś–Vaught test}}