Encyclopedia of Database Systems

2018 Edition
| Editors: Ling Liu, M. Tamer Özsu

• Gösta Grahne
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-8265-9_1254

Definition

Let T be a finite theory expressed in a language L, and φ an L-sentence. Then T finitely entails φ, in notation Tφ, if all finite models of T also are models of φ. A theory T is said to be complete in the finite if for each L-sentence φ either Tφ or T ⊧ ¬ φ. In particular, if T is incomplete (not complete in the finite), then there is an L-sentence φ, such that Tφ and T ⊭ ¬ φ. It follows from classical logic that a first order theory is complete in the finite if and only if all its finite models are isomorphic. Consider now a theory
$${T}_1\,{=}\!\left\{\!\begin{array}{c}\hfill R(a,b){\wedge} R(a,c),\hfill \\ {}\hfill \forall x,y{:}R(x,y){\to} (x,y){=}(a,b){\vee} (a,c),\hfill \\ {}\hfill a\ne b,a\ne c,b\ne c.\hfill \end{array}\right.$$
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