axiomatizable class

Let L be a first order language and T a theory in L. Recall that a model M is an L-structureMathworldPlanetmath such that M satisfies every sentenceMathworldPlanetmath in T. We say that the structure M is a model of T. Let us write Mod(T) the class of all L-structures that are models of T.

Definition. A class K of L-structures is said to be axiomatizable if there is a theory T such that K=Mod(T). Furthermore, K is a finitely axiomatizablePlanetmathPlanetmath or elemenary class if T is finite.

For example, the class of groups is elementary (and hence axiomatizable), because the set of group axioms is finite. However, the class of infinite groups is axiomatizable but not elementary. Similarly, the class of R-modules is elementary iff R is finite. The class of locally finite groups is an example of a non-axiomatizable class.


  • K is an elementary class iff there is a sentence φ such that K=Mod({φ}), for sentences φ1,,φn can be combined to form φ1φn, which is also a sentence since it has no free variablesMathworldPlanetmathPlanetmath.

  • A class is axiomatizable iff it is an intersectionMathworldPlanetmath of elementary classes. As such elementary class is sometimes abbreviated EC, and axiomatizable class ECΔ, where Δ means is another symbol for intersection.

  • A caution to the reader: some authors call an elementary class an axiomatizable class that is defined here.

Title axiomatizable class
Canonical name AxiomatizableClass
Date of creation 2013-03-22 17:34:38
Last modified on 2013-03-22 17:34:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 03C52
Synonym axiomatisable class
Synonym finitely axiomatizable
Synonym finitely axiomatisable
Synonym EC
Synonym ECΔ
Related topic Supercategories3
Related topic AxiomaticAndCategoricalFoundationsOfMathematicsII2
Defines elementary class