axiomatizable class (original) (raw)

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 axiomatizable 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.

Remarks.

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  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 φ1∧⋯∧φn, which is also a sentence since it has no free variables.
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  A class is axiomatizable iff it is an intersection of elementary classes. As such elementary class is sometimes abbreviated EC, and axiomatizable class ECΔ, where Δ means is another symbol for intersection.
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  A caution to the reader: some authors call an elementary class an axiomatizable class that is defined here.