Universal first-order theory

A universal theory can be axiomatized by sentences whose quantifiers are all universal. Such sentences often forbid specified finite configurations.

The universal consequences $T_{\forall }$ of a first-order theory $T$ are all universal first-order sentences entailed by $T$. Their model class is exactly the class of substructures of models of $T$, up to structure embedding. This characterization follows from the diagram embedding criterion for universal theories.

A first-order theory has its class of first-order models closed under substructures of a first-order structure exactly when it is equivalent to its universal consequences $T_{\forall }$. If $A\models T_{\forall }$, then $T\cup \mathrm{Diag}(A)$ is consistent. Otherwise a finite logical conjunction $\delta$ of signed diagram of a structure sentences yields a universal consequence $\forall \mathbf{x}\neg \delta (\mathbf{x})$ false in $A$. The compactness theorem gives a first-order model of $T$ containing an embedded copy of $A$, and closure under substructures of a first-order structure implies $A\models T$. The converse is preservation of universal sentences. The diagram of a structure must include negated as well as positive atomic sentences so that the resulting map is an embedding, not just a homomorphism.

A universal sentence has the form $\forall \mathbf{x}\psi (\mathbf{x})$ with quantifier-free formula $\psi$, including an empty quantifier block. Its truth passes from a first-order structure to every substructure of a first-order structure: tuples from the smaller domain are also tuples of the larger domain, and quantifier-free truth agrees. Therefore a first-order theory of universal sentences has a first-order model class closed under substructures of a first-order structure.

If failure of a property has a finite witness, one can often axiomatize the property by a universal sentence forbidding each possible finite witness. Compactness supplies finite witnesses when failure is expressed by inconsistency of an associated theory.