Stably Infinite Theory (Satisfiability Modulo Theory)

Definition

Stably Infinite Theory (Satisfiability Modulo Theory)

A theory ${\mathcal{T}}_i$ with signature ${\Sigma }_i$ is stably infinite if for every ${\mathcal{T}}_i$-satisfiable formula, there exists some ${\mathcal{T}}_i$-interpretation $I$ such that

$$I\models F\text{and}I\text{ has a domain of infinite cardinality}$$

Examples

• Theory of equality with uninterpreted functions
• Theory of arrays
• Theory of linear integer arithmetic