A theory is a set of sentences in a formal language, sometimes understood to be closed under deduction. A first-order theory uses a first-order language; consistency and completeness are separate properties of a theory.
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In mathematical logic, a **theory** is a formal system that consists of a set of sentences or propositions in a particular language, along with a set of axioms and inference rules that determine what can be derived or proven within that system. The sentences are typically formulated in first-order logic or another formal logical language, and they can express various mathematical statements or properties.