Theory (mathematical logic)

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. Usually a deductive system is understood from context. An element ${\displaystyle \phi \in T}$ of a theory ${\displaystyle T}$ is then called an axiom of the theory, and any sentence that follows from the axioms (${\displaystyle T\vdash \phi }$) is called a theorem of the theory. Every axiom is also a theorem. A first-order theory is a set of first-order sentences.

When defining theories for foundational purposes, additional care must be taken and normal set-theoretic language may not be appropriate.

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