Theory (mathematical logic)

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ${\displaystyle \phi \in T}$ of a deductively closed theory ${\displaystyle T}$ is then called a theorem of the theory. In many deductive systems there is usually a subset ${\displaystyle \Sigma \subseteq T}$ that is called "the set of axioms" of the theory ${\displaystyle T}$, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.