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In mathematical logic, a 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 ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory. In many deductive systems there is usually a subset Σ ⊆ T {\displaystyle \Sigma \subseteq T} that is called "the set of axioms" of the theory T {\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 recursively obtained by the inference rules of the system applied to the set of axioms.