linguistic:paraphrasing, structural:scholar_edition
Signatures

A signature consists of a set of function symbols S, a set of relation symbols S, and a function ar: S func ∪ S rel → N 0 {\displaystyle \operatorname {ar}:S_{\operatorname {func} }\cup S_{\operatorname {rel} }\rightarrow \mathbb {N} _{0}} representing the arity of function and relation symbols. (A nullary function symbol is called a constant symbol.) In the context of first-order logic, a signature is sometimes called a language. It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains.

A first-order theory consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature. Theories are often specified by giving a list of axioms that generate the theory, or by giving a structure and taking the theory to consist of the sentences satisfied by the structure.