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An axiom system written in the language of restricted predicate calculus (that is, in a first-order language). Examples of elementary axiom systems are those of formal arithmetic (cf. [[Arithmetic, formal|Arithmetic, formal]]), the Zermelo–Fraenkel system of set theory (see [[Axiomatic set theory|Axiomatic set theory]]), and the system of the theory of types (cf. [[Types, theory of|Types, theory of]]).