set of reals precedes another set of reals, then there exists at least one real number separating the two sets." The three axioms imply that R is a linear continuum. Axioms of addition (primitives: R, <, +): Axiom 4 x + (y + z) = (x + z) + y. Axiom 5 For all x, y, there exists a z such that x + z = y. Axiom 6 If x + y < z + w, then x < z or y < w. Axioms for one (primitives: R, <, +, 1): Axiom 7 1 ∈ R. Axiom 8 1 < 1 + 1. This axiomatization does not give rise to a first-order theory, because the formal statement of axiom 3 includes two universal quantifiers over all possible subsets of R. Tarski proved that these 8 axioms and 4 primitive notions are independent. How these axioms imply a field Otto Hölder showed that every Archimedean group is isomorphic (as an ordered group) to a subgroup of the Dedekind-complete Archimedean group with distinguished element , . Because is an Archimedean ordered field, let us define as the Dedekind completion of . The Dedekind completion of any Archimedean ordered field is terminal in the concrete category of Dedekind complete Archimedean ordered fields, Because is a Dedekind-complete Archimedean ordered field, every Archimedean group embeds into as well. As a result, the two sets and are isomorphic to each other, which means that is a field. Tarski stated, without proof, that these axioms gave a total ordering. The missing component was supplied in 2008 by Stefanie Ucsnay. See also References Real numbers Ordered groups Mathematical axioms Formal theories of arithmetic