In 1936, Alfred Tarski gave an axiomatization of the real numbers and their arithmetic, consisting of only the eight axioms shown below and a mere four primitive notions: the set of reals denoted R, a binary relation over R, denoted by infix <, a binary operation of addition over R, denoted by infix +, and the constant 1. Tarski's axiomatization, which is a second-order theory, can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered field; it is however made much more concise by avoiding multiplication altogether and using unorthodox variants of standard algebraic axioms and other subtle tricks. Tarski did not supply a proof that his axioms are sufficient or a definition for the multiplication of real numbers in his system. Tarski also studied the first-order theory of the structure (R, +, ·, <), leading to a set of axioms for this theory and to the concept of real closed fields.
The axioms
Axioms of order (primitives: R, <) Axiom 1 If x < y, then not y < x. [That is, "<" is an asymmetric relation. This implies that "<" is irreflexive, i.e., for all x, not x < x.] Axiom 2 If x < z, there exists a y such that x < y and y < z. Axiom 3 For all subsets X, Y ⊆ R, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if x ≠ z and y ≠ z, then x < z and z < y. [In other words, "<" is Dedekind-complete, or informally: "If a set of reals X precedes another set of reals Y, then there exists at least one real number z separating the two sets." This is a second-order axiom as it refers to sets and not just elements.]
Axioms of addition (primitives: R, <, +) Axiom 4 x + (y + z) = (x + z) + y. [Note that this is an unorthodox mixture of associativity and commutativity.] Axiom 5 For all x, y, there exists a z such that x + z = y. [This allows subtraction and also gives a 0.] Axiom 6 If x + y < z + w, then x < z or y < w. [This is the contrapositive of a standard axiom for ordered groups.]
Axioms for 1 (primitives: R, <, +, 1) Axiom 7 1 ∈ R. Axiom 8 1 < 1 + 1.
Discussion Tarski stated, without proof, that these axioms turn the relation < into a total ordering. The missing component was supplied in 2008 by Stefanie Ucsnay. The axioms then imply that R is a linearly ordered abelian group under addition with distinguished positive element 1, and that this group is Dedekind-complete, divisible, and Archimedean. Tarski never proved that these axioms and primitives imply the existence of a binary operation called multiplication that has the expected properties, so that R becomes a complete ordered field under addition and multiplication. It is possible to define this multiplication operation by considering certain order-preserving homomorphisms of the ordered group (R,+,<).
References