Tarski's axiomatization of the reals

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,&nbsp;+,&nbsp;·,&nbsp;<), 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&nbsp;<&nbsp;z, there exists a y such that x&nbsp;<&nbsp;y and y&nbsp;<&nbsp;z.

Axiom 3 :For all subsets X,&nbsp;Y&nbsp;⊆&nbsp;R, if for all x&nbsp;∈&nbsp;X and y&nbsp;∈&nbsp;Y, x&nbsp;<&nbsp;y, then there exists a z such that for all x&nbsp;∈&nbsp;X and y&nbsp;∈&nbsp;Y, if x&nbsp;≠&nbsp;z and y&nbsp;≠&nbsp;z, then x&nbsp;&lt;&nbsp;z and z&nbsp;&lt;&nbsp;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&nbsp;+&nbsp;(y&nbsp;+&nbsp;z)&nbsp;=&nbsp;(x&nbsp;+&nbsp;z)&nbsp;+&nbsp;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&nbsp;+&nbsp;z&nbsp;=&nbsp;y.

:[This allows subtraction and also gives a 0.]

Axiom 6 :If x&nbsp;+&nbsp;y&nbsp;<&nbsp;z&nbsp;+&nbsp;w, then x&nbsp;<&nbsp;z or y&nbsp;<&nbsp;w.

:[This is the contrapositive of a standard axiom for ordered groups.]

Axioms for 1 (primitives: R, <, +, 1)

Axiom 7 :1&nbsp;∈&nbsp;R.

Axiom 8 :1&nbsp;<&nbsp;1&nbsp;+&nbsp;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