Tarski's axiomatization of the reals

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In 1936, Alfred Tarski set out an axiomatization of the real numbers and their arithmetic, consisting of only the 8 axioms shown below and a mere four primitive notions: the set of reals denoted R, a binary total order over R, denoted by infix <, a binary operation of addition over R, denoted by infix +, and the constant 1.

The literature occasionally mentions this axiomatization but never goes into detail, notwithstanding its economy and elegant metamathematical properties. This axiomatization appears little known, possibly because of its second-order nature. Tarski's axiomatization can be seen as an optimized version of the more usual definition of real numbers as the unique Dedekind-complete ordered field.

The term "Tarski's axiomatization of real numbers" also refers to the theory of real-closed fields, which Tarski showed completely axiomatizes the first-order theory of the structure 〈R, +, ·, <〉.

[edit] The Axioms

Axioms of Order (primitives: R, <):

Axiom 1. "<" is an asymmetric relation.

Axiom 2. If x < z, there exists a y such that x < y and y < z. In other words, "<" is dense in R.

Axiom 3. "<" is Dedekind-complete.

Let X ⊆ R and Y ⊆ R. We now define two common English verbs in a particular way that suits our purpose:

X precedes Y if and only if for every x ∈ X and every y ∈ Y, x < y.

The real number z separates X and Y if and only if for every x ∈ X with x ≠ z and every y ∈ Y with y ≠ z, x < z and z < y.

Axiom 3 can then be stated as:

"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."

Axioms of Addition (primitives: R, <, +):

Axiom 4. x + (y + z) = (x + z) + y. Addition associates.

Axiom 5. For all x, y, there exists a z such that x + z = y. R is a divisible group under addition.

Axiom 6. If x + y < z + w, then x < z or y < w. R is a linearly ordered group under addition.

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

Axiom 7. 1 ∈ R.

Axiom 8. 1 < 1 + 1.

The Axioms for One assure that 1 is a distinguished element of R distinct from 0.

These axioms require but three existential quantifiers, one for each of Axioms 2, 3, and 5. This axiomatization does not give rise to a first-order theory, because the formal statement (not shown) of Axiom 3 includes two universal quantifiers over all possible subsets of R. Tarski proved these 8 axioms and 4 primitive notions independent.

[edit] How these axioms imply a field

Tarski (1994) sketched the (nontrivial) proof of how these axioms and primitives imply the existence of a binary operation called multiplication and having the expected properties, so that R is a complete ordered field under addition and multiplication. This proof builds crucially on addition being an abelian group over the integers.

A recent elegant derivation of this result, due to Arthan, A'Campo, and Ross Street, can be sketched as follows. An almost homomorphism is a map f:Z→Z such that {f(n+m)-f(m)-f(n): n,m∈Z} is finite. Two almost homomorphisms f,g are almost equal if {f(n)-g(n): n∈Z} is finite. This defines an equivalence relation on the set of almost homomorphisms, and the equivalence classes of that relation are simply the real numbers. The sum and product of two real numbers defined in this manner are simply the pointwise sum and composition, respectively, of the corresponding almost homomorphisms. Thus R is a complete ordered field with respect to "<" and the binary operations of addition and multiplication.