实数的塔尔斯基公理化系统

在我们中国，中学生都知道有一个“实数”系统，但是，对实数的了解只限于直观层面，不能深入理解。

进入20世纪，各种数学分支都公理化了，实数也不例外。但是，实数的公理化方案有多种，各有特色，塔尔斯基公理化系统也是备选方案之一。

塔尔斯基是上世纪与哥德尔齐名的最伟大的数理逻辑学家，模型论的奠基人，而且是无穷小微积分教材作者J,Keisler的博士导师。

   说明：塔尔斯基是个传奇人物，国人知之甚少。

袁萌  2月5日

附：Tarski's axiomatization of the reals

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 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 using unorthodox variants of standard algebraic axioms and other subtle tricks (see e.g. axioms 4 and 5, which combine together the usual four axioms of abelian groups).

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, +, ·, <</span>〉.

The axioms

Axioms of order (primitives: R, <):

Axiom 1

    If x < y, then not y < x. That is, "<" is an asymmetric （非对称）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. More formally, for all 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 z ≠ x and z ≠ y, then x < z and z < y.

To clarify the above statement somewhat, 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.

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.

These axioms imply that R is a linearly ordered abelian group under addition with distinguished element 1. R is also Dedekind-complete and divisible.

Tarski stated, without proof, that these axioms gave a total ordering. The missing component was supplied in 2008 by Stefanie Ucsnay.[2]

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 these 8 axioms and 4 primitive notions independent.

How these axioms imply a field

Tarski 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 the integers with addition being an abelian group and has its origins in Eudoxus' definition of magnituty.（全文完）