关于塔尔斯基“中学代数问题”的感言

对于初等数学中存在不可证明的恒等式，知晓这个“事实”，对于培养高中学生的核心数学素养是有助益的。

历史上，在1960年，塔尔斯基首先提出了这个问题，但是，过了20年之后，直到1980年才被严格数学证明这个“数学真理”。

请见:“Tarski's high school algebra problem”一文。由于文章篇幅较长，本文只摘抄其中一段文字，供读者参考。

袁萌  2月9日

附：

In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities （恒等式）involving addition, multiplication, and exponentiation（幂） over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level mathematics. The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist（一定存在）。

Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:

    x + y = y + x

    (x + y) + z = x + (y + z)

    x · 1 = x

    x · y = y · x

    (x · y) · z = x · (y · z)

    x · (y + z) = x · y + x ·z

    1x = 1

    x1 = x

    xy + z = xy · xz

    (x · y)z = xz · yz

    (xy)z = xy · z.

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