论文标题
华氏（和非轴心）数学
Axiomatic (and Non-Axiomatic) Mathematics
论文作者
论文摘要
数学结构和理论是数学逻辑的目标。如今，某些公理系统仅仅是定义，例如群体理论的公理。但是，有些系统要深得多，例如实际分析开始的完整有序字段的公理。小组在数学科学中比比皆是，而Dedekind的定理仅存在一个完整的有序领域，直到同构。 Cayley的抽象代数定理意味着，群体理论的公理完全将在组成和反转下封闭的置换组类别。在本文中，我们对各种数学结构的一阶公理性率进行了一些新的和新的结果。我们还将回顾一下在正实数集合中存在的添加，乘法和指数化的身份。
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of Complete Ordered Fields with which Real Analysis starts. Groups abound in mathematical sciences, while by Dedekind's theorem there exists only one complete ordered field, up to isomorphism. Cayley's theorem in Abstract Algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion. In this article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers.
