公理化（与非公理化）数学
Axiomatic (and Non-Axiomatic) Mathematics
AI总结：
本文综述了各种数学结构的一阶可公理化性，探讨了群论与完备有序域等公理系统的深浅差异，并回顾了正实数集上加法、乘法和指数运算的恒等式。
摘要：
公理化数学结构和理论是数理逻辑的目标之一。一些公理系统如今仅是定义，如群论的公理；但有些系统则深刻得多，如实分析所基于的完备有序域公理。群在数学科学中随处可见，而根据戴德金定理，在同构意义下仅存在唯一的完备有序域。抽象代数中的凯莱定理表明，群论公理完全公理化了在复合和逆运算下封闭的置换集合类。本文综述了关于各种数学结构一阶可公理化性的一些新旧结果。我们还将回顾在正实数集合中成立的关于加法、乘法和指数运算的恒等式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
