TL;DR
This paper surveys the axiomatization of mathematical structures, contrasting axiomatic and non-axiomatic approaches, and reviews identities in positive real numbers, highlighting foundational aspects of mathematical logic.
Contribution
It provides a comprehensive review of both classical and recent results on first-order axiomatizability and identities in key mathematical structures.
Findings
Unique axiomatization of complete ordered fields
Axioms of group theory fully characterize permutation groups
Identities over positive real numbers hold universally
Abstract
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.
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