DeepAI AI Chat
Log In Sign Up

Axiomatic (and Non-Axiomatic) Mathematics

by   Saeed Salehi, et al.

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.


page 1

page 2

page 3

page 4


On the Decidability of the Ordered Structures of Numbers

The ordered structures of natural, integer, rational and real numbers ar...

Decidability of the Multiplicative and Order Theory of Numbers

The ordered structures of natural, integer, rational and real numbers ar...

Adding an Abstraction Barrier to ZF Set Theory

Much mathematical writing exists that is, explicitly or implicitly, base...

A characterisation of ordered abstract probabilities

In computer science, especially when dealing with quantum computing or o...

Order in the chaos with examples from graph theory

In randomly created structures (be they natural or artificial) very ofte...

Towards solid abelian groups: A formal proof of Nöbeling's theorem

Condensed mathematics, developed by Clausen and Scholze over the last fe...

Revisit the Fundamental Theorem of Linear Algebra

This survey is meant to provide an introduction to the fundamental theor...