August 2022 Axiomatic (and non-axiomatic) mathematics
Saeed Salehi
Rocky Mountain J. Math. 52(4): 1157-1176 (August 2022). DOI: 10.1216/rmj.2022.52.1157


Axiomatizing mathematical structures and theories, or postulating them as Russell (1919) put it, is a goal of mathematical logic. Some axiomatic systems are mere definitions, such as the axioms of Group Theory; but some are much deeper, such as the axioms of complete ordered fields with which real analysis starts. Groups abound in the mathematical sciences, while by Dedekind’s theorem (1888) there exists only one complete ordered field, up to isomorphism. Cayley’s theorem (1854) 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 expository article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. As we will see, axiomatizability of some structures are still unsolved questions in mathematics, and several results have been open problems in the past. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers; and will have a look at Tarski’s high school problem (1969) and its solution.


Download Citation

Saeed Salehi. "Axiomatic (and non-axiomatic) mathematics." Rocky Mountain J. Math. 52 (4) 1157 - 1176, August 2022.


Received: 1 November 2021; Revised: 10 April 2022; Accepted: 13 April 2022; Published: August 2022
First available in Project Euclid: 26 September 2022

MathSciNet: MR4489153
zbMATH: 07598553
Digital Object Identifier: 10.1216/rmj.2022.52.1157

Primary: 03B25 , 03C10 , 03D35 , 03F40
Secondary: 11U05 , 12L05

Keywords: axiomatic system , first-order logic , identities

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.52 • No. 4 • August 2022
Back to Top