Axiomatic (and non-axiomatic) mathematics

Abstract

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.