Axioms of heterogeneous geometry (Q2281347)

Following his previous work, the author defines an \textit{arithmetic set} as a set of non-zero integers \(N\) such that (a) for any pair of different \(A\), \(B \in N\), \(A+B > 0\); and (b) for any triple of different \(A\), \(B\), \(C \in N\), \(AB+AC+BC > 0\). The conditions provide that for any different \(A\), \(B\), \(C \in N\), there exists a Euclidean triangle with side lengths \(\sqrt{A+B}\), \(\sqrt{A+C}\) and \(\sqrt{B+C}\). He then considers concrete examples of arithmetic sets \(N(D)\), \(D \leq -2\). The sets carry a natural metric and it is proved that it has certain non-Euclidean properties. The proofs are based on elementary geometric computations.