Introduction to Non-Diophantine Number Theory

Abstract

In the 19th century, non-Euclidean geometries were discovered and studied. In the 20th century, non-Diophantine
arithmetics were discovered and studied. Construction of non-Diophantine arithmetics is based on more general
mathematical structures, which are called abstract prearithmetics, as well as on the projectivity relation between
abstract prearithmetics. In a similar way, as set theory gives a foundation for mathematics, the theory of abstract
prearithmetics provides foundations for the theory of the Diophantine and non-Diophantine arithmetics. In this paper,
we use abstract prearithmetics for developing fundamentals of non-Diophantine number theory, which can be also
called non-Diophantine higher arithmetic as the conventional number theory is called higher arithmetic. In particular,
we prove the Fundamental Theorem of Arithmetic for a wide range of abstract prearithmetics.