by Jean Barbet | Dec 10, 2024 | Logic, Number Theory, Set Theory
We explore the foundation of natural arithmetic starting from Peano’s axioms within set theory, revealing an innovative approach to conceptualizing natural numbers. We question the traditional use of ordinals and propose an alternative formulation of the axiom... 
by Jean Barbet | Sep 26, 2023 | Logic, Non classé, Number Theory, Set Theory
Natural numbers have two faces: on one hand, they can be seen as sequences or “enumerations”—what we call ordinal numbers. On the other hand, they are perceived as “quantities,” which leads us to cardinal numbers. While this distinction is not... 
by Jean Barbet | Jul 9, 2023 | Logic, Number Theory, Set Theory
Natural arithmetic is the science of natural numbers: it is based on addition, multiplication, natural order and divisibility. Now, all these operations and relations are defined on the basis of the single successor function, whose properties are brought together in... 
by Jean Barbet | Mar 25, 2021 | Algebra, Geometry, Non classé, Number Theory
Descartes’ analytical method, which allows the Euclidean plane to be represented as the Cartesian product $ \mathbb{R}^2 $ through the theory of real numbers, also makes it possible to represent Euclidean space as the Cartesian product $ \mathbb{R}^3 =... 
by Jean Barbet | Mar 20, 2021 | Algebra, Geometry, Non classé, Number Theory
The complex multiplication naturally extends to a multiplication in four dimensions, which defines on the space $ \mathbb{R}^4 $ the structure of the algebra $ \mathbb{H} $ of Hamilton’s quaternions. This multiplication can be interpreted geometrically using the... 
by Jean Barbet | Mar 12, 2021 | Algebra, Non classé, Number Theory
Gaussian integers are complex numbers with integer coordinates. Thanks to their norm, a kind of integer measure of their size, we can describe some of their arithmetic properties. In particular, we can determine which are the usual prime numbers that...