Concerning three classes of non-Diophantine arithmetics

Abstract

We present three classes of abstract prearithmetics, $\{{\mathbf{A}}_M{\}}_{M\ge 1}$, $\{{\mathbf{A}}_{-M,M}{\}}_{M\ge 1}$, and $\{{\mathbf{B}}_M{\}}_{M>0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic ${\mathbf{R}}_{+}=({\mathbb{R}}_{+},+,\times ,{\le }_{{\mathbb{R}}_{+}})$, the second one is weakly projective with respect to the real Diophantine arithmetic $\mathbf{R}=(\mathbb{R},+,\times ,{\le }_{\mathbb{R}})$, while the third one is projective with respect to the extended real Diophantine arithmetic $\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times ,{\le }_{\overline{\mathbb{R}}})$. In addition, we have that every ${\mathbf{A}}_M$ and every ${\mathbf{B}}_M$ are a complete totally ordered semiring, while every ${\mathbf{A}}_{-M,M}$ is not. We show that the projection of any series of elements of ${\mathbb{R}}_{+}$ converges in ${\mathbf{A}}_M$, for any $M\ge 1$, and that the projection of any non-oscillating series series of elements of $\mathbb{R}$ converges in ${\mathbf{A}}_{-M,M}$, for any $M\ge 1$, and in ${\mathbf{B}}_M$, for all $M>0$. We also prove that working in ${\mathbf{A}}_M$ and in ${\mathbf{A}}_{-M,M}$, for any $M\ge 1$, and in ${\mathbf{B}}_M$, for all $M>0$, allows to overcome a version of the paradox of the heap.

References in corpus (2)

• Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus
• On the Safe Use of Inconsistent Mathematics