Concerning three classes of non-Diophantine arithmetics

Abstract

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