Involve 15 (5), 763-774, (2022) DOI: 10.2140/involve.2022.15.763
Michele Caprio, Andrea Aveni, Sayan Mukherjee
KEYWORDS: non-Diophantine arithmetics, convergence of series, paradox of the heap, 03H15, 03C62
We present three classes of abstract prearithmetics, , , and . The first is weakly projective with respect to the nonnegative real Diophantine arithmetic , the second is weakly projective with respect to the real Diophantine arithmetic , while the third is exactly projective with respect to the extended real Diophantine arithmetic . In addition, we have that every and every is a complete totally ordered semiring, while every is not. We show that the projection of any series of elements of converges in , for any , and that the projection of any nonindeterminate series of elements of converges in , for any , and in , for all . We also prove that working in and in , for any , and in , for all , allows us to overcome a version of the paradox of the heap.