Concerning three classes of non-Diophantine arithmetics
We present three classes of abstract prearithmetics, {AM}M≥1, {A-M,M}M≥1, and {BM}M>0. The first is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(ℝ+, +, ×, ≤ℝ+), the second is weakly projective with respect to the real Diophantine arithmetic R=(ℝ,+,×,≤ ℝ), while the third is exactly projective with respect to the extended real Diophantine arithmetic R=(ℝ,+,×,≤ ℝ). In addition, we have that every AM and every BM 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 AM, for any M≥1, and that the projection of any nonindeterminate series of elements of R converges in A-M,M, for any M≥1, and in BM, for all M>0. We also prove that working in AM and in A-M,M, for any M≥1, and in BM, for all M>0, allows us to overcome a version of the paradox of the heap.
