Abstract
We present two classes of abstract prearithmetics, AMM≥1 and BMM>0. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(R+,+,×,≤R+), and the second one is 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. 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 non-indeterminate series of elements of R converges in BM, for all M>0. We also prove that working in AM, for any M≥1, and in BM, for all M>0, allows to overcome a version of the paradox of the heap.
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