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Concerning three classes of non-Diophantine arithmetics

Michele Caprio, Andrea Aveni and Sayan Mukherjee

Vol. 15 (2022), No. 5, 763–774
Abstract

We present three classes of abstract prearithmetics, {AM}M1, {AM,M}M1, and {BM}M>0. The first is weakly projective with respect to the nonnegative real Diophantine arithmetic R+ = (+,+,×,+), the second is weakly projective with respect to the real Diophantine arithmetic R = (,+,×,), while the third is exactly projective with respect to the extended real Diophantine arithmetic R ¯ = ( ¯,+,×,¯). In addition, we have that every AM and every BM is a complete totally ordered semiring, while every AM,M is not. We show that the projection of any series of elements of + converges in AM, for any M 1, and that the projection of any nonindeterminate series of elements of converges in AM,M, for any M 1, and in BM, for all M > 0. We also prove that working in AM and in AM,M, for any M 1, and in BM, for all M > 0, allows us to overcome a version of the paradox of the heap.

Keywords
non-Diophantine arithmetics, convergence of series, paradox of the heap
Mathematical Subject Classification
Primary: 03H15
Secondary: 03C62
Milestones
Received: 6 January 2021
Revised: 27 October 2021
Accepted: 11 March 2022
Published: 3 March 2023

Communicated by Kenneth S. Berenhaut
Authors
Michele Caprio
Department of Statistical Science
Duke University
Durham, NC
United States
Andrea Aveni
Department of Statistical Science
Duke University
Durham, NC
United States
Sayan Mukherjee
Department of Statistical Science
Duke University
Durham, NC
United States