Concerning three classes of non-Diophantine arithmetics

Caprio, Michele; Aveni, Andrea; Mukherjee, Sayan

doi:10.2140/involve.2022.15.763

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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

DOI: 10.2140/involve.2022.15.763

Abstract

We present three classes of abstract prearithmetics, ${A_{M}}_{M \geq 1}$ , ${A_{- M, M}}_{M \geq 1}$ , and ${B_{M}}_{M > 0}$ . The first is weakly projective with respect to the nonnegative real Diophantine arithmetic $R_{+} = (ℝ_{+}, +, \times, \leq_{ℝ_{+}})$ , the second is weakly projective with respect to the real Diophantine arithmetic $R = (ℝ, +, \times, \leq_{ℝ})$ , while the third is exactly projective with respect to the extended real Diophantine arithmetic $\bar{R} = (\bar{ℝ}, +, \times, \leq_{\bar{ℝ}})$ . 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 $M \geq 1$ , and that the projection of any nonindeterminate series of elements of $ℝ$ converges in $A_{- M, M}$ , for any $M \geq 1$ , and in $B_{M}$ , for all $M > 0$ . We also prove that working in $A_{M}$ and in $A_{- M, M}$ , for any $M \geq 1$ , and in $B_{M}$ , for all $M > 0$ , allows us to overcome a version of the paradox of the heap.

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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

Vol. 15, No. 5, 2022