A Constructive Field of Infinitesimals: Chunk and Permeate Approach

Abstract

While intuitive, na\"ive infinitesimal reasoning is classically inconsistent, and rigorous nonstandard analysis relies on non-constructive machinery. We resolve this tension by constructing an explicit, totally ordered field $\mathbb{R}^{\mathbb{Z}_{<}}$ using only real sequences and Cauchy convolution. We model the combined real and hyperreal axioms via the Chunk and Permeate strategy, a paraconsistent technique that isolates contradictions without global collapse. Equipping $\mathbb{R}^{\mathbb{Z}_{<}}$ with a two-tier topology, we develop a calculus where infinitesimal derivatives and integrals permeate cleanly to their classical counterparts. We further introduce a $(k,n)$-continuity hierarchy capturing infinitesimal smoothness invisible to standard or transfer-based models. Finally, $\mathbb{R}^{\mathbb{Z}_{<}}$ yields a direct algebraic consistency proof for Sergeyev's Grossone arithmetic, and we establish strict computability bounds on field operations. By guaranteeing infinitesimal contradictions never reach the classical chunk, this work bridges paraconsistent logic, constructive mathematics, and nonstandard analysis into a transparent, computationally tractable framework for infinitesimal reasoning.