AI 中文总结
研究通过构建全序域\(R^{Z_<}\)解决无穷小推理问题，基于“分块与渗透”策略，用实序列和卷积构造，配备拓扑并发展微积分，为格罗索内算术建模，架起多领域桥梁，提供易于处理的无穷小推理框架。
AI 中文摘要
朴素的无穷小推理虽直观，但在经典逻辑中不一致，而严格的非标准分析依赖非构造性超滤子和转移原理。我们通过构建明确的全序域\(R^{Z_<}\)解决此矛盾，它作为实数和超实数公理（不一致）并集的模型，仅用实序列和卷积，包含实数、无穷和无穷小。该构造以“分块与渗透”（C&P）策略为基础，配备两层拓扑，发展微稳定函数的微积分，引入精细的\((k,n)\)-连续性层次结构，还直接为谢尔盖耶夫的格罗索内算术建模并分析域运算的可计算性。此工作架起了弗协调逻辑、构造性数学和非标准分析之间的桥梁，为无穷小推理提供了一个透明且计算上易于处理的框架，在逆数学和物理学中有潜在应用。
英文摘要
While intuitive, naïve 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.
Commentsv3: Updated abstract and content
