Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field
Year: 2026
Type: article
Abstract: Infinite Numbers, introduced in preceding studies as functional expressions φ(ξ) of a distinguished infinity unit ξ, are here given a rigorous germ-based foundation within a Hardy-field setting. The real system is realized as an ordered non-Archimedean field canonically isomorphic to the chosen Hardy field, while complex Infinite Numbers arise by complexification. Equality is defined by eventual equality rather than asymptotic equivalence, thereby retaining lower-order and nonzero infinitesimal contributions. Within stated admissibility conditions, the principal arithmetic, differential, partially defined integral, and retained-transform operations of the preceding framework receive rigorous interpretations. The theory further establishes conditions relating discrete accumulation to asymptotic antiderivative representations and proves that the retained bilateral-transform family is injective on C(ℝ) along every fixed vertical line, yielding a well-defined inverse on its range. Two applications illustrate the framework. For a nonlinear boundary-value problem on the real line, retained finite symmetric transforms remain applicable despite the absence of a common classical bilateral region of convergence, leading to an explicit heteroclinic solution and nonzero infinitesimal endpoint corrections. A discrete-series limit illustrates the accumulation theorem and is independently verified by a classical estimate.
Source: Mathematics
DOI: 10.3390/math14193594
Author: Emmanuel Thalassinakis
Institution: Technical University of Crete
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7220442747