Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field
- 1Applied Mathematics and Computers Laboratory (AMCL), Technical University of Crete, Akrotiri Campus, 73100 Chania, Greece
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.
Related
- Generalized functions in a Non-Archimedean field
- The infinite sum of the cubes of reciprocal balancing numbers
- Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
- Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer
- On the infinite sum of reciprocals of the fourth powers of balancing numbers
BibTeX
@article{thalassinakis2026infinite,
title = {{Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field}},
author = {Emmanuel Thalassinakis},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193594},
url = {https://www.mdpi.com/2227-7390/14/19/3594},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Infinite Numbers: A Rigorous Germ-Based Foundation in a Non-Archimedean Field
AU - Emmanuel Thalassinakis
PY - 2026
DA - 2026/10/03
JO - Mathematics
DO - 10.3390/math14193594
UR - https://www.mdpi.com/2227-7390/14/19/3594
AB - 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.
SN - 2227-7390
ER -