Banishing divergence Part 1: Infinite numbers as the limit of sequences of real numbers (1108.5081v1)

Abstract: Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences that head off to infinity. It begins by defining Archimedean classes of infinite numbers. Each class is denoted by a prototype sequence. These prototypes are used as asymptotes for determining leading term limits of sequences. By subtracting off leading term limits and repeating, limits are obtained for a subset of sequences called here ``smooth sequences". $\mathbb{I}_n$ is defined as the set of ratios of limits of smooth sequences. It is shown that $\mathbb{I}_n$ is an ordered field that includes real, infinite and infinitesimal numbers.

• The paper introduces a novel non-Archimedean framework that defines infinite numbers as limits of smooth sequences, resolving divergence using structured asymptotic limits.
• The paper develops leading term limits and organizes Archimedean classes to form an ordered field that unifies real, infinite, and infinitesimal numbers.
• The paper highlights significant theoretical and computational implications, offering fresh insights into managing divergent sequences in mathematical research.

Overview of "Banishing Divergence Part 1: Infinite Numbers as the Limit of Sequences of Real Numbers"

This paper by David Alan Paterson introduces a novel framework for understanding infinite numbers as the limits of sequences. The proposed framework diverges from existing methods, such as nonstandard analysis using hyperreals, surreal numbers, and geometric continuum. The central contribution lies in constructing a non-Archimedean system where infinite numbers emerge as asymptotic limits akin to the way real numbers are limits of Cauchy sequences. This new framework attempts to resolve the divergence in sequences heading off to infinity by defining appropriate asymptotic limits.

Key Contributions

1. Archimedean Classes and Non-Archimedean Framework: The paper begins by cataloging Archimedean classes within a non-Archimedean ordered field. This novel field includes real, infinite, and infinitesimal numbers, each identified through Archimedean class prototypes. By using commutative Big O notation, Paterson defines these classes and provides a method for expanding them.
2. Leading Term and Asymptotic Limits: The introduction of leading term limits offers an alternative to classical Cauchy limits, specifically for sequences that tend towards infinity but do not oscillate. Through this framework, the paper suggests a hierarchy of infinite and infinitesimal numbers by iterating limits and removing leading terms.
3. Ordered Field of Infinite Numbers: The set I, comprising ratios of limits of smooth sequences, is established as an ordered field. This field encompasses infinite, infinitesimal, and real numbers, providing closure under standard operations of addition, multiplication, and division. The paper further suggests that these smooth sequences, although not forming a complete field under addition, can extend the existing set of real numbers.

Implications and Speculations

The establishment of infinite numbers through asymptotic limits opens new potential avenues for both theoretical mathematics and applied disciplines where understanding divergence is crucial. The proposal is particularly promising for fields like computational mathematics, where handling infinite sequences is a common requirement. However, while the introduction of smooth sequences and their limits presents a potential shift in dealing with infinite numbers, it raises several questions and speculative scenarios in the broader mathematical landscape.

1. Theoretical Implications: The paper invites further inquiry into whether the defined infinite numbers are a subset of known surreal numbers or constitute a new class requiring additional characterization. Moreover, the proposed framework challenges traditional notions of divergence and convergence, potentially redefining them in terms of asymptotic limits.
2. Speculative Future Directions: Possible extensions of this work might explore applying this framework to topological spaces, field theory, and other branches of mathematics. There's also potential for advancing computational techniques for managing infinite sequences in purely applied contexts.
3. Comparisons to Existing Constructions: While similar to the surreal numbers, Paterson’s approach distinguishes itself by focusing on the asymptotic limits of smooth sequences, potentially offering unique properties absent in other non-Archimedean fields.

The paper lays a foundation for further exploration in Part 2, which promises to explore oscillatory sequences and possible applications in elementary mathematics, bridging the gap between infinite numerical systems and their practical utilizations. Overall, this work sets the stage for a reevaluation of how mathematics conceptualizes and operationalizes the concept of infinity.