[Paper Review] Alternative Mathematics without Actual Infinity

Tōru Tsujishita|arXiv (Cornell University)|Apr 9, 2012

Mathematical and Theoretical Analysis22 references3 citations

TL;DR

This paper proposes an alternative mathematical framework that replaces actual infinity with a qualitative plurality of finiteness, using accessible finite sets and vague concepts like 'indistinguishability' to construct continua. It develops a nonstandard-style mathematics independent of ZFC, establishing real numbers, calculus, and measure theory through internal measurement and sorites-based equivalence, yielding a coherent, intuitionistic foundation for analysis without infinite sets.

ABSTRACT

An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose separation axiom is restricted to definite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable elements and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. Ascoli-Arzela type theorem is given as an example indicating the feasibility of treating function spaces. The real numbers are defined to be the points on the linear continuum and have indefiniteness. Exponentiation is introduced by the Eulerian style and basic properties are established. Basic calculus is developed and the differentiability is captured by the behavior on a point. Main tools of Lebesgue measure theory is obtained in a similar way as Loeb measure. Differences from the current mathematics are examined, such as the indefiniteness of natural numbers, qualitative plurality of finiteness, mathematical usage of vague concepts, the continuum as a primary inexhaustible entity and the hitherto disregarded aspect of "internal measurement" in mathematics.

Motivation & Objective

• To develop a nonstandard mathematics independent of Cantorian infinite set theory, avoiding reliance on ZFC as a foundation.
• To resolve the philosophical and mathematical tensions between discrete and continuous by treating infinity as a phenomenon of large-scale finiteness.
• To formalize vague concepts like 'accessibility' and 'indistinguishability' coherently within finite set theory.
• To reconstruct real numbers, calculus, and measure theory using sorites relations and points as proper classes of indistinguishable elements.
• To introduce 'internal measurement' as a foundational principle, reflecting cognitive limitations in mathematical observation.

Proposed method

• Uses restricted separation axioms based on definite objective conditions to define semisets and classes within finite set theory.
• Defines weak equivalence relations via sorites phenomena, where small changes do not alter identity, forming the basis of continua.
• Constructs continua as collections equipped with indistinguishability relations, where points are proper classes of mutually indistinguishable elements.
• Introduces a new definition of real numbers as points on a linear continuum, inheriting indefiniteness from the underlying structure.
• Applies Euler-style exponentiation and develops calculus via difference quotients and infinitesimal Taylor expansions.
• Establishes measure theory analogously to Loeb measure, using null semisets and integration over measurable semisets.

Experimental results

Research questions

• RQ1 Can nonstandard analysis be fully reconstructed without appealing to actual infinity or ZFC?
• RQ2 How can vague concepts like 'accessibility' and 'indistinguishability' be made logically coherent within finite mathematics?
• RQ3 What is the role of the sorites paradox in defining continuity and the continuum?
• RQ4 How can calculus and measure theory be developed when the totality of natural numbers is indefinite?
• RQ5 In what way does internal measurement—reflecting cognitive limitations—reshape the foundations of mathematical objects?

Key findings

• A consistent mathematical framework is developed that avoids actual infinity by grounding mathematics in qualitative plurality of finiteness and accessible finite sets.
• The real numbers are defined as points on a linear continuum, inheriting indefiniteness from the underlying sorites-based indistinguishability structure.
• A version of the Ascoli-Arzela theorem is established, demonstrating the feasibility of treating function spaces in this system.
• Exponentiation is introduced in an Euler-style manner, and basic algebraic and analytic properties are derived without infinite sets.
• Measure theory is reconstructed via null semisets and integration over measurable semisets, mirroring Loeb measure but within finite set theory.
• The framework supports a novel view of mathematical objects as dynamically refined through theorems, reflecting internal measurement and cognitive limitations.