Set Theory and Logic

2506 Submissions

[3] ai.viXra.org:2506.0132 [pdf] submitted on 2025-06-29 01:28:02

Textures of the Continuum: A Mirror Hierarchy of Infinitesimals and Infinities

Authors: Moninder Singh Modgil, Dhyandeo Dattatray Patil
Comments: 50 Pages.

This paper presents a novel framework for understanding the continuum through a dual hierarchy of infinitesimals, denoted by ϵi, in one-to-one correspondence with Cantor’s aleph numbers ℵi. While traditional set theory emphasizes hierarchies of cardinality and size, we introduce a mirrored structure capturing the granularity ofresolution and infinitesimal differentiation. The ϵ-hierarchy is developed analogously to the ℵ-hierarchy, extending through ordinal-indexed infinitesimals and enabling rigorous mathematical treatment of non-Archimedean textures of the continuum. We build this framework into various domains: logic, differential geometry, category theory, quantum field theory, and set-theoretic foundations. The paper formalizes ϵ-indexedversions of forcing, sheaf structures, internal toposes, and large cardinals, while offeringapplications to black hole singularities, entropy bounds, and homotopy type theory. By grounding infinitesimal behavior in a formal logical and categorical context, this work proposes a comprehensive foundation for exploring resolution-aware mathematics and physics, with potential to bridge analytic and transfinite regimes.
Category: Set Theory and Logic

[2] ai.viXra.org:2506.0112 [pdf] submitted on 2025-06-24 02:29:18

Brace Height and Ordinal Length: A Duality in the Von Neumann Hierarchy

Authors: David Selke
Comments: 2 Pages.

We explore a structural duality within the von Neumann conception of ordinal numbers: between the flat, extensional enumeration of ordinals and the internal, intensional brace-nesting required to construct them. We argue that the transitivity of membership implies a direct tradeoff: the longer the ordinal list, the taller the brace structure required to individuate its elements. This perspective motivates a critique of large countable ordinals such as $omega_1$ and questions the ontological legitimacy of their construction under the iterative conception of set.
Category: Set Theory and Logic

[1] ai.viXra.org:2506.0060 [pdf] submitted on 2025-06-16 01:50:44

The Indecidability of the Existence of Weakly Inaccessible Cardinals in ZFC: A Gödel-Inspired Logical Approach

Authors: Javier Muñoz de la Cuesta
Comments: 5 Pages.

This paper explores the undecidability of the existence of weakly inaccessible cardinalsin the von Neumann universe under Zermelo-Fraenkel set theory with the Axiom of Choice(ZFC). Through a novel logical framework inspired by Gödel’s incompleteness theorems,we demonstrate that the statement "there exists a weakly inaccessible cardinal" is neitherprovable nor disprovable in ZFC, assuming ZFC is consistent. The framework introducesintuitive concepts such as the intrinsic distinctness of "super-large" objects, a "functional escape" from the properties of smaller objects, and the limits of formal systems. These ideas are formalized to show that weakly inaccessible cardinals transcend ZFC’s constructive capabilities, aligning with Gödelian incompleteness. The paper provides technical details on forcing and the constructible universe L, clarifies the original contribution of the logical approach, and contextualizes weakly inaccessible cardinals among other large cardinals. Implications for set theory and future research directions are discussed.
Category: Set Theory and Logic