| ai.viXra.org > Set Theory and Logic |
|
 |
| ai.viXra.org > Set Theory and Logic |
|
|
Previous months:
2025 - 2505(1) - 2506(3) - 2507(3) - 2508(1) - 2509(1) - 2511(2)
Any replacements are listed farther down
[11] ai.viXra.org:2511.0044 [pdf] submitted on 2025-11-14 21:35:32
Authors: Alin Setar
Comments: 4 Pages.
This paper presents a critical review of Chang Hee Kim’s 2025 preprint, "Cantor’s Continuum Hypothesis Is Proved Wrong" (ai.viXra:2505.0211v1). The work proposes an alternative approach to infinite cardinality by decomposing the set of natural numbers into infinitely many symmetric subsets. We examine the logical structure of Kim’s argument, the role of "infinite decomposition," and its implications for Cantor’s diagonal method and the Continuum Hypothesis (CH). In particular, we demonstrate that Kim’s framework operates outside the standard Zermelo—Fraenkel set theory with the Axiom of Choice (ZFC), since it redefines core notions such as bijection, power set, and cardinality.
Category: Set Theory and Logic
[10] ai.viXra.org:2511.0026 [pdf] submitted on 2025-11-09 00:02:23
Authors: Jianbo Li
Comments: 4 Pages. Degenerates from quantization methods to classical equations
This paper investigates the proposition of natural number addition, $1 + 1 = 2$. First, a rigorous proof using the classical CSUM method is presented within the frameworks of set theory and the Peano axioms. Next, the traditional proof based on formal logic by Bertrand Russell and Alfred North Whitehead in textit{Principia Mathematica} is reviewed. Finally, the two approaches are compared in detail, analyzing their differences in conceptual approach, complexity, and scalability, and discussing the advantages of the CSUM method in terms of algebraic intuition and category-theoretic extensions. This study aims to demonstrate how modern methods can complement traditional formal logic, thereby enriching perspectives on foundational mathematics research.
Category: Set Theory and Logic
[9] ai.viXra.org:2509.0063 [pdf] submitted on 2025-09-23 16:54:45
Authors: Chang Hee Kim
Comments: 7 Pages. [License] CC BY (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper presents a constructive definition of the real numbers R directly from the non-negative integers. Every real number is defined as a definite, finite value expressed through finite or generative sums of rationals, where infinity is treated as a process rather than a completed entity. This framework explains why Weierstrass’ epsilon—delta formulation works, resolves classical paradoxes as in set theory and topology, and establishes a foundation for analysis that is simple, intuitive, and contradiction-free.
Category: Set Theory and Logic
[8] ai.viXra.org:2508.0052 [pdf] submitted on 2025-08-19 05:18:37
Authors: John Augustine McCain
Comments: 10 Pages. CC-BY-NC 4.0
We present a reductio ad absurdum argument demonstrating the epistemological absurdity of requiring exhaustive verification for building confidence in mathematical conjectures. Through the development of a counterexample-collapse engine for the Goldbach conjecture, we show that demanding exhaustive verification leads to unreasonable epistemic commitments: if exhaustive verification were truly required, then discovering a single potential counterexample should rationally collapse all confidence to near-zero indefinitely. Since this behavior is clearly unreasonable, we conclude that exhaustive verification requirements are epistemologically unjustified, and that probabilistic evidence accumulation represents superior reasoning about mathematical reality. This argument has profound implications for mathematical epistemology, computational verification, and the nature of reasonable belief formation.
Category: Set Theory and Logic
[7] ai.viXra.org:2507.0046 [pdf] submitted on 2025-07-07 01:14:14
Authors: David Selke
Comments: 3 Pages.
The standard Foundation Axiom of ZFC forbids infinite descending membership chains, ensuring that all sets are well-founded in a bottom-up sense. However, we argue that this condition is not sufficient to prevent problematic constructions. In particular, we show that the structure of the von Neumann ordinals permits internal constructions of unbounded membership height, and we propose a converse principle---a kind of "reverse foundation"---to rule out such behavior. Our critique focuses on ω2, whose construction involves a transfinite accumulation of nested sets, each introducing a new local "top level" of membership. We introduce three core principles that make these issues apparent: Extensional Parsing, Length/Height Duality, and Local Top-Level Saturation.
Category: Set Theory and Logic
[6] ai.viXra.org:2507.0022 [pdf] submitted on 2025-07-04 18:50:57
Authors: Peter MacDonald Phillips
Comments: 2 Pages.
We investigate the Generalized Goldbach Conjecture (GGC) in the context of Infinite CommutativeRings with Identity (ICRI). The conjecture states that every nonzero element in theeven ideal, defined as the ideal generated by the sum of two units, can be expressed as the sumof two irreducible elements. We show that while GGC holds in the ring of integers Z, it failsin the product ring Z × Z, an infinite commutative ring with identity. This demonstrates thatGGC is independent of the axioms of ICRI.
Category: Set Theory and Logic
[5] ai.viXra.org:2507.0021 [pdf] submitted on 2025-07-04 20:07:54
Authors: Hamid Javanbakht
Comments: 19 Pages.
This paper undertakes a sequential analysis of five foundational contributions to the intersection of algebraic topology, higher category theory, and constructive type theory. Beginning with the structure of Dyer—Lashof operations in bordism and extending through the formal coactions underlying the Nishida relations, the category of π-finite spaces, the architecture of the effective 2-topos, and finally the internal groupoid semantics of Martin-Löf type theory, we trace a coherent development of stratified constructive structure. Each work articulates a distinct dimension of what we term telic stratification—the generation of mathematical types, spaces, and operations through staged, internally coherent constructions. We interpret the categorical universes that arise in these theories as classifying logoi: higher-categorical contexts that organize and reflect such stratification. An extended appendix develops this synthesis, proposing a unifying framework in which homotopy, computability, and internal logic are not merely compatible but mutually reinforcing. The resulting perspective suggests a foundational program grounded in the operational semantics of higher categories and constructive homotopy theory.
Category: Set Theory and Logic
[4] ai.viXra.org:2506.0132 [pdf] submitted on 2025-06-29 01:28:02
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
[3] ai.viXra.org:2506.0112 [pdf] submitted on 2025-06-24 02:29:18
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
[2] ai.viXra.org:2506.0060 [pdf] submitted on 2025-06-16 01:50:44
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
[1] ai.viXra.org:2505.0211 [pdf] submitted on 2025-05-31 21:19:45
Authors: Chang Hee Kim
Comments: 23 Pages. License: CC-BY.
This paper demonstrates that Cantor's Continuum Hypothesis is fundamentally flawed. The argument begins by showing that the set M = {0} U N can be decomposed into infinitely many subsets, each of which is infinite and pairwise disjoint from the other subsets of M. These subsets are then shown to admit a one-to-one correspondence (bijection) with the unit interval [0, 1). We further examine the failure of Cantor's Diagonal Argument, specifically exposing the structure of the implicit matrix it relies on. By decomposing M into such disjoint infinite subsets, we construct a direct bijection to the rows (real decimals) and columns (decimal digits) of this matrix. Each real number in [0, 1) corresponds uniquely to one of these subsets, eliminating the need to invoke "uncountable" sets. Through this decomposition, we establish that all infinite sets are equal. The very notion of comparing sizes of infinite sets — so-called "cardinality" — becomes unnecessary.
Category: Set Theory and Logic
[3] ai.viXra.org:2509.0063 [pdf] replaced on 2025-10-10 19:00:02
Authors: Chang Hee Kim
Comments: 100 Pages. License CC BY (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references)leased under Creative Commons CC BY 4.0 license.
This paper introduces a constructive definition of the real numbers R derived directly from the non-negative integers Z. Each real number is defined as a definite and finite value, represented either by a finite sum or by a generative, convergent process of rationals. By rejecting bijection between infinite sets—the core assumption underlying Cantorian set theory—the proposed framework eliminates self-contradictions inherent in ZFC and resolves classical paradoxes in set theory, measure theory, and topology. The result is a coherent, contradiction-free foundation for analysis that restores mathematics to constructive logic and physical intelligibility.
Category: Set Theory and Logic
[2] ai.viXra.org:2507.0022 [pdf] replaced on 2025-09-22 21:05:21
Authors: Peter MacDonald Phillips
Comments: 2 Pages.
We investigate the Generalized Goldbach Conjecture (GGC) in the context of Infinite Commutative Rings with Identity (ICRI). The conjecture states that every nonzero element in the even ideal, defined as the ideal generated by the sum of two units, can be expressed as the sum of two irreducible elements. If GGC is true in the ring of integers Z, we show that it fails in the product ring Z×Z, an infinite commutative ring with identity. This demonstrates that if GGC is true it is independent of the axioms of ICRI.
Category: Set Theory and Logic
[1] ai.viXra.org:2505.0211 [pdf] replaced on 2025-10-21 16:13:39
Authors: Chang Hee Kim
Comments: 22 Pages. CC BY 4.0
This paper demonstrates that Cantor’s Continuum Hypothesis is fundamentally flawed. The argument begins by showing that the set M = {0} U N can be decomposed into infinitely many subsets, each of which is infinite and pairwise disjoint from the other subsets of M. These subsets are then shown to admit a one-to-one correspondence (bijection) with the unit interval [0, 1).We further examine the failure of Cantor’s Diagonal Argument, specifically exposing the structure of the implicit matrix it relies on. By decomposing M into such disjoint infinite subsets, we construct a direct bijection to the rows (real decimals) and columns (decimal digits) of this matrix. Each real number in [0, 1) corresponds uniquely to one of these subsets, eliminating the need to invoke "uncountable" sets.Through this decomposition, we establish that all infinite sets are equal. The very notion of comparing sizes of infinite sets — so-called "cardinality" — becomes unnecessary. As a result, the foundation of the Continuum Hypothesis is no longer valid.
Category: Set Theory and Logic