Set Theory and Logic

2507 Submissions

[3] ai.viXra.org:2507.0046 [pdf] submitted on 2025-07-07 01:14:14

Toward a Reverse Foundation Axiom: Implicit Upward Unboundedness in the Von Neumann Ordinals

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

[2] ai.viXra.org:2507.0022 [pdf] replaced on 2025-09-22 21:05:21

On the Independence of the Generalized Goldbach Conjecture in Infinite Commutative Rings with Identity

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:2507.0021 [pdf] submitted on 2025-07-04 20:07:54

Telic Stratification and Classifying Logoi I: Bordism, π-Finitude, and Internal Cohesion

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