An Expansion on Paraconsistent Micro-Geometry: A Non-Standard Approach to Subgroup-Stratified Vitali Measures via the Logic of da Costa
George Yeats Lazou·Google: Gemini 3.5 Flash-Lite
Abstract
The existence of so-called pathological sets with respect to certain contexts of measure theory has often demanded a treatment that better satisfies our intuition or provides greater insight into their nature. The non-measurable sets of Vitali, in particular, seem to exhibit the intuitive properties of infinitesimals, and indeed, to some extent this can be for malized. While the standard Loeb measure yields real-valued extensions, the underlying hyperfinite constructions naturally suggest that Vitali sets possess infinitesimal inner and outer measures. However, a fully explicit bottom-up construction of these infinitesimal assignments has remained elusive, making it difficult to systematically distinguish between differ ent Vitali sets—a gap that this paper seeks to rectify. Thus, given the top-down guaranteed existence of a non-standard framework for Vitali sets, we provide a formal construction from the bottom up that allows for a stratification of measures reflected as a stratification of functions and operators. It is hoped that this stratified construction will find diverse applications from capturing micro-geometric efficiency to ergodic theory and dynamical systems, and be generally seen as a pleasing addition to the literature.Versions
v2this version4 Oct 2026
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