An Expansion on Paraconsistent Micro-Geometry: A Dialethic Non-Standard Approach to Subgroup-Stratified Vitali Measures
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 formalized. The Loeb measure, named after its creator Peter A. Loeb, seems to imply that Vitali sets have infinitesimal measure, at least in the hyperreals, without explicitly providing a construction. Unfortunately, this lack of construction means that there is no real way of distinguishing between different Vitali sets, and it is this that we try to rectify in this paper. Thus, given the top-down guaranteed existence of a non-standard infinitesimal Loeb measure 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.Discussion
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