Skands and Coskands: Exploring Non-Founded Set Theory

This paper introduces and explores the concepts of 'skands' and 'coskands' within the framework of non-founded set theory, specifically NBG without the axiom of regularity. It aims to extend set theory by allowing for non-well-founded sets, which are sets that can contain themselves or form infinite descending membership chains. The paper's significance lies in its exploration of alternative set-theoretic foundations and its potential implications for understanding mathematical structures beyond the standard ZFC axioms. The introduction of skands and coskands provides new tools for modeling and reasoning about non-well-founded sets, potentially opening up new avenues for research in areas like computer science and theoretical physics where such sets may be relevant.