Omega-inconsistency and the Universe of Sets
Willard Mittelman, Theodore E. Simos, George Psihoyios · AIP conference proceedings · 2008
An ω‐inconsistent theory of the cumulative hierarchy of ZFC is presented, using an expanded language that includes a truth‐predicate, with formulas of the language of ZF being represented by urelements. This ω‐inconsistency makes it possible to distinguish, within the cumulative hierarchy itself, between sets and proper classes, taking the latter to be collections so large that they lead to ω‐inconsistency.