Cantor's Diagonal, Inverse Powerset, and Multilevel Successor Models

We present a unified framework showing how Cantor’s diagonal argument can be viewed as a construction of the power set, how one can introduce a partial inverse‐powerset operator P^{-1} into ZFC without inventing new “negative‐rank” sets, and how a tower of countable models (each equipped with a unique nonstandard successor) collapses via P^{-1} to a single “first infinitesimal” successor in the base model. This article collects these ideas into a coherent exposition suitable for readers familiar with basic set theory, model theory, and nonstandard analysis.