Well-Ordered Model Universes

Abstract

In this paper we show how to build a model of $ZFC$ such that all its inner models satisfying the Axiom of Choice are well-ordered with respect to inclusion, and that said ordering is of arbitrary height (including possibly $Ord$ high). We do this by iterating $\kappa$-Sacks forcing for ever-increasing $\kappa$, while showing that such forcings do not add any unexpected intermediate inner models.