# Well-Ordered Model Universes

## TL;DR

This paper constructs models of ZFC where all inner models with the Axiom of Choice are well-ordered by inclusion, with the order's height being arbitrarily large, using iterated κ-Sacks forcing.

## Contribution

It introduces a method to create models of ZFC with a well-ordered hierarchy of inner models satisfying AC, extending the possible height of this hierarchy to any ordinal.

## Key findings

- Inner models satisfying AC are well-ordered by inclusion
- The height of the well-ordering can be arbitrarily large
- Iterated κ-Sacks forcing does not produce unexpected intermediate models

## 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.