Intermediate models and Kinna--Wagner Principles

Abstract: Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $\alpha$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $\mathsf{ZF+KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $\mathsf{ZF}$, that is $V\subseteq W\subseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $\mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets.