Intermediate models and Kinna–Wagner Principles

Kinna–Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write K W P KWP to denote that there is some α α for which this holds. The Kinna–Wagner Conjecture, formulated by the first author [Bull. Symb. Log., arXiv:2006.04514], states that if V V is a model of Z F + K W P ZF+KWP and G G is a V V -generic filter, then whenever W W is an intermediate model of Z F ZF , that is V ⊆ W ⊆ V [ G ] V⊆ W⊆ V[G] , then W = V ( x ) W=V(x) for some x x if and only if W W satisfies K W P 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.