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 KWP to denote that there is some α for which this holds. The Kinna–Wagner Conjecture, formulated by the first author [Bull. Symb. Log., arXiv:2006.04514], states that if V is a model of ZF + KWP and G is a V -generic filter, then whenever W is an intermediate model of ZF, that is V ⊆ W ⊆ V [G], then W = V (x) for some x if and only if W satisfies 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.