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Boolean-valued class forcing

Volume 255 / 2021

Carolin Antos, Sy-David Friedman, Victoria Gitman Fundamenta Mathematicae 255 (2021), 231-254 MSC: 03E30, 03E05, 03E40. DOI: 10.4064/fm20-7-2021 Published online: 11 October 2021

Abstract

We show that the Boolean algebras approach to class forcing can be carried out in the theory Kelley–Morse plus the Choice Scheme (${\rm KM} +{\rm CC} $) using hyperclass Boolean completions of class partial orders. Applying the Boolean algebras approach, we show that every intermediate model between a model $\mathscr V \models {\rm KM} +{\rm CC} $ and one of its class forcing extensions is itself a class forcing extension if and only if it is simple—generated by the classes of $\mathscr V $ together with a single new class. We show that there can be non-simple intermediate models between a model of ${\rm KM} +{\rm CC} $ and its class forcing extension, and so the full Intermediate Model Theorem can fail for models of ${\rm KM} +{\rm CC} $.

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