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DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS
Part of:
Set theory
Published online by Cambridge University Press: 06 December 2023
Abstract
We construct divergent models of 
$\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis (
$\mathsf {CH}$) under various assumptions. Divergent models of 
$\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in 
$\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of 
$\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying 
$\neg \mathsf {CH}$, like 
$\mathsf {MM}$, that divergent models shed light on, see Question 5.1.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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