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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

NAM TRANG
NAM TRANG*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS DENTON, TX 76205, USA
*
E-mail: [email protected]
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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.

Keywords

divergent modelsinner model theorydescriptive set theory

MSC classification

Primary: 03E15: Descriptive set theory 03E45: Inner models, including constructibility, ordinal definability, and core models 03E60: Determinacy principles
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 11
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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