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# Non-tame Mice from Tame Failures of the Unique Branch Hypothesis

Published:2013-11-01

• Grigor Sargsyan,
Department of Mathematics, Rutgers University, New Brunswick, NJ, 08854 USA
• Nam Trang,
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213 USA
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## Abstract

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing $Ord \cup \mathbb{R}$ such that $M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this implies the existence (in $V$) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.
 Keywords: mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH
 MSC Classifications: 03E15 - Descriptive set theory [See also 28A05, 54H05] 03E45 - Inner models, including constructibility, ordinal definability, and core models 03E60 - Determinacy principles

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