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

  • 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 mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH
MSC Classifications: 03E15, 03E45, 03E60 show english descriptions Descriptive set theory [See also 28A05, 54H05]
Inner models, including constructibility, ordinal definability, and core models
Determinacy principles
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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