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

  • Grigor Sargsyan (a1) and Nam Trang (a2)

Abstract

In this paper, we show that the failure of the unique branch hypothesis $\left( \text{UBH} \right)$ 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 $M\,\vDash \,\text{A}{{\text{D}}^{+}}\,+\,\Theta \,>\,{{\theta }_{0}}$ . 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.

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References

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[1] Jech, T., Set theory. The third millennium ed., revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[2] Ketchersid, R. O., Toward AD from the continuum hypothesis and an ω1-dense ideal. Ph.D. Thesis, University of California, Berkeley, ProQuest LLC, Ann Arbor, MI, 2000.
[3] Martin, D. A. and Steel, J. R., Iteration trees. J. Amer. Math. Soc. 7(1994), no. 1, 173. http://dx.doi.org/10.1090/S0894-0347-1994-1224594-7
[4] Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees. Lecture Notes in Logic, 3, Springer-Verlag, Berlin, 1994.
[5] Neeman, I., Inner models in the region of a Woodin limit of Woodin cardinals. Ann. Pure Appl. Logic 116(2002), no. 1–3, 67155. http://dx.doi.org/10.1016/S0168-0072(01)00103-8
[6] Sargsyan, G., Descriptive inner model theory. Bull. Symbolic Logic, to appear. http://math.rutgers.edu/_gs481. http://dx.doi.org/10.2178/bsl.1901010
[7] Sargsyan, G., On the strength of PFAI. http://math.rutgers.edu/_gs481.
[8] Sargsyan, G., A tale of hybrid mice. http://math.rutgers.edu/_gs481/.
[9] Schindler, R. and Steel, J. R., The core model induction. http://math.berkeley.edu/_steel.
[10] Schlutzenberg, F. and Trang, N., Scales in LpΣ(ℝ). http://math.cmu.edu/_namtrang.
[11] Steel, J. R., PFA implies ADL(ℝ). J. Symbolic Logic 70(2005), no. 4, 12551296. http://dx.doi.org/10.2178/jsl/1129642125
[12] Steel, J. R., Derived models associated to mice. In: Computational prospects of infinity. Part I. Tutorials, Lect. Notes Ser. Inst.Math. Sci. Natl. Univ. Singap., 14,World Sci. Publ., Hackensack, NJ, 2008, pp. 105193. http://dx.doi.org/10.1142/9789812794055 0003
[13] Steel, J. R., Scales in K(ℝ). In: Games, scales, and Suslin cardinals. The Cabal Seminar. Vol. I, Lect. Notes Log., 31, Assoc. Symbol. Logic, Chicago, IL, 2008, pp. 176208. http://dx.doi.org/10.1017/CBO9780511546488.011
[14] Steel, J. R., Scales in K(ℝ) at the end of a weak gap. J. Symbolic Logic, 73(2008), no. 2, 369390. http://dx.doi.org/10.2178/jsl/1208359049
[15] Steel, J. R., The derived model theorem. In: Logic Colloquium 2006, Lect. Notes Log., Assoc. Symbol. Logic, Chicago, IL, 2009, pp. 280327. http://dx.doi.org/10.1017/CBO9780511605321.014
[16] Steel, J. R., Core models with more Woodin cardinals. J. Symbolic Logic, 67(2002), no. 3, 11971226. http://dx.doi.org/10.2178/jsl/1190150159
[17] Woodin, W. H., Suitable extender models I. J. Math. Log. 10(2010), no. 1–2, 101339. http://dx.doi.org/10.1142/S021906131000095X
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Non-tame Mice from Tame Failures of the Unique Branch Hypothesis

  • Grigor Sargsyan (a1) and Nam Trang (a2)

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