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Large cardinal axioms from tameness in AECs


Authors: Will Boney and Spencer Unger
Journal: Proc. Amer. Math. Soc. 145 (2017), 4517-4532
MSC (2010): Primary 03C45, 03E55, 03C48
DOI: https://doi.org/10.1090/proc/13555
Published electronically: April 7, 2017
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Abstract: We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions. For instance, we show that if $ \kappa $ is an uncountable cardinal such that $ \mu ^\omega < \kappa $ for every $ \mu < \kappa $ and every AEC with Löwenheim-Skolem number less than $ \kappa $ is $ <\kappa $-tame, then $ \kappa $ is almost strongly compact. This is done by isolating a class of AECs that exhibits tameness exactly when sufficiently complete ultrafilters exist.


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  • [1] Joan Bagaria and Menachem Magidor, Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857-1877. MR 3152715, https://doi.org/10.1090/S0002-9947-2013-05871-0
  • [2] John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009. MR 2532039
  • [3] John T. Baldwin and Saharon Shelah, Examples of non-locality, J. Symbolic Logic 73 (2008), no. 3, 765-782. MR 2444267, https://doi.org/10.2178/jsl/1230396746
  • [4] T. Beke and J. Rosický, Abstract elementary classes and accessible categories, Ann. Pure Appl. Logic 163 (2012), no. 12, 2008-2017. MR 2964883, https://doi.org/10.1016/j.apal.2012.06.003
  • [5] Will Boney, Tameness from large cardinal axioms, J. Symb. Log. 79 (2014), no. 4, 1092-1119. MR 3343531, https://doi.org/10.1017/jsl.2014.30
  • [6] A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1317-1327. MR 3589328, https://doi.org/10.1090/proc/13190
  • [7] Rami Grossberg and Monica VanDieren, Categoricity from one successor cardinal in tame abstract elementary classes, J. Math. Log. 6 (2006), no. 2, 181-201. MR 2317426, https://doi.org/10.1142/S0219061306000554
  • [8] Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings. MR 1994835
  • [9] M. Lieberman and J. Rosický, Classification theory for accessible categories, J. Symb. Log. 81 (2016), no. 1, 151-165. MR 3471133, https://doi.org/10.1017/jsl.2014.85
  • [10] Michael J. Lieberman, Category-theoretic aspects of abstract elementary classes, Ann. Pure Appl. Logic 162 (2011), no. 11, 903-915. MR 2817563, https://doi.org/10.1016/j.apal.2011.05.002
  • [11] Michael Makkai and Robert Paré, Accessible categories: the foundations of categorical model theory, Contemporary Mathematics, vol. 104, American Mathematical Society, Providence, RI, 1989. MR 1031717
  • [12] Michael Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514-538. MR 0175782
  • [13] Saharon Shelah, Maximal failures of sequence locality in aec, Preprint.
  • [14] Saharon Shelah, Categoricity for abstract classes with amalgamation, Ann. Pure Appl. Logic 98 (1999), no. 1-3, 261-294. MR 1696853, https://doi.org/10.1016/S0168-0072(98)00016-5

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Additional Information

Will Boney
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: wboney@math.harvard.edu

Spencer Unger
Affiliation: Department of Mathematics, University of California-Los Angeles, Los Angeles, California 90095
Email: sunger@math.ucla.edu

DOI: https://doi.org/10.1090/proc/13555
Received by editor(s): October 6, 2015
Received by editor(s) in revised form: July 5, 2016, October 7, 2016, and October 19, 2016
Published electronically: April 7, 2017
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society

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