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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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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