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A choice function on countable sets, from determinacy


Author: Paul B. Larson
Journal: Proc. Amer. Math. Soc. 143 (2015), 1763-1770
MSC (2010): Primary 03E25, 03E40, 03E60
DOI: https://doi.org/10.1090/S0002-9939-2014-12349-5
Published electronically: November 19, 2014
MathSciNet review: 3314088
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Abstract: We prove that $ \mathrm {AD}_{\mathbb{R}}$ implies the existence of a definable class function which, given a countable set $ X$, a tall ideal $ I$ on $ \omega $ containing $ \mathrm {Fin}$ and a function from $ I \setminus \mathrm {Fin}$ to $ X$ which is invariant under finite changes, selects a nonempty finite subset of $ X$. Among other applications, this gives an alternate proof of the fact (previously established by Di Prisco-Todorcevic) that there is no selector for the $ E_{0}$ degrees in the $ \mathcal {P}(\omega )/\mathrm {Fin}$-extension of a model of $ \mathrm {AD}_{\mathbb{R}}$.


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Additional Information

Paul B. Larson
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: larsonpb@miamioh.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12349-5
Received by editor(s): March 28, 2013
Received by editor(s) in revised form: July 8, 2013, and September 2, 2013
Published electronically: November 19, 2014
Additional Notes: This research was supported by NSF Grants DMS-0801009 and DMS-1201494. The results were obtained in February 2011. The author thanks Andrés Caicedo and Grigor Sargsyan for help with the bibliographic references.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.