A choice function on countable sets, from determinacy
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- by Paul B. Larson PDF
- Proc. Amer. Math. Soc. 143 (2015), 1763-1770 Request permission
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 $\mathrm {AD}_{\mathbb {R}}$.References
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Additional Information
- Paul B. Larson
- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
- MR Author ID: 646854
- Email: larsonpb@miamioh.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3314088