A choice function on countable sets, from determinacy

Larson, Paul

doi:10.1090/S0002-9939-2014-12349-5

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 $\mathrm {AD}_{\mathbb {R}}$.

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