BASES AND BOREL SELECTORS FOR TALL FAMILIES

Abstract

Abstract Given a family ${\cal C}$ of infinite subsets of ${\Bbb N}$ , we study when there is a Borel function $S:2^{\Bbb N} \to 2^{\Bbb N} $ such that for every infinite $x \in 2^{\Bbb N} $ , $S\left( x \right) \in {\Cal C}$ and $S\left( x \right) \subseteq x$ . We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an $F_\sigma $ tall ideal on ${\Bbb N}$ without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a ${\bf{\Pi }}_2^1 $ tall ideal on ${\Bbb N}$ without a tall closed subset.

Locations

• Journal of Symbolic Logic - View - PDF