Definable Maximal Independent Families

Abstract

We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a ${\mathbf{\Sigma }}_2^1$ m.i.f. is equivalent to the existence of a ${\mathbf{\Pi }}_1^1$ m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a ${\mathbf{\Pi }}_1^1$ m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.

Accepted for publication, 2019