# Definable Maximal Independent Families

## TL;DR

This paper investigates the existence and properties of maximal independent families within the projective hierarchy, establishing equivalences, model-specific results, and introducing a related cardinal invariant.

## Contribution

It proves the equivalence of certain definability levels for maximal independent families and analyzes their existence in different set-theoretic models, also introducing a new cardinal invariant.

## Key findings

- Existence of a $oldsymbol{oldsymbol{ imes}1}_2$ m.i.f. is equivalent to a $oldsymbol{oldsymbol{ imes}1}_1$ m.i.f.
- No projective m.i.f. exists in the Cohen model.
- A $oldsymbol{oldsymbol{ imes}1}_1$ m.i.f. exists in the Sacks model.

## Abstract

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