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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Sigma 2 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="bold"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\boldsymbol {\Sigma }^1_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> m.i.f. is equivalent to the existence of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Pi 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="bold"> Π </mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\boldsymbol {\Pi }^1_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Pi 1 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="bold"> Π </mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\boldsymbol {\Pi }^1_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.