Definable maximal independent families

Brendle, Jörg; Fischer, Vera; Khomskii, Yurii

doi:10.1090/proc/14497

Definable maximal independent families
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by Jörg Brendle, Vera Fischer and Yurii Khomskii PDF

Proc. Amer. Math. Soc. 147 (2019), 3547-3557 Request permission

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.

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