Good projective witnesses

Abstract

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality ${\mathfrak{a}}_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between ${\aleph }_1$ and $\mathfrak{c}$, and there is a ${\hat{I}}_2^1$-definable MCG of this cardinality. Here ${\hat{I}}_2^1$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

33 pages