Good projective witnesses

TL;DR

This paper introduces a new forcing method to construct models where the minimal size of a maximal cofinitary group is strictly between  and the continuum, with a _2-definable example, extending results on MAD families.

Contribution

It develops a novel forcing technique to produce models with a maximal cofinitary group of intermediate size that is _2-definable, linking group theory and descriptive set theory.

Findings

Consistently,  < _g < \u00c3

Existence of a _2-definable MCG of intermediate size

Extension of results on MAD families and definability

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 $\Pi^1_2$-definable MCG of this cardinality. Here $\Pi^1_2$ 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.