Cofinitary groups and projective well-orders

Abstract

We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideal. Introducing a new robust coding technique, we establish the relative consistency of ${\mathfrak{a}}_g=\mathfrak{d}<\mathfrak{c}={\aleph }_2$ alongside the existence of a $\hat{I}{”}_3^1$-wellorder of the reals and a co-analytic witness for ${\mathfrak{a}}_g$.

21 pages, revision