Strong Projective Witnesses

Abstract

We show Shelah's original creature forcing from 1984 strongly preserves tight mad families. In particular, answering questions of Fischer and Friedman and Friedman and Zdomskyy, we show the constellation $\aleph_1 = \mathfrak{a} < \mathfrak{s} = \aleph_2$ is consistent with the existence of a $\Delta_3^1$ wellorder of the reals and tight mad families of sizes $\aleph_1, \aleph_2$ which are $\Pi_1^1, \Pi_2^1$-definable, respectively. Each of these projective definitions is of minimal possible complexity.