Strong Projective Witnesses
Vera Fischer, Julia Millhouse
TL;DR
This work analyzes Shelah's creature forcing to show strong preservation of tight mad families under countable-support iterations, enabling refined consistency results for cardinal characteristics. By integrating Sacks coding, club shooting, localization, and $S$-proper iterations, the authors produce a $ obreak \Delta_3^1$ wellorder of the reals together with coexisting, definable tight mad witnesses of sizes $ obreak obreak aleph_1$ and $ obreak obreak aleph_2$, while maintaining $ obreak aleph_1 = \mathfrak{a} < \mathfrak{s} = \mathfrak{c} = obreak aleph_2$. They further examine the definable spectra of mad witnesses and extend prior results on projective complexity, showing $ obreak \Pi_1^1$ and $ obreak \Pi_2^1$ definable tight mad families of the expected sizes, all under optimal descriptive set-theoretic bounds. The methods hinge on outer hull arguments and precise coding to preserve definability through iterations, contributing to the understanding of how forcing interacts with cardinal characteristics and projective definability. Overall, the paper advances the synthesis of forcing, definability, and reals-structure, producing sharp, definable witnesses to cardinal characteristics in a carefully controlled model.
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 $Δ_3^1$ wellorder of the reals and tight mad families of sizes $\aleph_1, \aleph_2$ which are $Π_1^1, Π_2^1$-definable, respectively. Each of these projective definitions is of minimal possible complexity.