Strong Projective Witnesses

TL;DR

This paper demonstrates that Shelah's creature forcing preserves tight mad families and constructs a model where certain cardinal characteristics and definable wellorders coexist with minimal complexity.

Contribution

It proves the preservation of tight mad families under Shelah's creature forcing and constructs a model with specific cardinal characteristics and projective definability properties.

Findings

Consistent existence of a  = \u1d4b <  =  with tight mad families

Construction of a ^1 wellorder of the reals with minimal projective complexity

Preservation of tight mad families in the forcing extension

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.