Coanalytic families of functions

TL;DR

This work addresses the definability gap between $\Sigma^1_2$ and $\Pi^1_1$ witnesses for several combinatorial families of reals. By a uniform coding framework using $\chi_0,\chi_1$ and a $\Pi^1_1$ graph $H$ uniformizing a given witness, the authors convert $\Sigma^1_2$ witnesses into $\Pi^1_1$ witnesses of the same size for Van Douwen families, maximal ED families of permutations, maximal cofinitary groups (under extra assumptions), and maximal ideal independent families. They establish $\Pi^1_1$ definability via the Spector–Gandy theorem and discuss how forcing can yield coanalytic witnesses in certain models. The results provide a unified method to obtain coanalytic witnesses for these four classes, with additional insights into the minimal possible descriptive-set-theoretic complexity and consistency consequences such as $\aleph_1 = \mathfrak s_{mm} < \mathfrak c$ alongside a $\Delta^1_3$-wellorder of the reals.

Abstract

For Van Douwen families, maximal families of eventually different permutations and maximal ideal independent families we show that the existence of a $Σ^1_2$ family implies the existence of a $Π^1_1$ family of the same size. We also prove a similar, but slightly weaker result for generating sets of cofinitary groups.

Theorems & Definitions (24)

• Theorem 1
• proof
• Corollary 2
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Lemma 7
• proof