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Good projective witnesses

Vera Fischer, Sy David Friedman, David Schrittesser, Asger Törnquist

TL;DR

This work introduces a forcing framework to adjoin self-coding permutations to cofinitary groups, enabling the construction of maximal cofinitary groups with intermediate cardinalities and $oldsymbol{oldsymbol{\ ext{Pi}}^1_2}$-definable witnesses. The authors develop a forcing $bQ$ that codes ground-model data into new generators and show how a cardinal-preserving iteration yields a maximal cofinitary group of size $oldsymbol{ olinebreak aleph_M}$ with continuum $oldsymbol{ olinebreak aleph_N}$ for $2\le M<N$, while ensuring the group’s definition is projectively definable at the level $oldsymbol{oldsymbol{\ ext{Pi}}^1_2}$. The construction generalizes to maximal almost disjoint (MAD) families, producing a $oldsymbol{oldsymbol{\ ext{Pi}}^1_2}$ MAD witness and drawing parallels to the Horowitz–Shelah MCG. The results provide a natural counterpart to Borel MCGs and reveal new tools for coding and definability in forcing, with several open questions about the scope and limits of definable witnesses in related combinatorial structures.

Abstract

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $Π^1_2$-definable MCG of this cardinality. Here $Π^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

Good projective witnesses

TL;DR

This work introduces a forcing framework to adjoin self-coding permutations to cofinitary groups, enabling the construction of maximal cofinitary groups with intermediate cardinalities and -definable witnesses. The authors develop a forcing that codes ground-model data into new generators and show how a cardinal-preserving iteration yields a maximal cofinitary group of size with continuum for , while ensuring the group’s definition is projectively definable at the level . The construction generalizes to maximal almost disjoint (MAD) families, producing a MAD witness and drawing parallels to the Horowitz–Shelah MCG. The results provide a natural counterpart to Borel MCGs and reveal new tools for coding and definability in forcing, with several open questions about the scope and limits of definable witnesses in related combinatorial structures.

Abstract

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality of a maximal cofinitary group (MCG) is strictly between and , and there is a -definable MCG of this cardinality. Here is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

Paper Structure

This paper contains 13 sections, 22 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Adding cofinitary groups of coding permutations
  3. Finite partial extensions of cofinitary groups
  4. The Conjugated Subwords Lemma
  5. Two Lemmas on free groups
  6. Computable Coding
  7. The forcing
  8. The forcing iteration
  9. Preparing the Universe
  10. Preparing the continuum
  11. Adding the MCG
  12. Definability and maximality of the group
  13. Questions

Key Result

Theorem 1.3

It is relatively consistent with $\textup{ZFC}$ that $\mathfrak c\geq\aleph_3$ and there is an MCG $\mathcal{G}$ of size $\aleph_2$, $\mathcal{G}\in L({\mathbb R})$.

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 2.2
  • Remark 2.4
  • Example 2.5
  • ...and 46 more