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Definable discrete sets with large continuum

David Schrittesser

TL;DR

The paper tackles the problem of producing definable maximal $\mathcal{R}$-discrete sets for $\Sigma^1_1$ binary relations by using an iteration of Sacks forcing. It develops a Ramsey-theoretic framework for iterated Sacks forcing, including topological devices, density arguments, and fusion, and proves Galvin-type and Mycielski-type theorems to control definability of colorings. A key absoluteness result transfers definability between ground models and iterated extensions, enabling the construction of a $\Delta^1_2$ maximal $\mathcal{R}$-discrete set, which, in turn, yields a $\Pi^1_1$ maximal orthogonal family of Borel probability measures in the extension. Consequently, the existence of such definable discrete sets is compatible with $\neg\mathrm{CH}$ and provides a robust interaction between descriptive set theory and forcing, including consequences for maximal orthogonal families and related questions about definability under forcing extensions.

Abstract

Let $\mathcal R$ be a $Σ^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a $Δ^1_2$ maximal $\mathcal R$-discrete set, and thus the existence of such sets is compatible with the negation of the continuum hypothesis. As an application we find a $Π^1_1$ maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.

Definable discrete sets with large continuum

TL;DR

The paper tackles the problem of producing definable maximal -discrete sets for binary relations by using an iteration of Sacks forcing. It develops a Ramsey-theoretic framework for iterated Sacks forcing, including topological devices, density arguments, and fusion, and proves Galvin-type and Mycielski-type theorems to control definability of colorings. A key absoluteness result transfers definability between ground models and iterated extensions, enabling the construction of a maximal -discrete set, which, in turn, yields a maximal orthogonal family of Borel probability measures in the extension. Consequently, the existence of such definable discrete sets is compatible with and provides a robust interaction between descriptive set theory and forcing, including consequences for maximal orthogonal families and related questions about definability under forcing extensions.

Abstract

Let be a binary relation and call a set -discrete iff no two distinct of its elements are -related. We show that in the extension of by iterated Sacks forcing, there is a maximal -discrete set, and thus the existence of such sets is compatible with the negation of the continuum hypothesis. As an application we find a maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.

Paper Structure

This paper contains 11 sections, 19 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ramsey theory of iterated Sacks forcing
  4. Topologically determined conditions and their branch space
  5. A simpler description of iterated Sacks conditions
  6. The Polarized Mycielski's Theorem for product of Sacks forcing
  7. A Galvin-type theorem for iterated Sacks forcing
  8. An absoluteness result
  9. Definable discrete sets in the iterated Sacks extension
  10. A co-analytic maximal orthogonal family in the iterated Sacks extension
  11. Questions

Key Result

Theorem 1.1

For every $\bar{p}$ from a dense subset of $\mathop{\mathrm{\mathbb{P}}}\nolimits$ and every universally Baire coloring $c$ of pairs from $[\bar{p}]$ there is $\bar{q} \in \mathop{\mathrm{\mathbb{P}}}\nolimits$ such that $\bar{q} \leq \bar{p}$ and for each $\xi \in \mathrm{ supp }(\bar{q})$, $c \mat

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.4
  • Definition 3.1: The set $D^{\textbf{t}}_{{}}$ of topologically determined conditions
  • Definition 3.2: The branch space
  • ...and 56 more