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Reals in the Matet and Willow Models

Raiean Banerjee

TL;DR

The work investigates the $\boldsymbol{\Delta}^1_2$-level regularities of Matet ($\boldsymbol{\Delta}^1_2(\mathbb{T})$) and Willow ($\boldsymbol{\Delta}^1_2(\mathbb{W})$) within Brendle's Uniform Forcings Diagram. It develops a fusion-based machinery for Matet forcing, proving pure decision and that Matet adds reals of minimal degree, and uses these tools to control quasi-generics for closed locally countable graphs; these methods extend to iterated Matet forcing up to $\omega_2$ with faithful fusion sequences. The paper then derives separations between Willow and Matet regularities in various forcing models: the Willow model disrupts $\boldsymbol{\Delta}^1_2(\mathbb{T})$, while Sacks and Miller models yield corresponding non-hold results for Matet-regularity via fusion and height-control arguments. Finally, it shows that $\boldsymbol{\Delta}^1_2(\mathbb{W})$ does not imply $\boldsymbol{\Delta}^1_2(E_0)$, by leveraging minimal-degree phenomena in $E_0$ forcing and ground-model Borel constructions. Together, these results contribute to the (incomplete) completeness of the Uniform Regularities Diagram and illuminate the interactions between tree-forcing notions and projective regularities.

Abstract

In this article, we try to complete the regularity implications between the regularitites of the well-known tree forcing notions at the $\boldsymbolΔ^1_2$ level of the projective hierarchy. The missing links in this case were the regularities corresponding to Matet and Willow tree Forcings. Some of the techniques yield more general results related to locally countable closed graphs too, which we mention as corollaries.

Reals in the Matet and Willow Models

TL;DR

The work investigates the -level regularities of Matet () and Willow () within Brendle's Uniform Forcings Diagram. It develops a fusion-based machinery for Matet forcing, proving pure decision and that Matet adds reals of minimal degree, and uses these tools to control quasi-generics for closed locally countable graphs; these methods extend to iterated Matet forcing up to with faithful fusion sequences. The paper then derives separations between Willow and Matet regularities in various forcing models: the Willow model disrupts , while Sacks and Miller models yield corresponding non-hold results for Matet-regularity via fusion and height-control arguments. Finally, it shows that does not imply , by leveraging minimal-degree phenomena in forcing and ground-model Borel constructions. Together, these results contribute to the (incomplete) completeness of the Uniform Regularities Diagram and illuminate the interactions between tree-forcing notions and projective regularities.

Abstract

In this article, we try to complete the regularity implications between the regularitites of the well-known tree forcing notions at the level of the projective hierarchy. The missing links in this case were the regularities corresponding to Matet and Willow tree Forcings. Some of the techniques yield more general results related to locally countable closed graphs too, which we mention as corollaries.
Paper Structure (7 sections, 16 theorems, 14 equations)

This paper contains 7 sections, 16 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Minimal degree and avoiding quasi-generics
  3. Iteration of Matet forcing
  4. Separating Willow and Matet regularities
  5. Willow and Matet regularities in the Sacks Model
  6. Willow and Matet regularities in the Miller model
  7. $E_{0}$ and Willow

Key Result

Theorem 2.1

Let $\theta$ be a sentence and $(s,A)$ a Matet condition. Then there is an extension $(s,B)$ such that for any $t \in \mathrm{FU}(B)$, $(s,B \mathrm{past} t)$ decides $\theta$.

Theorems & Definitions (30)

  • proof
  • Theorem 2.1
  • proof
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 20 more