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Aronszajn trees and maximality

Omer Ben-Neria, Siiri Kivimäki, Menachem Magidor, Jouko Väänänen

TL;DR

The article proves, relative to the existence of a weakly compact cardinal, that there is a forcing extension in which κ = μ^+ and there exists a maximal wide κ-Aronszajn tree T into which every wide κ-Aronszajn tree embeds. The construction uses Levy collapse to build a wide universal tree T and a σ-closed, side-condition–driven forcing to embed all wide κ^+-Aronszajn trees into T, while preserving κ^+ and ensuring no new branches are added. The technique hinges on strong properness and a robust quotient analysis to maintain genericity and universality throughout the iteration. The work highlights fundamental differences between trees with no infinite branches and those without branches of length κ, and concludes with several intriguing open questions for future research.

Abstract

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $μ$, we prove the consistency of the existence of a wide $μ^+$-Aronszajn tree, i.e. a tree of height and cardinality $μ^+$ with no branches of length $μ^+$, into which every wide $μ^+$-Aronszajn tree can be embedded.

Aronszajn trees and maximality

TL;DR

The article proves, relative to the existence of a weakly compact cardinal, that there is a forcing extension in which κ = μ^+ and there exists a maximal wide κ-Aronszajn tree T into which every wide κ-Aronszajn tree embeds. The construction uses Levy collapse to build a wide universal tree T and a σ-closed, side-condition–driven forcing to embed all wide κ^+-Aronszajn trees into T, while preserving κ^+ and ensuring no new branches are added. The technique hinges on strong properness and a robust quotient analysis to maintain genericity and universality throughout the iteration. The work highlights fundamental differences between trees with no infinite branches and those without branches of length κ, and concludes with several intriguing open questions for future research.

Abstract

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal , we prove the consistency of the existence of a wide -Aronszajn tree, i.e. a tree of height and cardinality with no branches of length , into which every wide -Aronszajn tree can be embedded.
Paper Structure (8 sections, 30 theorems, 66 equations)

This paper contains 8 sections, 30 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Building the Wide Tree $T$
  4. The embedding poset
  5. Strong Properness
  6. Strong properness of quotients
  7. No new branches
  8. Open Problems

Key Result

Theorem 1

Suppose that $\kappa$ is a weakly compact cardinal and $\mu < \kappa$ is regular uncountable. There is a (set) forcing extension of the universe in which $\kappa = \mu^+$ and there is a maximal wide $\kappa$-Aronszajn tree i.e. a wide $\kappa$-Aronszajn tree $T$ such that any other wide $\kappa$-Aro

Theorems & Definitions (104)

  • Theorem 1
  • Definition 2: Reflecting structures
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Definition 6
  • Definition 7: Collapse index
  • Remark 9
  • Lemma 10
  • Definition 11
  • ...and 94 more