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The tree property on long intervals of regular cardinals

James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman, Dima Sinapova, Spencer Unger

TL;DR

The paper advances the long-standing program of obtaining the tree property at regular cardinals by showing it can hold on a long interval that overlaps a strong limit cardinal, under a suitable large-cardinal hypothesis. The authors develop a sophisticated forcing construction, combining an Easton-style preparation, a diagonal Prikry-type forcing, and intricate absorption/quotient-to-term techniques to align continuum values and preserve the tree property across a wide range of cardinals. Central to the argument are universal Laver functions, interleaved forcing posets (A,B,U,C,S), and robust branch-preservation lemmas that collectively guarantee that no new Aronszajn trees arise in the final model. The result yields a concrete instance where the tree property holds for all regular cardinals in an interval up to much larger cardinals, marking a key milestone toward the broader aim of tree property everywhere and highlighting the deep interaction between large cardinals, forcing, and cardinal arithmetic.

Abstract

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.

The tree property on long intervals of regular cardinals

TL;DR

The paper advances the long-standing program of obtaining the tree property at regular cardinals by showing it can hold on a long interval that overlaps a strong limit cardinal, under a suitable large-cardinal hypothesis. The authors develop a sophisticated forcing construction, combining an Easton-style preparation, a diagonal Prikry-type forcing, and intricate absorption/quotient-to-term techniques to align continuum values and preserve the tree property across a wide range of cardinals. Central to the argument are universal Laver functions, interleaved forcing posets (A,B,U,C,S), and robust branch-preservation lemmas that collectively guarantee that no new Aronszajn trees arise in the final model. The result yields a concrete instance where the tree property holds for all regular cardinals in an interval up to much larger cardinals, marking a key milestone toward the broader aim of tree property everywhere and highlighting the deep interaction between large cardinals, forcing, and cardinal arithmetic.

Abstract

In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.
Paper Structure (42 sections, 79 theorems, 71 equations)

This paper contains 42 sections, 79 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A fact about $\kappa$-cc forcing
  4. Laver functions and Laver indestructibility
  5. Trees and systems
  6. A branch lemma
  7. Another branch lemma
  8. Term forcing
  9. Projection and absorption
  10. Easton sets in Easton extensions
  11. Robustness of chain condition
  12. A technical fact
  13. $\mathbb{A}$, $\mathbb{B}$, $\mathbb{U}$, $\mathbb{C}$ and $\mathbb{S}$
  14. The basic forcing
  15. Further analysis
  16. ...and 27 more sections

Key Result

Theorem 1.1

Modulo a suitable large cardinal assumption, it is consistent that $\aleph_{\omega^2}$ is strong limit and the tree property holds for all regular cardinals $\kappa$ such that $\aleph_2 \le \kappa \le \aleph_{\omega^2 + 3}$.

Theorems & Definitions (290)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • proof
  • Remark 2.7
  • proof
  • ...and 280 more