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Full Souslin trees at small cardinals

Assaf Rinot, Shira Yadai, Zhixing You

Abstract

A $κ$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $κ$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be $\aleph_3$ many full $\aleph_2$-trees such that the product of any countably many of them is an $\aleph_2$-Souslin tree.

Full Souslin trees at small cardinals

Abstract

A -tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full -Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be many full -trees such that the product of any countably many of them is an -Souslin tree.
Paper Structure (9 sections, 17 theorems, 37 equations)

This paper contains 9 sections, 17 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Organization of this paper
  3. Preliminaries
  4. Abstract trees
  5. Streamlined trees
  6. Coherent sequences
  7. Diamond-type prediction principles
  8. Full Souslin trees at strongly inaccessibles
  9. Full Souslin trees at successors of regulars

Key Result

Theorem A

Suppose that $\kappa$ is a subtle cardinal and that $\boxtimes^-(\kappa)$ holds. Then there exists a full $\kappa$-Souslin tree.

Theorems & Definitions (75)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 65 more