Abstract
Ben-David and Shelah proved that if λ is a singular strong-limit cardinal and 2 λ = λ + , then ∗ λ entails the existence of a normal λ-distributive λ + -Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis ∗ λ by (λ + ,<λ). As (λ + ,<λ) does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for κ regular uncountable, (κ) entails the existence of a partition of κ into κ many fat sets. When contrasted with a classical model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that ω2 cannot be split into two fat sets.
| Original language | English |
|---|---|
| Pages (from-to) | 217-291 |
| Number of pages | 75 |
| Journal | Fundamenta Mathematicae |
| Volume | 245 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2019 |
Bibliographical note
Funding Information:Proof. Appeal to Theorem 5.1 with (µ, χ, Ω) := (λ+, ℵ0, ∅) to obtain a C-sequence satisfying clause (1) of Theorem 5.4. Then, let D⃗ be the C-sequence provided by clause (2) of Theorem 5.4. Since forcing with the Acknowledgments. This work was partially supported by the Israel Science Foundation (Grant #1630/14). During the revision of this paper, the first author was supported by the Center for Absorption in Science, Ministry of Aliyah and Integration, State of Israel.
Publisher Copyright:
c Instytut Matematyczny PAN, 2019
Keywords
- And phrases: Aronszajn tree
- C-sequence
- Club guessing
- Distributive tree
- Fat set
- Nonspecial Aronszajn tree
- Postprocessing function
- Square principle
- Uniformly coherent Souslin tree
- Walks on ordinals