Higher Souslin trees and the GCH, revisited

Assaf Rinot

doi:10.1016/j.aim.2017.03.002

Higher Souslin trees and the GCH, revisited

Assaf Rinot

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

Abstract

It is proved that for every uncountable cardinal λ, GCH+□(λ⁺) entails the existence of a cf(λ)-complete λ⁺-Souslin tree. In particular, if GCH holds and there are no ℵ₂-Souslin trees, then ℵ₂is weakly compact in Gödel's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ₂and ℵ₃Souslin trees, then the Axiom of Determinacy holds in L(R).

Original language	English
Pages (from-to)	510-531
Number of pages	22
Journal	Advances in Mathematics
Volume	311
DOIs	https://doi.org/10.1016/j.aim.2017.03.002
State	Published - 30 Apr 2017

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Funding

This work was partially supported by the Israel Science Foundation (grant #1630/14).

Funders	Funder number
Israel Science Foundation	1630/14

Keywords

Microscopic approach
Souslin tree
Square
Weakly compact cardinal

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10.1016/j.aim.2017.03.002

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abstract = "It is proved that for every uncountable cardinal λ, GCH+□(λ+) entails the existence of a cf(λ)-complete λ+-Souslin tree. In particular, if GCH holds and there are no ℵ2-Souslin trees, then ℵ2is weakly compact in G{\"o}del's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ2and ℵ3Souslin trees, then the Axiom of Determinacy holds in L(R).",

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AB - It is proved that for every uncountable cardinal λ, GCH+□(λ+) entails the existence of a cf(λ)-complete λ+-Souslin tree. In particular, if GCH holds and there are no ℵ2-Souslin trees, then ℵ2is weakly compact in Gödel's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ2and ℵ3Souslin trees, then the Axiom of Determinacy holds in L(R).

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ER -

Higher Souslin trees and the GCH, revisited

Abstract

Bibliographical note

Funding

Keywords

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