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

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

20 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
Externally published	Yes

Keywords

Microscopic approach
Souslin tree
Square
Weakly compact cardinal

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2017.03.002

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title = "Higher Souslin trees and the GCH, revisited",

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).",

keywords = "Microscopic approach, Souslin tree, Square, Weakly compact cardinal",

author = "Assaf Rinot",

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AU - Rinot, Assaf

PY - 2017/4/30

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N2 - 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).

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).

KW - Microscopic approach

KW - Souslin tree

KW - Square

KW - Weakly compact cardinal

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SP - 510

EP - 531

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Higher Souslin trees and the GCH, revisited

Abstract

Keywords

ASJC Scopus subject areas

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