scholarly journals Guessing models imply the singular cardinal hypothesis

2019 ◽  
Vol 147 (12) ◽  
pp. 5427-5434 ◽  
Author(s):  
John Krueger
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

A definable failure of the singular cardinal hypothesis

2012 ◽  
Vol 192 (2) ◽  
pp. 719-762 ◽  
Author(s):  
Sy-David Friedman ◽  
Radek Honzik
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis

scholarly journals ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS

2009 ◽  
Vol 09 (01) ◽  
pp. 139-157 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Singular Cardinal Hypothesis ◽  
Aronszajn Trees ◽  
Strongly Compact Cardinals

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ.

scholarly journals The negation of the singular cardinal hypothesis from o(K)=K++

1989 ◽  
Vol 43 (3) ◽  
pp. 209-234 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis

scholarly journals Sigma-Prikry forcing II: Iteration Scheme

2021 ◽  
pp. 2150019
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Blow Up ◽  
General Scheme ◽  
Iteration Scheme ◽  
Singular Cardinal ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

scholarly journals Weakly normal ideals ou PKl and the singular cardinal hypothesis

1993 ◽  
Vol 143 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Yoshihiro Abe
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis ◽  
Normal Ideals
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