The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2
نویسنده
چکیده
We 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 .
منابع مشابه
Aronszajn Trees and Failure of the Singular Cardinal Hypothesis
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...
متن کاملThe tree property and the failure of SCH at uncountable cofinality
Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ.
متن کاملAronszajn trees and the successors of a singular cardinal
From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ++ has the tree property. In particular this model has no special κ+-trees.
متن کاملTree Property at Successor of a Singular Limit of Measurable Cardinals
Assume λ is a singular limit of η supercompact cardinals, where η ≤ λ is a limit ordinal. We present two forcing methods for making λ+ the successor of the limit of the first η measurable cardinals while the tree property holding at λ+. The first method is then used to get, from the same assumptions, tree property at אη2+1 with the failure of SCH at אη2 . This extends results of Neeman and Sina...
متن کاملThe tree property at double successors of singular cardinals of uncountable cofinality
Assuming the existence of a strong cardinal κ and a measurable cardinal above it, we force a generic extension in which κ is a singular strong limit cardinal of any given cofinality, and such that the tree property holds at κ++.
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ورودعنوان ژورنال:
- J. Symb. Log.
دوره 77 شماره
صفحات -
تاریخ انتشار 2012