Aronszajn trees and the successors of a singular cardinal
نویسنده
چکیده
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.
منابع مشابه
Souslin trees and successors of singular cardinals
The questions concerning existence of Aronszajn and Souslin trees are of the oldest and most dealt-with in modern set theory. There are many results about existence of h+-Aronszajn trees for regular cardinals A. For these cases the answer is quite complete. (See Jech [6] and Kanamory & Magidor [8] for details.) The situation is quite different when A is a singular cardinal. There are very few r...
متن کاملThe tree property at successors of singular cardinals
Assuming some large cardinals, a model of ZFC is obtained in which אω+1 carries no Aronszajn trees. It is also shown that if λ is a singular limit of strongly compact cardinals, then λ carries no Aronszajn trees.
متن کاملAronszajn 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 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++.
متن کاملReview on Todd Eisworth’s Chapter for the Handbook of Set Theory: “successors of Singular Cardinals”
This chapter offers a comprehensive and lucid exposition of the questions and techniques involved in the study of combinatorics of successors of singular cardinals. What is so special about successors of singular cardinals? They are successor cardinals, but are also similar to inaccessible cardinals, in the sense that there is no maximal regular cardinal below them, meaning for instance that th...
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ورودعنوان ژورنال:
- Arch. Math. Log.
دوره 52 شماره
صفحات -
تاریخ انتشار 2013