Search for: All records

The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's diamond principle implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-sigma-scattered linear orders of cardinality greater than aleph1, as given a successor cardinal kappa+, we obtain such linear orderings of cardinality kappa+ with the additional property that their square is the union of kappa-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-sigma-scattered linear orders of cardinality kappa+ exist for every infinite cardinal kappa in Gödel's constructible universe, and also (using work of Rinot [28]) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal mu of Mitchell order mu++. 

The space of normal measures on a measurable cardinal is naturally ordered by the Mitchell ordering. In the first part of this paper we show that the Mitchell ordering can be linear on a strong cardinal where the Generalised Continuum Hypothesis fails. In the second part we show that a supercompact cardinal at which the Generalised Continuum Hypothesis fails may carry a very large number of normal measures of Mitchell order zero. 

Abstract We present an alternative proof that from large cardinals, we can force the tree property at $$\kappa ^+$$ and $$\kappa ^{++}$$ simultaneously for a singular strong limit cardinal $$\kappa $$ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $$\kappa =\aleph _{\omega ^2}$$ .