Abstract
AbstractWe show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$ . We show that under $\kappa ^{<\kappa }> \kappa $ , club $\kappa $ -Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $ . Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.
Highlights
Many of the tree forcings on the Baire space over have various analogues for higher cardinals
Under an hypothesis on a κ-evasion number, we show that club Miller forcing collapses 2κ to the bounding number at κ
We present an equivalent version of the forcing and show that under κ a κ-version of Axiom A
Summary
♦κ can be replaced by (Dl )κ as a sufficient condition that the κ-support iteration of a tree forcing iterands with Axiom A preserves κ+ This is a weak partial answer to Kanamori’s question. (1) For a regular uncountable κ and a stationary S ⊆ κ we let (Dl )S mean the following: There is a sequence F = F : ∈ S such that F ⊆ is of cardinality < κ and for every f ∈ κκ there are stationarily many ∈ S such that f ∈ F. We assume that κ is regular and uncountable and S ⊆ κ is stationary. (1) Assume that κ is strongly inaccessible and S ⊆ κ is stationary and P is a κ-c.c. forcing notion of cardinality κ. We show that P forces that P : ∈ S has the guessing property of a (Dl )∗S -sequence. have to show that
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