A version of kappa -Miller forcing

Heike Mildenberger,Saharon Shelah

doi:10.1007/s00153-020-00721-y

Abstract

We consider a version of kappa -Miller forcing on an uncountable cardinal kappa . We show that under 2^{<kappa } = kappa this forcing collapses 2^kappa to omega and adds a kappa -Cohen real. The same holds under the weaker assumptions that {{,mathrm{cf},}}(kappa ) > omega , 2^{2^{<kappa }}= 2^kappa , and forcing with ([kappa ]^kappa , subseteq ) collapses 2^kappa to omega .

Highlights

Many of the tree forcings on the classical Baire space have various analogues for higher cardinals
A tree is superperfect if each node has an extension that has infinitely many immediate tree successors
We can assume that each node has just one direct successor or infinitely many

Summary

Introduction

Many of the tree forcings on the classical Baire space have various analogues for higher cardinals. For a κ-version of Miller forcing, superperfectness and splitting are usually interpreted as follows: Above each node t ∈ p ⊆ κκ that are κ superperfect: for each s ∈ T there is s t such that t is a κ-splitting node of T. ΚQ>1κκh. aWs easwariwteePakestφeilfemtheenwt 1eaQk1κe=st κ, and Q2κ has as a condition forces φ

Results about Q1Ä

Transfer to Q2Ä

Ä-Cohen reals and the Levy collapse