The consistency of a club-guessing failure at the successor of a regular cardinal

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Abstract

I answer a question of Shelah by showing that if $\k$ is a regular cardinal such that $2^{{<}\k}=\k$, then there is a ${<}\k$--closed partial order preserving cofinalities and forcing that for every club--sequence $\la C_\d\mid \d\in \k^+\cap\cf(\k)\ra$ with $\ot(C_\d)=\k$ for all $\d$ there is a club $D\sub\k^+$ such that $\{\a<\k\mid \{C_\d(\a+1), C_\d(\a+2)\}\sub D\}$ is bounded for every $\d$. This forcing is built as an iteration with ${<}\k$--supports and with symmetric systems of submodels as side conditions.
Original languageEnglish
Title of host publicationInfinity, computability, and metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch
EditorsStefan Geschke, Benedikt Loewe, Philipp Schlicht
Place of PublicationLondon
PublisherCollege Publications
Pages5-27
Number of pages23
VolumeTributes 23
ISBN (Electronic)978-1848901308
ISBN (Print)1848901305
Publication statusPublished - 2014

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