2019/02/11 by Brent Cody, Cody, Brent, Victoria Gitman +4
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #math.LO #msc:03E35 #msc:03E55
paper · pdf · doi:10.48550/arxiv.1902.04146
Changed title and added citations to Brickhill-Welch
arxiv created 2020/01/29 · arxiv updated 2020/01/31
Hellsten \citeMR2026390 proved that when κ is Π1n-indescribable, the n-club subsets of κ provide a filter base for the Π1n-indescribability ideal, and hence can also be used to give a characterization of Π1n-indescribable sets which resembles the definition of stationarity: a set S⊆κ is Π1n-indescribable if and only if S∩ C≠∅ for every n-club C⊆κ. By replacing clubs with n-clubs in the definition of \Box(κ), one obtains a \Box(κ)-like principle \Boxn(κ), a version of which was first considered by Brickhill and Welch \citeBrickhillWelch. The principle \Boxn(κ) is consistent with the Π1n-indescribability of κ but inconsistent with the Π1n+1-indescribability of κ. By generalizing the standard forcing to add a \Box(κ)-sequence, we show that if κ is κ+-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ+-weakly compact and \Box1(κ) holds. If κ is Π12-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ+-weakly compact, \Box1(κ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π12-indescribable cardinal, it is consistent that κ is κ+-weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S0 and S1 of κ such that there is no β<κ for which both S0∩β and S1∩β are weakly compact.