2023/08/25 by Radek Honzík, Šárka Stejskalová, Honzik, Radek +1
Computer Science · Mathematics · #03E05 #03E17 #03E35 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2308.13478
openalex publication_date 2023/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that if the existence of a supercompact cardinal κ with a weakly compact cardinal λ above κ is consistent, then the following are consistent as well (where \mathfrakt(κ) and \mathfraku(κ) are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal κ such that κ+ < \mathfrakt(κ)= \mathfraku(κ)< 2κ and SR(κ++) hold, and (ii) There is an inaccessible cardinal κ such that κ+ = \mathfrakt(κ) < \mathfraku(κ)< 2κ and SR(κ++), TP(κ++) and ¬ wKH(κ+) hold. The cardinals \mathfraku(κ) and 2κ can have any reasonable values in these models. We obtain these results by combining the forcing construction from Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results for compactness principles. Apart from \mathfraku(κ) and \mathfrakt(κ) we also compute the values of \mathfrakb(κ), \mathfrakd(κ), \mathfraks(κ), \mathfrakr(κ), \mathfraka(κ), cov(Mκ), add(Mκ), non(Mκ), cof(Mκ) which will all be equal to \mathfraku(κ). In (ii), we compute \mathfrakp(κ) = \mathfrakt(κ) = κ+ by observing that the κ+-distributive quotient of the Mitchell forcing adds a tower of size κ+. Finally, we observe that (i) and (ii) hold also for the traditional invariants on κ= ω, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property DSS(ω2), which implies the negation of the approachability property ¬ AP(ω2).