Some cardinal invariants of the generalized Baire spaces

Amaya Diana Carolina Montoya
2017 unpublished
The central theme of the research in this dissertation is the well-known Cardinal invariants of the continuum. This thesis consists of two main parts which present the results obtained in joint work with (alphabetically): Jörg Brendle, Andrew Brooke-Taylor, Vera Fischer, Sy-David Friedman and Diego Mejía. The first part focuses on the generalization of the classical cardinal invariants of the continuum to the generalized Baire spaces κ^κ , when κ is a regular uncountable cardinal. First, we
more » ... ent a generalization of some of the cardinals in Cichoń's diagram to this context and some of the ZFC relationships that are provable between them. Further, we study their values in some generic extensions corresponding to < κ-support and κ-support iterations of generalized classical forcing notions. We point out the similarities and differences with the classical case and explain the limitations of the classical methods when aiming for such generalizations. Second, we study a specific model where the ultrafilter number at κ is small, 2^κ is large and in which a larger family of cardinal invariants can be decided and proven to be < 2^κ. The second part focuses exclusively on the countable case: We present a generalization of the method of matrix iterations to find models where various constellations in Cichoń's diagram can be obtained and the value of the almost disjointness number can be decided. The method allows us also to find a generic extension where seven cardinals in Cichoń's diagram can be separated.
doi:10.25365/thesis.47024 fatcat:carzzxpf3fav3mspiljb5cwxju