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On the Distributivity of Perfect Tree Forcings for Singulars of Uncountable Cofinality

Maxwell Levine

Tuesday, 2021-11-30 14:30

Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is $(\aleph_0,\aleph_1)$-distributive but not $(\aleph_0,\aleph_2)$-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing $P_\kappa$ is $(\omega,\nu)$-distributive for $\nu<\kappa$ if $\kappa$ is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e. if $P_\kappa$ is (cf$(\kappa),\nu)$-distributive for $\nu<\kappa$ if $\kappa$ has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case $P_\kappa$ is not (cf$(\kappa),2)$-distributive.