Lietz, A. T. (2024, May 31). Characterizing sequences generic for variants of Namba forcing [Conference Presentation]. 2nd Wroclaw Logic Conference, Wrocław, Poland.
There are many different variations of Namba forcing, for example the standard formulation Nm consisting of ω2-perfect trees and the variation
Nm′ ⊆ Nm of those ω2-perfect tree which split everywhere above their stem. Magidor-Shelah have shown, assuming CH, that Nm is essentially different from Nm′. Jensen proved an even stronger theorem in which he differentiates Nm and Nm′ further from the variant of Nm′ consisting of those trees in Nm′ all of whose nodes above the stem have stationarily many immediate successors. Jensen also assumed CH. We generalize these theorems by removing the CH assumption and taking into account many more variations of Namba forcing. Further, we show that all “natural” variations of Namba forcing generate extensions which are minimal conditioned on cof(ωV 2 ) = ω and moreover, we analyze exactly which and how many other sequences in such an extension are generic for a variation of Namba forcing. This answers a question of Gunter Fuchs.