Some remarks on Namba-type forcings
For the purpose of this abstract, let “a Namba-type forcing” be any forcing that forces ω2 to get cofinality ω and doesn’t collapse ω1.
It is well known that the existence of a semiproper Namba-type forcing is equivalent to a Strong Chang’s Conjecture, but that instead the existence of a stationary set preserving Namba-type forcing is provable in plain ZFC. However, in the context of questions on iterated-forcing-using-side-conditions, it is natural to ask whether one can demand more than mere stationary set preservation and get provably in ZFC a Namba-type forcing that allows many (but not necessarily club many) models for which there exist sufficient semi-generic conditions.
In this talk I will discuss a “side-condition version” ℙ of Namba forcing and explain that there exists a very natural projective stationary family of countable elementary submodels of Hθ such that ℙ is semiproper with respect to these models. In fact, we can consider a notion of strong semiproperness, in analogy to the notion of strong properness and verify that ℙ satisfies it, again with respect to these distinguished models.
As an application of this approach towards Namba forcing, we discuss a particularly natural presentation of an “ersatz iterated Namba forcing” which, given an increasing sequence (κα: α < γ) of regular cardinals ≥ ω2, adds for every α < γ a countable cofinal subset of κα, while at the same time preserving stationarity of stationary subsets of ω1. In the proof, we will make strong use of a technique involving labelled trees and games played on such trees that appears in [1].
Finally, we will mention closely related ongoing work and remaining questions.
This talk is based on joint work with my thesis supervisor Boban Veličković.
References
- [1]
- Matthew Foreman and Menachem Magidor, Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on Pκ(λ), Acta Mathematica, vol. 186 (2001), no. 2, pp. 271–300.
- [2]
- Saharon Shelah, Proper and Improper Forcing, Perspectives in Logic, vol. 5, Cambridge University Press, 1998.
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