The existence of a saturated ideal on ℵ2 and the ℵ-tree property are interesting properties in their own right, with the literature filled with results involving one or the other. However, it seems that all known techniques for producing a model with one of these two properties destroys the other property. The question of constructing a model with both properties has appeared in the literature.
In Chapter 1, we cover the technical background needed throughout the rest of the dissertation. In Chapter 2, we present a variant of the Kunen-Magidor construction and use it to construct a model that has a saturated ideal on ℵ2 and satisfies ☐ω1. In Chapter 3, we prove that ☐ω1,ω and ☐ (ω2) fail in the original Kunen-Magidor model, and we also show that the original Kunen-Magidor model has simultaneous stationary set reflection. The question of constructing a model that has a presaturated/strong ideal---instead of a saturated ideal---on ℵ2 and has the ℵ-tree property may be more approachable because it seems that some variant of the Kunen-Magidor construction may work in the presaturated/strong case. Thus, in Chapter 4, we present another variant of the Kunen-Magidor construction and use it to prove a partial result towards the goal of constructing a model that has a presaturated/strong ideal on ℵ2 and has the ℵ-tree property. We then discuss a potential strategy for using the variant to construct such a model.
|Commitee:||Foreman, Matthew, Goldbring, Isaac|
|School:||University of California, Irvine|
|Department:||Mathematics - Ph.D.|
|School Location:||United States -- California|
|Source:||DAI-B 82/6(E), Dissertation Abstracts International|
|Keywords:||Kameran, Kolahi, Kunen Magidor, Saturated ideal, Set theory, Tree property|
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