Fragility and indestructibility of the tree property

Spencer Unger

doi:10.1007/s00153-012-0287-6

Fragility and indestructibility of the tree property

Spencer Unger^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove various theorems about the preservation and destruction of the tree property at ω ₂. Working in a model of Mitchell [9] where the tree property holds at ω ₂, we prove that ω ₂ still has the tree property after ccc forcing of size א ₁ or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that the tree property at ω ₂ can be indestructible under ω ₂-directed closed forcing.

Original language	English
Pages (from-to)	635-645
Number of pages	11
Journal	Archive for Mathematical Logic
Volume	51
Issue number	5-6
DOIs	https://doi.org/10.1007/s00153-012-0287-6
State	Published - Aug 2012
Externally published	Yes

Keywords

Forcing
Fragility
Indestructibility
Large cardinals
Tree property

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10.1007/s00153-012-0287-6

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AB - We prove various theorems about the preservation and destruction of the tree property at ω 2. Working in a model of Mitchell [9] where the tree property holds at ω 2, we prove that ω 2 still has the tree property after ccc forcing of size א 1 or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that the tree property at ω 2 can be indestructible under ω 2-directed closed forcing.

KW - Forcing

KW - Fragility

KW - Indestructibility

KW - Large cardinals

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Fragility and indestructibility of the tree property

Abstract

Keywords

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