Fragility and indestructibility of the tree property

Spencer Unger*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)635-645
Number of pages11
JournalArchive for Mathematical Logic
Issue number5-6
StatePublished - Aug 2012
Externally publishedYes


  • Forcing
  • Fragility
  • Indestructibility
  • Large cardinals
  • Tree property


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