The eightfold way

James Cummings, Sy David Friedman, Menachem Magidor, Assaf Rinot, Dima Sinapova

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at κ++ assuming that κ = κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be Nωin the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a κ++-Souslin tree, variants of Mitchell's forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into Nω.

Original languageEnglish
Pages (from-to)349-371
Number of pages23
JournalJournal of Symbolic Logic
Issue number1
StatePublished - 1 Mar 2018

Bibliographical note

Publisher Copyright:
© 2018 The Association for Symbolic Logic.


  • Aronszajn tree
  • Mitchell forcing
  • Prikry forcing
  • approachability
  • square
  • stationary reflection
  • tree property


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