Abstract
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 <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 language | English |
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Pages (from-to) | 349-371 |
Number of pages | 23 |
Journal | Journal of Symbolic Logic |
Volume | 83 |
Issue number | 1 |
DOIs | |
State | Published - 1 Mar 2018 |
Bibliographical note
Publisher Copyright:© 2018 The Association for Symbolic Logic.
Keywords
- approachability
- Aronszajn tree
- Mitchell forcing
- Prikry forcing
- square
- stationary reflection
- tree property