The Eightfold Way

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${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$.

A note on the eightfold way

“…AndṪ i is a nice A i -name for a subset of ω 2 ∩ cof(ω). 1 Suppose that M is a transitive model of ZFC − which is closed under ω 1 -sequences. Then if M models that A i ,Ṫ j : i ≤ α, j < α is a suitable iteration, then in fact it is.…”

“…A solution to the problem of [1] addressed in this article was originally discovered by the first author, using a mixed support forcing iteration similar to the forcings appearing in [1] and [2]. Later, the second author found a different proof using the idea of a disjoint stationary sequence.…”

“…This is a natural assumption to make, since the ω 2 -tree property implies that ω 2 is weakly compact in L. On the other hand, Harrington and Shelah [4] proved that stationary reflection at ω 2 is equiconsistent with the existence of a Mahlo cardinal. Cummings et al [1] asked whether a Mahlo cardinal is sufficient to prove the consistency of the existence of an ω 2 -Aronszajn tree, the failure of the ω 1 -approachability property, and stationary reflection at ω 2 . In this article we answer this question in the affirmative.…”

“…We begin by reviewing the relevant definitions and facts. We refer the reader to [1] for a more detailed discussion of these ideas and their history. A stationary set S ⊆ ω 2 ∩ cof(ω) is said to reflect at an ordinal β ∈ ω 2 ∩ cof(ω 1 ) if S ∩ β is a stationary subset of β.…”

Indestructibility of the Tree Property

“…It is known that if T is a tree of height with cf( ) = κ + (with no limit on the size of the levels of T ), then a forcing P which is κ + -square-cc 6 does not add a new cofinal branch to T (see for instance [24]).…”