THE EIGHTFOLD WAY

@article{Cummings2018THEEW,
  title={THE EIGHTFOLD WAY},
  author={James Cummings and Sy-David Friedman and Menachem Magidor and Assaf Rinot},
  journal={The Journal of Symbolic Logic},
  year={2018},
  volume={83},
  pages={349 - 371}
}
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 above κ. If in addition κ is supercompact then we can force κ to be ${\aleph _\omega }$ in the extension. The proofs… 
View on Cambridge Press
homepages.math.uic.edu
32 Citations
Highly Influential Citations
4
Background Citations
11
Methods Citations
3
Results Citations
1

32 Citations

Citation Type

Citation Type

Small $$\mathfrak {u}(\kappa )$$ u ( κ ) at singular
We show that the tree property, stationary reflection and the failure of approachability at $$\kappa ^{++}$$ κ + + are consistent with $$\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa $$ u ( κ ) = κ +…
Tree property at all regular even cardinals
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $\delta$, $\delta$ is strong limit,…
INDESTRUCTIBILITY OF THE TREE PROPERTY
TLDR
The tree property at theTree property at V, a generic extension of V, is indestructible under all forcing notions living in V which are “closed and “liftable” in a prescribed sense (such as ${\kappa + + }}$-directed closed forcings or well-met forcings which are -closed with the greatest lower bounds).
SOME LIFTING ARGUMENTS FOR LARGE AND SMALL CARDINALS
Suppose M and N are transitive models of set theory, P is a forcing notion in M and G is P-generic over M . We say that an elementary embedding j : (M,∈) → (N,∈) lifts to M [G] if there is j : (M…
A note on the eightfold way
Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $\omega_2$-Aronszajn tree, the $\omega_1$-approachability property fails, and every stationary subset of…
TREES AND STATIONARY REFLECTION AT DOUBLE SUCCESSORS OF REGULAR CARDINALS
We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals κ, updating some classical constructions in the process. This includes…
TERNARY GENERALIZATIONS OF GRADED ALGEBRAS WITH SOME PHYSICAL APPLICATIONS
We discuss cubic and ternary algebras which are a direct generalization of Grassmann and Cli ord algebras, but with Z3-grading replacing the usual Z2-grading. Elementary properties and structures of…
SMALL u(κ) AT SINGULAR κ WITH COMPACTNESS AT κ++
We show that the tree property, stationary reflection and the failure of approachability at κ are consistent with u(κ) = κ < 2, where κ is a singular strong limit cardinal with the countable or…
Spinors in $\mathbb{K}$-Hilbert Spaces
We consider a structure of the K -Hilbert space, where K ≃ R is a field of real numbers, K ≃ C is a field of complex numbers, K ≃ H is a quaternion algebra, within the framework of division rings of…
Particular Symmetries: Group Theory of the Periodic System
To this day, a hundred and fifty years after Mendeleev’s discovery, the overal structure of the periodic system remains unaccounted for in quantum-mechanical terms. Given this dire situation, a…
...
...

References

SHOWING 1-10 OF 40 REFERENCES
The tree property at successors of singular cardinals
TLDR
It is shown that if $\ lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.
A relative of the approachability ideal, diamond and non-saturation
  • Assaf Rinot
  • Mathematics
    The Journal of Symbolic Logic
  • 2010
TLDR
It is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.
The consistency strength of the tree property at the double successor of a measurable cardina
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ; (2) κ is a weakly compact hypermeasurable cardinal. From the…
Extender Based Forcings
TLDR
It is shown that it is consistent to have “ c f κ = ℵ 0 . GCH below κ, 2 κ > κ + , and ” and answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.
Reflecting stationary sets and successors of singular cardinals
TLDR
It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets, and that under suitable assumptions it is consistent that REF and there is a κ which is κ+n-supercompact.
THE TREE PROPERTY AT ${\aleph _{{\omega ^2} + 1}}$ AND ${\aleph _{{\omega ^2} + 2}}$
TLDR
It is shown that from large cardinals it is consistent to have the tree property simultaneously at ${\aleph _{{\omega ^2} + 1}} and ${\ aleph _ {{\omegas ^2}}$ with strong limit.
Prikry-Type Forcings
One of the central topics of set theory since Cantor has been the study of the power function κ→2 κ . The basic problem is to determine all the possible values of 2 κ for a cardinal κ. Paul Cohen…
A new class of order types
Higher Souslin trees and the GCH, revisited
Successors of Singular Cardinals
Successors of singular cardinals are a peculiar—although they are successor cardinals, they can still exhibit some of the behaviors typically associated with large cardinals. In this chapter, we…
...
...