# Subtle and Ineffable Tree Properties

@inproceedings{Wei2010SubtleAI, title={Subtle and Ineffable Tree Properties}, author={Christoph Wei{\ss}}, year={2010} }

In the style of the tree property, we give combinatorial principles that capture the concepts of the so-called subtle and ineffable cardinals in such a way that they are also applicable to small cardinals. Building upon these principles we then develop a further one that even achieves this for supercompactness.
We show the consistency of these principles starting from the corresponding large cardinals. Furthermore we show the equiconsistency for subtle and ineffable. For supercompactness…

## 36 Citations

### The ineffable tree property and failure of the singular cardinals hypothesis

- Mathematics
- 2020

A long standing project in set theory is to analyze how much compactness can be obtained in the universe. Compactness is the phenomenon where if a certain property holds for all small substructures…

### The super tree property at the successor of a singular

- MathematicsIsrael Journal of Mathematics
- 2020

For an inaccessible cardinal κ , the super tree property (ITP) at κ holds if and only if κ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP…

### THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CARDINALS

- MathematicsThe Journal of Symbolic Logic
- 2014

It is proved that if there is a model of ZFC with infinitely many supercompact cardinals, then there is the strong tree property of Z FC where ${\aleph _{\omega + 1}}$ has theStrong tree property.

### Characterizing large cardinals through Neeman's pure side condition forcing

- MathematicsFundamenta Mathematicae
- 2021

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side…

### Small embedding characterizations for large cardinals

- MathematicsAnn. Pure Appl. Log.
- 2019

### The Strong and Super Tree Property at Successors of Singular Cardinals

- Mathematics
- 2022

. The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That…

### Split Principles, large cardinals, splitting families and split ideals

- Mathematics
- 2017

We introduce split principles and show that their negations provide simple combinatorial characterizations of large cardinal properties. As examples, we show how inaccessiblity, weak compactness,…

### MRP, tree properties and square principles

- MathematicsThe Journal of Symbolic Logic
- 2011

It is shown that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2.

## References

SHOWING 1-10 OF 22 REFERENCES

### The tree property at successors of singular cardinals

- MathematicsArch. Math. Log.
- 1996

It is shown that if $\ lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.

### Hierarchies of forcing axioms I

- MathematicsJournal of Symbolic Logic
- 2008

The results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subCompactness and Γ-indescribability.

### Forcing indestructibility of set-theoretic axioms

- EconomicsJournal of Symbolic Logic
- 2007

It is shown in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to ℵ1.

### Reflecting stationary sets and successors of singular cardinals

- MathematicsArch. Math. Log.
- 1991

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.

### Combinatorial characterization of supercompact cardinals

- Mathematics
- 1974

It is proved that supercompact cardinals can be characterized by combinatorial properties which are generalizations of ineffability. 0. Introduction. A A B is the symmetric difference of A and B.…

### Combinatorial Principle in the Core Model for one Woodin Cardinal

- MathematicsAnn. Pure Appl. Log.
- 1995

### Combinatorial principles in the core model for one Woodin

- Mathematics
- 1995

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form L[B]. We generalize to L[E] some combinatorial principles…

### Fragments of Martin's Maximum in generic extensions

- Chemistry, MathematicsMath. Log. Q.
- 2004

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω1 + 1)‐game‐closed forcings. PFA can be destroyed by a strongly (ω1 + 1)‐game‐closed…

### A general Mitchell style iteration

- MathematicsMath. Log. Q.
- 2008

We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω2. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

### The proper forcing axiom and the singular cardinal hypothesis

- PhilosophyJournal of Symbolic Logic
- 2006

Abstract We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].