# Strong tree properties for two successive cardinals

@article{Fontanella2012StrongTP, title={Strong tree properties for two successive cardinals}, author={Laura Fontanella}, journal={Archive for Mathematical Logic}, year={2012}, volume={51}, pages={601-620} }

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously $${(\aleph_2, \mu)}$$ -ITP and $${(\aleph_3, \mu')}$$ -ITP hold, for all $${\mu\geq \aleph_2}$$ and $${\mu'\geq \aleph_3}$$ .

## 8 Citations

### Strong tree properties for small cardinals

- Mathematics, PhilosophyThe Journal of Symbolic Logic
- 2013

It is proved that if there is a model of ZFC with infinitely many supercompact cardinals, then there is one where for every n ≥ 2 and μ ≥ ℕn, there is (ℕ n, μ)-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.

### 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…

### Strong Tree Properties at Successors of Singular Cardinals

- Mathematics
- 2012

We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.

### Guessing models and the approachability ideal

- MathematicsJ. Math. Log.
- 2021

This paper produces a generic extension of the universe in which the principles of supercompact cardinals and ISP hold simultaneously, and the restriction of the approachability ideal $I[\omega_2]$ to the set of ordinals of cofinality $\omega-1$ is the non stationary ideal on this set.

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