## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 58, Number 3 (2017), 409-432.

### Two Upper Bounds on Consistency Strength of $\neg {\square}_{{\aleph}_{\omega}}$ and Stationary Set Reflection at Two Successive ${\aleph}_{n}$

#### Abstract

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a ${\kappa}^{+}$-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ${\aleph}_{\omega}$ and make the principle ${\square}_{{\aleph}_{\omega},<\omega}$ fail in the generic extension. We also show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ${\aleph}_{2}$ and arrange in the generic extension that simultaneous reflection holds at ${\aleph}_{2}$, and at the same time, every stationary subset of ${\aleph}_{3}$ concentrating on points of cofinality $\omega $ has a reflection point of cofinality ${\omega}_{1}$.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 58, Number 3 (2017), 409-432.

**Dates**

Received: 30 July 2012

Accepted: 31 December 2014

First available in Project Euclid: 1 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1491012044

**Digital Object Identifier**

doi:10.1215/00294527-2017-0005

**Mathematical Reviews number (MathSciNet)**

MR3681102

**Zentralblatt MATH identifier**

06761616

**Subjects**

Primary: 03E05: Other combinatorial set theory

Secondary: 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals

**Keywords**

subcompact cardinal square sequence stationary set reflection modified Prikry forcing iterated club shooting

#### Citation

Zeman, Martin. Two Upper Bounds on Consistency Strength of $\neg\square_{\aleph_{\omega}}$ and Stationary Set Reflection at Two Successive $\aleph_{n}$. Notre Dame J. Formal Logic 58 (2017), no. 3, 409--432. doi:10.1215/00294527-2017-0005. https://projecteuclid.org/euclid.ndjfl/1491012044