## Notre Dame Journal of Formal Logic

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

Martin Zeman

#### 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 $\square_{\aleph_{\omega},\lt \omega}$ 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

#### 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

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