## Journal of Symbolic Logic

### Forcing closed unbounded subsets of $\aleph_{\omega_{1}+1}$

M. C. Stanley

#### Abstract

Using square sequences, a stationary subset $S_T$ of $\aleph_{\omega_{1}+1}$ is constructed from a tree $T$ of height $\omega_1$, uniformly in $T$. Under suitable hypotheses, adding a closed unbounded subset to $S_T$ requires adding a cofinal branch to $T$ or collapsing at least one of $\omega_1$, $\aleph_{\omega_{1}}$, and $\aleph_{\omega_1+1}$. An application is that in ZFC there is no parameter free definition of the family of subsets of $\aleph_{\omega_1+1}$ that have a closed unbounded subset in some $\omega_1$, $\aleph_{\omega_{1}}$, and $\aleph_{\omega_1+1}$ preserving outer model.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 681-707.

Dates
First available in Project Euclid: 6 January 2014

https://projecteuclid.org/euclid.jsl/1389032271

Digital Object Identifier
doi:10.2178/jsl.7803010

Mathematical Reviews number (MathSciNet)
MR3135494

Zentralblatt MATH identifier
1348.03043

Subjects
Primary: 03E05, 03E35, 03E40, 03E45

#### Citation

Stanley, M. C. Forcing closed unbounded subsets of $\aleph_{\omega_{1}+1}$. J. Symbolic Logic 78 (2013), no. 3, 681--707. doi:10.2178/jsl.7803010. https://projecteuclid.org/euclid.jsl/1389032271

#### References

• A. Beller, R. B. Jensen, and P. Welch Coding the universe, London Mathematical Society Lecture Note Series 47, Cambridge University Press,1982.
• J. Cummings, M. Foreman, and M. Magidor Canonical structure in the universe of set theory: part two, Annals of Pure and Applied Logic, vol. 142(2006), pp. 55–75.
• K. J. Devlin Constructibility, Springer-Verlag,1984.
• S. Shelah Cardinal arithmetic, Oxford Science Publications,1994.
• M. C. Stanley Forcing closed unbounded subsets of $\aleph_{\omega+1}$, Sets and proofs (S. B. Cooper and J. K. Truss, editors), London Mathematical Society Lecture Note Series, vol. 258, Cambridge University Press,1999, pp. 365–382.
• –––– Forcing closed unbounded subsets of $\omega_{2}$, Annals of Pure and Applied Logic, vol. 110(2001), pp. 23–87.