Journal of Symbolic Logic

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

M. C. Stanley

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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

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

First available in Project Euclid: 6 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

stationary set closed unbounded set tree square strong covering pattern forcing class forcing


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.

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