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Proceedings of the American Mathematical Society

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Forcing with copies of countable ordinals


Author: Miloš S. Kurilić
Journal: Proc. Amer. Math. Soc. 143 (2015), 1771-1784
MSC (2010): Primary 03E40, 03E10, 03C15; Secondary 03E35, 03E17, 06A06
DOI: https://doi.org/10.1090/S0002-9939-2014-12360-4
Published electronically: December 4, 2014
MathSciNet review: 3314089
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Abstract: Let $\alpha$ be a countable ordinal and $\mathbb {P}(\alpha )$ the collection of its subsets isomorphic to $\alpha$. We show that the separative quotient of the poset $\langle \mathbb {P}(\alpha ), \subset \rangle$ is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form $P(\omega ^\gamma )/\mathcal {I}_{\omega ^\gamma }$, where $\gamma \in \mathrm {Lim}\cup \{ 1 \}$ and $\mathcal {I}_{\omega ^\gamma }$ is the corresponding ordinal ideal. Moreover, the poset $\langle \mathbb {P} (\alpha ), \subset \rangle$ is forcing equivalent to a two-step iteration of the form $(P(\omega )/\mathrm {Fin})^+ \ast \pi$, where $[\omega ] \Vdash$ “$\pi$ is an $\omega _1$-closed separative pre-order” and, if $\mathfrak {h}=\omega _1$, to $(P(\omega )/\mathrm {Fin})^+$. Also we analyze the quotients over ordinal ideals $P(\omega ^\delta )/\mathcal {I}_{\omega ^\delta }$ and the corresponding cardinal invariants $\mathfrak {h}_{\omega ^\delta }$ and $\mathfrak {t}_{\omega ^\delta }$.


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Additional Information

Miloš S. Kurilić
Affiliation: Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
Email: milos@dmi.uns.ac.rs

Received by editor(s): April 29, 2013
Received by editor(s) in revised form: September 6, 2013
Published electronically: December 4, 2014
Additional Notes: This research was supported by the Ministry of Education and Science of the Republic of Serbia (Project 174006).
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.