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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
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

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.