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Forcing with matrices of countable elementary submodels


Authors: Borisa Kuzeljevic and Stevo Todorcevic
Journal: Proc. Amer. Math. Soc. 145 (2017), 2211-2222
MSC (2010): Primary 03E57
DOI: https://doi.org/10.1090/proc/13133
Published electronically: January 26, 2017
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Abstract: We analyze the forcing notion $ \mathcal P$ of finite matrices whose rows consist of isomorphic countable elementary submodels of a given structure of the form $ H_{\theta }$. We show that forcing with this poset adds a Kurepa tree $ T$. Moreover, if $ \mathcal P_c$ is a suborder of $ \mathcal P$ containing only continuous matrices, then the Kurepa tree $ T$ is almost Souslin, i.e., the level set of any antichain in $ T$ is not stationary in $ \omega _1$.


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Additional Information

Borisa Kuzeljevic
Affiliation: Mathematical Institute SANU, Kneza Mihaila 36, 11001 Belgrade, Serbia
Email: borisa@mi.sanu.ac.rs

Stevo Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada, M5S 2E4 – and – Institut de Mathématiques de Jussieu, UMR 7586, 2 pl. Jussieu, Case 7012, 75251 Paris Cedex 05, France – and – Mathematical Institute SANU, Kneza Mihaila 36, 11001 Belgrade, Serbia
Email: stevo@math.toronto.edu, stevo.todorcevic@imj-prg.fr, stevo@mi.sanu.ac.rs

DOI: https://doi.org/10.1090/proc/13133
Keywords: Proper forcing, side condition, countable elementary submodel
Received by editor(s): March 28, 2015
Received by editor(s) in revised form: June 19, 2015, and September 2, 2015
Published electronically: January 26, 2017
Additional Notes: This paper was conceived during the Summer of 2014 when the first author was visiting Institut de Mathématiques de Jussieu. The first author would like to thank the Equipe de Logique of that Institute for support which made this visit possible. The first author was partially supported by the MPNTR grant ON174006
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society