Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 421–428 | Cite as

Maximal trees

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Abstract

We show that, consistently, there can be maximal subtrees of \(\mathcal{P}(\omega )\) and \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) of arbitrary regular uncountable size below the size of the continuum \({\mathfrak c}\). We also show that there are no maximal subtrees of \(\mathcal{P}(\omega ) / {\mathrm {fin}}\) with countable levels. Our results answer several questions of Campero-Arena et al. (Fund Math 234:73–89, 2016).

Keywords

Tree Maximal tree Tree number ccc forcing Iterated forcing Mathias forcing 

Mathematics Subject Classification

Primary 03E35 Secondary 03E17 03E05 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Graduate School of System InformaticsKobe UniversityKobeJapan

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