Some combinatorial principles defined in terms of elementary submodels

Sakaé Fuchino ¹   ; Stefan Geschke ²

¹ Department of Natural Science and Mathematics College of Engineering Chubu University Kasugai, Aichi 487-8501, Japan
² II. Mathematisches Institut Freie Universität Berlin Arnimallee 3 14195 Berlin, Germany

Fundamenta Mathematicae, Tome 181 (2004) no. 3, pp. 233-255

Résumé

We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese–Nation property of ${\mathcal P}(\omega )$ studied in [6] already follow from SEP. We show that it is consistent that SEP holds while ${\mathcal P}(\omega )$ fails to have the $(\aleph _1,\aleph _0)$-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle, answering a question of Blass.

Détail

DOI : 10.4064/fm181-3-3

Keywords: equivalent simpler formulation axiom sep which introduced order capture combinatorial behaviour models set theory obtained adding cohen reals model formulation shows many consequences weak freese nation property mathcal omega studied already follow sep consistent sep holds while mathcal omega fails have aleph aleph ideal property introduced answers question addressed independently fuchino kunen consider natural variants sep certain changes definition sep lead different principle answering question blass