Some combinatorial principles defined in terms of elementary submodels
Volume 181 / 2004
Abstract
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.
