Canonical models for \(\aleph_1\)-combinatorics (Q1302295)

The authors consider statements which are \(\Sigma_2\) in the structure \(\langle H_{\aleph_2},{\in},{\mathcal I}\rangle\), where \(H_{\aleph_2}\) is the collection of sets of hereditary cardinality \(\aleph_1\) and \(\mathcal I\) is a predicate for the non-stationary subsets of \(\omega_1\). They remark that such statements tend to assert that the combinatorics of \(\aleph_1\) is complex. Therefore, given a sentence \(\phi\) about sets, it is interesting to to look for models where \(\phi\) and as few as possible \(\Sigma_2\) statements hold, in order to see the effect of \(\phi\) on the combinatorics of \(\aleph_1\). For a sentence \(\phi\), following the \(P_{\max}\) method developed in Woodin's book [\textit{W. H. Woodin}, The axiom of determinacy, forcing axioms, and the nonstationary ideal (de Gruyter, Berlin) (1999)], the authors seek a \(\sigma\)-closed forcing \(P_{\phi}\) definable in \(L({\mathbb R})\) with properties based on the above idea. (Large cardinals are necessary.) When this exists, the sentence \(\phi\) is said to be \(\Pi_2\)-compact. Typical examples of sentences \(\phi\) shown to be \(\Pi_2\)-compact are: ``dominating number\({} =\aleph_1\)'', ``cofinality of the meager ideal\({}=\aleph_1\)'', ``bounding number\({}=\aleph_1\)'' and the existence of various types of Souslin trees.