Ideals on \(\omega\) which are obtained from Hausdorff-gaps (Q1192583)

Consider a Hausdorff gap in \(\omega^ \omega\), that is, a pair \[ {\mathcal G}=\bigl\langle\langle f_ \alpha: \alpha<\omega_ 1\rangle, \langle g_ \alpha: \alpha<\omega_ 1\rangle\bigr\rangle \] of sequences of functions from \(\omega\) to \(\omega\) such that \(f_ \beta<^* f_ \alpha<^* g_ \alpha<^* g_ \beta\) for all \(\beta<\alpha<\omega_ 1\) and such that there is no \(h: \omega\to\omega\) with \(f_ \alpha<^* h<^* g_ \alpha\) for all \(\alpha\). Here \(f<^* g\) means that \(\bigl\{n: g(n)\leq f(n)\bigr\}\) is finite. The gap \(\mathcal G\) determines an ideal \(I_{\mathcal G}\) on \(\omega\), namely the set of all \(x\subseteq\omega\) over which the gap is filled, that is, there is an \(h: x\to\omega\) such that \(f_ \alpha\upharpoonright x<^* h<^* g_ \alpha\upharpoonright x\) for all \(\alpha\). The author shows that 1) under CH every ideal on \(\omega\) is of the form \(I_{\mathcal G}\) for some \(\mathcal G\) and 2) if \(V\) is obtained from a model of CH by adding Cohen reals then in \(V\) an ideal is of the form \(I_{\mathcal G}\) iff it is \(\leq\omega_ 1\)-generated.