Iterated forcing and normal ideals on \(\omega _ 1\)
Let \(D_{\omega_ 1}\) denote the stationary ideal on \(\omega_ 1\) and for and \(S\subset \omega_ 1\), let \(D_{\omega_ 1}+S\) denote the ideal generated by adding S. The main results of this paper are that if one assumes that there are enough supercompact cardinals then one can get the consistency of GCH together with there being a stationary \(S\subset \omega_ 1\) such that either \({\mathcal P}(\omega_ 1)/(D_{\omega_ 1}+S)\) is isomorphic to the completion of the Levy collapse \(Lv(\aleph_ 0,<\aleph_ 2)\) or that it has the ``Ulam property''. In both cases the ideal \(D_{\omega_ 1}+S\) is layered. In this instance the Ulam property means that there are \(\omega_ 1\)-many \(\omega_ 1\)-complete filters extending \(D_{\omega_ 1}+S\) such that each positive set belongs to at least one of the filters. The technique employed is (roughly!) to use a revised countable support iteration of semi-proper posets so as to preserve \(\omega_ 1\). A maximal antichain of \({\mathcal P}(\omega_ 1)/(D_{\omega_ 1}+S)\) (for a suitably chosen S) is trapped and its maximality is ``sealed'' by using a suitable semi-proper forcing. This technique is very recent (originating in the Foreman, Magidor and Shelah articles on Martin's Maximum) and is quite distinct from the previous methods of obtaining ideals on small cardinals with nice saturation properties - namely those which start with such an ideal on a large cardinal in the ground model and collapse this cardinal and preserve the properties of the ideal.
scientific article; zbMATH DE number 4068859