Morasses and the Lévy-collapse

Abstract

For several old problems in combinatorial set theory A. Hajnal and the present author [2] showed that on collapsing a sufficiently Mahlo cardinal to ω1 by the Lévy-collapse one gets a model where these problems are solved in the “counter-example” direction. The authors of [2] have speculated that the theorems of that paper should hold in L, and this, in fact, was shown for some of the results by Todorčević and Velleman [7,8]. The observation that collapsing a large cardinal to ω1 may give rise to L-like constructions is not new. As it was shown long ago by Silver and Rowbottom, there is a Kurepa-tree if a strongly inaccessible cardinal is Lévy-collapsed to ω1. In [5] it is proved that even Silver's W holds in that model. Here we show that even a quagmire exists there, but not necessarily a morass. To be more exact, we show that if κ < λ are the first two strongly inaccessible cardinals, first λ is Lévy-collapsed to κ+, and then κ is Lévy-collapsed to then there is no ω1-morass with built-in diamond in the resulting model (GCH is assumed). If λ is Mahlo, there is not even a morass.Our notations are standard. For excellent survey papers on morass-like principles and their uses in combinatorial set theory see [4,5,6].