On iterated forcing for successors of regular cardinals

Todd Eisworth¹

¹ Department of Mathematics University of Northern Iowa Cedar Falls, IA 50614, U.S.A.

Fundamenta Mathematicae, Tome 179 (2003) no. 3, pp. 249-266

Résumé

We investigate the problem of when ${\leq}\lambda$-support iterations of ${}\lambda$-complete notions of forcing preserve $\lambda^+$. We isolate a property— properness over diamonds—that implies $\lambda^+$ is preserved and show that this property is preserved by $\lambda$-support iterations. Our condition is a relative of that presented by Rosłanowski and Shelah in \cite{RoSh:655}; it is not clear if the two conditions are equivalent. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{\delta\lambda^+:\mathop{\rm cf}(\delta)=\lambda\}$ that complements a theorem of Shelah \cite{Sh:f}.

Détail

DOI : 10.4064/fm179-3-4

Keywords: investigate problem leq lambda support iterations lambda complete notions forcing preserve lambda isolate property properness diamonds implies lambda preserved property preserved lambda support iterations condition relative presented ros anowski shelah cite rosh clear conditions equivalent close application technology presenting consistency result uniformizing colorings ladder systems delta lambda mathop delta lambda complements theorem shelah cite