Large ideals on small cardinals (Q1314645)

It is shown that under the assumption of the existence of large cardinals, it is consistent with ZFC+GCH to assume that various ``large'' ideals of sets exist. The following main theorem is proved. Assume the existence of an elementary embedding \(j\) of the universe of all sets, \(V\), into itself and let \(\kappa\) be the least ordinal which is moved by \(j,\kappa< j(\kappa)\), let \(\tau<\kappa\) and \(\kappa\leq\nu < j(\kappa)\) be regular. Assume further that \(V\) satisfies ZFC+GCH, that \(j(\kappa)\) is a Mahlo cardinal in \(V\) and that \(M:= \{j(x)\); \(x\in V\}\) is closed under all functions from \(j(\nu)\) into \(M\). Then there is a \(\tau\)- distributive generic extension \(V'\) of \(V\) such that: (1) \(V'\models \kappa= \tau^ +\), \(j(\kappa)= v^ +\), GCH. (2) In \(V'\) for every regular cardinal \(\delta\in \{\kappa\}\cup \{j(\xi)\); \(\kappa\leq \xi<\nu\}\) there is a \(\kappa\)-complete, \((\nu,\tau)\)-centered, normal fine uniform ideal \(I_ \delta\) on \(X_ \delta\), where \(X_ \kappa= [\kappa]^ \tau\) and \(X_ \delta= \{X\subseteq \delta\); \(| X|= j^{-1} (\delta)\}\) if \(\delta> \kappa\). For all \(\delta\geq \nu\), \(I_ \delta\) is \(\nu\)-layered. For all \(\delta> \kappa\), \(I_ \delta\) is weakly normal. Moreover \([X_ \delta]^{<\delta} \subseteq \delta\).