Precipitous Towers of Normal Filters

Abstract

We prove that every tower of normal filters of height $\gd$ ($\gd$ supercompact) is precipitous assuming that each normal filter in the tower is the club filter restricted to a stationary set. We give an example to show that this assumption is necessary. We also prove that every normal filter can be generically extended to a well-founded $V$-ultrafilter (assuming large cardinals).