Abstract: In this paper we investigate towers of normal filters. These towers were first used by Woodin. Woodin proved that if $\delta$ is a Woodin cardinal and P is the full stationary tower up to $\delta$ or the countable version Q then the generic ultrapower is closed under $<\delta$ sequences (so the generic ultrapower is well-founded). We show that if R is a tower of height $\delta$, $\delta$ supercompact, and the filters generating R are the club filter restricted to a stationary set, then R is precipitous. We also give some examples of non-precipitous towers. We also show that every normal filter can be extended to a V-ultrafilter with well-founded ultrapower in some generic extension of V (assuming large cardinals). Similarly for any tower of inaccessible height. This is accomplished by showing that there is a stationary set that projects to the filter or the tower and then forcing with P below this stationary set.
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