Precipitous Towers of Normal Filters

We prove that every tower of normal filters of height δ (δ 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). In this paper we investigate towers of normal filters. These towers were first used by Woodin in [W88]. Woodin proved that if δ is a Woodin cardinal and P is the full stationary tower up to δ (P<δ) or the countable version (Q<δ) then the generic ultrapower is closed under < δ sequences (so the generic ultrapower is well-founded). We show that if P is a tower of height δ, δ supercompact, and the filters generating P are the club filter restricted to a stationary set, then P is precipitous. We give an example (assuming large cardinals) of a non-precipitous tower. 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<δ below this stationary set. An important idea in our proof of precipitousness (Theorem 6.4) has the following form in Woodin’s proof. If A〉 ⊆ P<δ are maximal antichains (i ∈ ω and δ Woodin) then there is a κ < δ such that {a ≺ Vκ+1 | |a| < κ & (∀i ∈ ω) ∃b ≺ Vκ+1 such that 1) a ⊆ b, b end extends a ∩ Vκ 2) ∃x ∈ A〉 ∩ Vκ ∩ ⌊ (⌊ ∩ ∪§ ∈ §) 