Collapsing functions

We define what it means for a function on ω 1 to be a collapsing function for λ and show that if there exists a collapsing function for (2   ω   1) + , then there is no precipitous ideal on ω 1 . We show that a collapsing function for ω 2 can be added by forcing. We define what it means to be a weakly ω 1 ‐Erdös cardinal and show that in L [ E ], there is a collapsing function for λ iff λ is less than the least weakly ω 1 ‐Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω 1 . We also show that the following statements hold in L [ E ]. The least cardinal λ with the Chang property ( λ, ω 1 ) ↠ ( ω 1 , ω ) is equal to the least ω 1 ‐Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non‐stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j ( ω 1 ) is the least ω 1 ‐Erdös cardinal. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)