Collapsing functions

摘要

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~(ω1))~+, 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-Erdos cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω_1-Erdos 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 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-Erdps 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-Erdos cardinal.