✍ Scribed by Ernest Schimmerling; Boban Velickovic
No coin nor oath required. For personal study only.
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)^+^, 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)
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