科研之友
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摘要
There are several examples in the literature showing that compactness-like properties of a cardinal. cause poor behavior of some generic ultrapowers which have critical point. (Burke [1] when. is a supercompact cardinal; Foreman-Magidor [6] when k = omega 2 in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if (I) over right arrow is a tower of ideals which concentrates on the class GIC(omega 1) of omega(1)-guessing, internally club sets, then (I) over right arrow is not presaturated (a set is omega(1)-guessing iff its transitive collapse has the omega(1)-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA(+) or MM holds and there is an inaccessible cardinal, then there is a tower with critical point omega(2) which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin%26apos;s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at omega 2 has similar implications for towers of ideals which concentrate on the wider class GIS(omega 1) of omega(1)-guessing, internally stationary sets. %26lt;br%26gt;Finally, we show that the word %26quot;presaturated%26quot; cannot be replaced by %26quot;precipitous%26quot; in the theorems above: Martin%26apos;s Maximum (which implies SRP and the Tree Property at omega 2) is consistent with a precipitous tower on GIC(omega 1).
- 出版日期2013-10
