2011/10/07 by Sean Cox, Matteo Viale, Cox, Sean +1
Computer Science · Mathematics · #03 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1110.1584
openalex publication_date 2011/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke \citeMR1472122 when κ is a supercompact cardinal; Foreman-Magidor \citeMR1359154 when κ= ω2 in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if I is a tower of ideals which concentrates on the class GICω1 of ω1-guessing, internally club sets, then I is not presaturated (a set is ω1-guessing iff its transitive collapse has the ω1-approximation property as defined in Hamkins \citeMR2540935). This theorem, combined with work from \citeVWISP, shows that if PFA+ or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω2 which is not presaturated; moreover this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor \citeMR1359154) to exist in all models of Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω2 has similar implications for towers of ideals which concentrate on the wider class GISω1 of ω1-guessing, internally stationary sets. Finally, we show that the word "presaturated" cannot be replaced by "precipitous" in the theorems above: Martin's Maximum (which implies SRP and the Tree Property at ω2) is consistent with a precipitous tower on GICω1.