naa a22 4500 510783791 CHVBK 20180411083255.0 cr unu---uuuuu 180411e20131001xx s 000 0 eng 10.1007/s11856-013-0004-0 doi (NATIONALLICENCE)springer-10.1007/s11856-013-0004-0 Martin's Maximum and tower forcing [Elektronische Daten] [Sean Cox, Matteo Viale] 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 [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω 2 in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $$\overrightarrow I $$ is a tower of ideals which concentrates on the class $$GI{C_{{\omega _1}}}$$ of ω 1-guessing, internally club sets, then $$\overrightarrow I $$ is not presaturated (a set is ω 1-guessing iff its transitive collapse has the ω 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 ω 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's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω 2 has similar implications for towers of ideals which concentrate on the wider class $$GI{C_{{\omega _1}}}$$ of ω 1-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 ω 2) is consistent with a precipitous tower on $$GI{C_{{\omega _1}}}$$ . Hebrew University Magnes Press, 2013 Cox Sean Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany aut Viale Matteo Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany aut Israel Journal of Mathematics Springer US; http://www.springer-ny.com 197/1(2013-10-01), 347-376 0021-2172 197:1<347 2013 197 11856 https://doi.org/10.1007/s11856-013-0004-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11856-013-0004-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Cox Sean Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany aut NATIONALLICENCE 700 1- Viale Matteo Institut für mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstrasse 62, 48149, Münster, Germany aut NATIONALLICENCE 773 0- Israel Journal of Mathematics Springer US; http://www.springer-ny.com 197/1(2013-10-01), 347-376 0021-2172 197:1<347 2013 197 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer