Simplicity of the automorphism group of fields with operators
Dienstag, 8.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly applies to the theory of difference closed fields, which is simple and hence has possibly no saturated models in their uncountable cardinality. \n \n
Failure of GCH on a Measurable Cardinal
Dienstag, 15.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Let GCH hold in \(V\), and let \(\bkappa\) be a cardinal with a definable elementary embedding \(j:V\brightarrow M\) such that \({\brm crit}(j)=\bkappa\), \({}^{\bkappa}M\bsubseteq M\) and \(\bkappa^{++}=(\bkappa^{++})^{M}\) (in particular, \(\bkappa\) is measurable). H. Woodin proved that there is a cofinality preserving generic extension in which \(\bkappa\) stays measurable and GCH fails on it. This is achieved by using an Easton support iteration of Cohen forcings for having \(2^{\balpha}=\balpha^{++}\) for every inaccessible \(\balpha\bleq\bkappa\), and then adding an additional forcing to ensure the elementary embedding extends to the generic extension. Y. Ben Shalom proved in his thesis that this last forcing is unnecessary for the construction, and further extended the result to get \(2^{\bkappa}=\bkappa^{+\bgamma}\) assuming \(\bkappa^{+\bgamma}=(\bkappa^{+\bgamma})^{M}\), for any successor ordinal \(1<\bgamma<\bkappa\). We will present these results in some detail, and further extend the result of Ben Shalom for \(\bgamma=\bkappa+1\) assuming \(\bkappa^{+\bkappa+1}=(\bkappa^{+\bkappa+1})^{M}\).
Predicting with Diamond Sequences and with Ostaszewski Club Sequences
Dienstag, 22.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
An Ostaszewski club sequence is a weakening of Jensen's diamond.\nIn contrast to the diamond, the club does not imply the continuum hypothesis.\nNumerous questions about the club stay open, and we know only few models in which\nthere is just a club sequence but no diamond sequence. In recent joint\nwork with Shelah we found that a winning strategy for the completeness player\nin a bounding game on a forcing order does not suffice to establish the club\nin the extension.