Friedman's and other Reflection Properties
Dienstag, 4.6.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In 1975, Friedman introduced the property \(F(\bkappa)\),\nstating that every subset of \(\bkappa\) either contains or is disjoint\nfrom a closed set of ordertype \(\bomega_1\). Famously, this property\nfollows from the power forcing axiom ``Martin's Maximum''. In this\ntalk, we introduce posets which force the negation of this property and\nother related notions and investigate the patterns in which these\nproperties can fail in connection to large cardinals.
Developments in Namba Forcing
Dienstag, 11.6.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
One way to study the properties of the infinite cardinals is to examine the extent to which they can be changed by forcing. In 1969 and 1970, Bukovsk{\b'y} and Namba independently showed that \(\baleph_2\) can be forced to be an ordinal of cofinality \(\baleph_0\) without collapsing \(\baleph_1\). The forcings they used and their variants are now known as Namba forcing. Shelah proved that Namba forcing collapses \(\baleph_3\) to an ordinal of cardinality \(\baleph_1\). In a 1990 paper, Bukovsky and Coplakova asked whether there can be an extension that collapses \(\baleph_2\) to an ordinal of cardinality \(\baleph_1\) without collapsing \(\baleph_3\). We will show that a slight strengthening of local precipitousness on \(\baleph_2\) due to Laver allows us to construct such an extension.\n