# Killing Inaccessible Cardinals Softly

Set theory seminarFriday, October 25, 201310:00 am

# Killing Inaccessible Cardinals Softly

I shall introduce the killing-them-softly phenomenon among large cardinals by showing how it works for inaccessible and Mahlo cardinals. A large cardinal is killed softly whenever, by forcing, one of its large cardinal properties is destroyed while as many as possible weaker large cardinal properties, below this one, are preserved. I shall also explore the various degrees of inaccessibility and show Mahlo cardinals are $alpha$-hyper$^{beta}$-inaccessible and beyond.