Ernest Schimmerling

### Mathematical logic seminar - March 5, 2003

 Speaker: Dr. Thomas E. Forster Fellow and Director of Studies in Mathematics Department of Pure Mathematics and Mathematical Statistics University of Cambridge Title: Models of stratified fragments of ZF : work in progress Abstract:
If $G$ is a group of permutations of $V_\omega$ it has countably many different actions as follows. For each $n < \omega$ it can move $x$ by acting on $\bigcup^n x$. A set that is fixed by everything in $G$ under the $n$th action of $G$ is said to be $n$-symmetric; if it is $n$-symmetric for all sufficiently large $n$ it is just plain symmetric. The class of hereditarily symmetric sets is a model for the stratified axioms of ZF but contains no wellordering of $V_\omega$! There is also a stratified analogue of $L$ too, but the construction is extremely fragile.