Joel Friedman in 1974 invented these things called Spinozistic partitionings [sic] (of a set) where the pieces are all pairwise isomorphic as binary structures (piece, \in).  He shows that $V_\omega$ has one. The idea seems to me to hold possibilities for exercises for a set theory course, but beyond that i can't see any real reason why i should care about them.  Is there some model theory angle i'm missing..?