The axiom of foundation forbids cycles in the membership relation, you are of course allowed to talk about other relations that do have cycles. For example, R={

The axiom of foundation forbids cycles in the membership relation, you are of course allowed to talk about other relations that do have cycles. For example, R={(1,2),(2,3),(3,1)} is a relation defined in ZFC that relates 1 to 2, 2 to 3 and 3 to 1. If you want to study cycles in directed graphs it never matters that you can't have membership cycles and I'm confused about why anyone would think it does.