[Edit: as Francois Dorais points out, adding a minimum is not enough; still a partially ordered set satisfying DCC need not have infima. So what I am asking about really is different from Noetherian induction.]
After a little thought I was optimistic that there should be a version of induction partially ordered sets with a minimum element. I even thought that the right definition of inductive subset should be essentially the one given above, with (POI2) modified slightly to
Neither can I think of some small modification of (POI2') which evades this example. I still think there should be some kind of principle of induction in partially ordered sets with infima, but I don't know what to do. Can anyone state such a principle which recovers as special cases as a special casescase the principle of ordered induction and the principle of Noetherian inductionand the principle of Noetherian induction?
