> Assume (a,b) not in S and let let c be the smallest c in N for which (c,b) not > > in S. Next let d be the smallest d in N for which (c,d) not in S. > > > > Case c = 1. Then (1,d) not in S and 1 < d. Thus (1,d-1) in S and > > by hypthesis, (1,d) = (1, (d-1)+1) in S, a contradiction. > > > > Case 1 < c. Then (c-1,d) in S. Thus by hypothesis, (c,d) = ((c-1)+1,d) in S > > a contradiction. > > > > Exercise. State and prove the base and induction steps > > of n-fold induction. Can there be an aleph_0-fold induction?