Okay, now I'm wondering if a set of S any cardinality K > aleph_0 has standard formulas for the maximum cardinality C(K) of a chain of its subsets, and A(K) of an antichain of its subsets. For finite K: * C(K) is clearly K+1. * The non-obvious formula for A(K) is given by Sperner's Theorem. There's a nice discussion and short proof of it in "Proofs from The Book", 2nd ed. (there is a 3rd). But you may enjoy thinking about it before reading about it. --Dan