Here's where I am on this right now. Generalize the 2n, 2n+1 antichain: choose an arbitrary partition of Z into blocks of 2^k consecutive values, k>0, where a block of size 2^k must start on a multiple of 2^k. (We are allowing different blocks to use different k's.) For each block, arbitrarily put either the first half or the second half into the set. This produces an anti-chain. (I'll skip the details, at least for now.) It's still not maximal, though; for example, {n : n = 0,1,2,4 (mod 8)} could be added. Franklin T. Adams-Watters -----Original Message----- From: dasimov@earthlink.net Franklin wrote: << OK. I'm still refining this problem. The problem now is to describe a maximal uncountable antichain of subsets of Z, where no component set is either finite or counter-finite (i.e., Z-S must not be finite). And yes, I chose the word describe deliberately; the existence of such a set is easy to establish
Hey, you could've said "nice example" when I solved your last question (:-)>. Anyhow, I've found as cute a little antichain as you could ask for that consists solely of subsets of Z that are each infinite and co-infinite. (I'm not sure how hard this is, but it seemed hard to me.) Without Zorn's Lemma, however, I don't yet see how to extend it to a maximal antichain. --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ___________________________________________________ Try the New Netscape Mail Today! Virtually Spam-Free | More Storage | Import Your Contact List http://mail.netscape.com