It is perhaps easier to think about this if you make it positive instead of negative: 1. a set X is infinite if there is a surjection s: X -> X u {X}, or 2. a set X is infinite if there is an injection i: X u {X} -> X. Again, you want to show that these are equivalent. I'll just note that the proof in one direction is trivial; the other way is much harder. One other note: both of these formulations assume that X is not an element of itself. The normal axioms of set theory don't require this. (There is an axiom called the Axiom of Descent which does, but proofs depending on it usually note that it is required, just as proofs using the Continuum Hypothesis do.) A formulation that avoids this problem is: 1. a set X is infinite if there is a surjection s: X -> X which is not an injection. 2. a set X is infinite if there is an injection i: X -> X which is not a surjection. Franklin T. Adams-Watters -----Original Message----- From: dasimov@earthlink.net Maybe a bit simpler is to say either 1. a set X is finite if there is no surjection s: X -> X u {X}, or 2. a set X is finite if there is no injection i: X u {X} -> X. Exercise: Show 1. <=> 2. --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