From: utzoo!decvax!harpo!npoiv!npois!houxm!5941ux!machaids!hocda!spanky!ka Newsgroups: net.math Title: Re: I am trying to comprehend [set theory] Article-I.D.: spanky.194 Posted: Mon Feb 7 22:00:54 1983 Received: Wed Feb 9 05:21:53 1983 References: ihuxr.321 To answer the second question first, I have never seen the notation "E !y", but I assume from the context that the author means "there exists exactly one y." I intermingled the formal and English proofs of the nonexistence of the set of all set in the hopes that each version would clarify the other: Proof is by contradiction. Assume that the set of all sets exists. 1) Ax x zz z ~(z ~(y x < z ^ phi) where phi is a formula NOT INVOLVING y. The difference is that in (2a), the restriction on phi is written in English, while in (2) the restriction is enforced by allowing only the property "p", rather than the entire formula "phi", to vary. I think Lew's confusion stems from a misunderstanding of this re- striction. In (2a), phi should not depend on y; but that doesn't prevent us from later substituting y for x, as I do in (6). The reason I prefer (2) to (2a) is that the restriction is implicit in to formula, which avoids such confusion. Kenneth Almquist