ka (02/08/83)
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<U (U defined here) The axiom schema of comprehension says that if a set X exists, the set Y consisting of all the elements of X having a given property also exists; 2) Ap Ax Ey Az z<y <-> z<x ^ p(z) thus if the set of all sets exists, then the set of all sets which are not elements of themselves also exists. 3) Ax Ey Az x<y <-> z<x ^ ~(z<z) (from (2), p = ~(z<z)) 4) Ey Az z<y <-> z<U ^ ~(z<z) (from (3), x = U) 5) Ey Az z<y <-> ~(z<z) (from (1) and (4)) Call this set V. Is V an element of V? If so, then since V contains no sets which are elements of themselves, V is not an element of V and we have a contradiction. Similarly, if V is not an element of V, then V must be an element of V and we also have a contradiction. 6) Ey y<y <-> ~(y<y) (from (5), z = y) Since the assumption that the set of all sets exists leads to a con- tradiction, the set of all sets must not exist. QED. Because formula (2) has a property as a variable, it is a second order predicate calculus statement rather than a first order state- ment. It is identical to the formula folloing formula given by Lew Mammel (I think; my understanding of higher order predicate calculus is a bit shaky): 2a) Ey Ax (x < 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