From: utzoo!decvax!harpo!ihnp4!ihuxr!lew
Newsgroups: net.math
Title: I am trying to comprehend
Article-I.D.: ihuxr.321
Posted: Mon Feb  7 14:13:21 1983
Received: Wed Feb  9 02:02:07 1983
Reply-To: lew@ihuxr.UUCP (Lew Mammel, Jr.)

First, let me apologize for my horrible mangling of the Continuum Hypothesis
in a recent posting. Partly as a result of this, I sought to gain an
elementary understanding of Axiomatic Set Theory. I am stuck on the
"Axiom of Comprehension", and in particular, its relation to Russell's Paradox.

Using "A" for "for all", "E" for "there exists", "<" for "is an element of",
"^" for "and", and "<->" for "if and only if", Kunen states this axiom
in "Set Theory, an Introduction to Independence Proofs" as the
universal closure of:

Ey Ax (x < y <-> x < z ^ phi) where phi is a formula NOT INVOLVING y.

(Emphasis mine) Kunen then proceeds to use Russell's paradox to prove
the nonexistence of the universal set. He first lets phi = x ~< x
and then resorts to english description. I can't complete the proof
without using y in phi.

To complete my confusion, Monk in "Intro. to Set Theory" uses this
axiom to prove the non-existence of "the set of all non-self-members"
and then uses the same axiom to define the universal set! Monk uses
"x is a set" in place of "x < z" in the axiom.

Who out there comprehends Comprehension?

As a footnote, Kunen uses an undefined notation in the "Replacement
Scheme" axiom.  He doesn't define "!" anywhere, but states the axiom:

A (x < B) E !y phi(x,y) -> E Y A (x < B) E (y < Y) phi(x,y)

Can anybody explain this?

	Lew Mammel, Jr. ihuxr!lew