On Dec 17, 2010, at 10:37 AM, John F. Sowa wrote:
There are many different versions of set theory, but the most
common ones assume the following axiom or something equivalent:
JFS:
For any monadic predicate P, there exists a set of all x
such that P(x) is true.
Some logicians use the word 'class' for set-like things that don't satisfy this axiom.
Well, then that's just to say that most common set theories do NOT have the above axiom, usually called the Naive Comprehension schema. That axiom (a schema, really), is exactly the source of paradox in so-called naive set theory. Modern set theories like ZF modify the schema so that one can no longer conjure sets ex nihilo from any predicate expressible in the language of set theory. Alternatively, some theories do as you suggest and distinguish between sets and (proper) classes. In VNBG, for example, the above is modified to say that, for any predicate P, there is a CLASS of all SETs s such that P(x) is true. There is, therefore, in VNBG, a CLASS R of all non-self-membered SETs. Since it is provable that R is not itself a set, it follows that R ∉ R, but we cannot infer from that that R ∈ R, since R is a class of SETs.
MW:
Yes. I'm happy to drop that in favour of a set exists if its members
exist and can be enumerated (explicitly or implicitly for infinite sets).
That raises a question about how you enumerate the elements. ZF set
theory has a systematic layering that requires each set to be composed
of elements from the layers beneath it. With that definition, nothing
called a "ZF set" can have itself as a member. Therefore, ZF set theory
can retain the above axiom.
No, it can't. The reason it fails is that there are predicates (e.g., x ∉ x) that are true of sets in arbitrarily high "layers" and, hence, there is no layer in which some set contains them all. What we get in ZF instead of the schema above is the schema of separation, roughly:
For any set A and monadic predicate P, there exists a set of all x∈A such that P(x) is true.
Thus, this schema requires a set to begin with, from which we can separate out the elements that satisfy P(x). It is the problematic capacity of Naive Comprehension for creatio ex nihilo with regard to sets that yields Russell's paradox.
JFS:
And by the way, Common Logic manages to avoid contradictions
by a method similar to what I just described: the examples
that create paradoxes in some versions of logic just produce
CL statements in which the paradoxical entity doesn't exist.
MW:
And what do you call that version of set theory?
Common Logic does not include any built-in set theory. It's possible
to state axioms in CL for all the common versions of set theory.
It's also possible to use CL to state new versions. If you want
to do so, you could call it "MW set theory".
Right.