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Mar 23, 2006, 1:51:03 PM3/23/06

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Page 15 of Barwise and Moss's "Vicious Circles", a brief survey of Set

Theory, says "If a class is a member of a class then it is a set".

Theory, says "If a class is a member of a class then it is a set".

Can anyone tell me what that means. I understand a class to be an

informal way to talk about predicates. Eg. I understand "x is in class

C" to mean "x satisfies predicate C" and so on. But then how could a

class be a member of a class? In the following paragraph the authors

talk as if this is absolutely standard. I'm mystified.

--

Dan

Mar 23, 2006, 2:08:53 PM3/23/06

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Strictly speaking, ZFC doesn't have classes, but uses predicates to accomplish

approximately the same thing.

Other set theories such as NBG have classes. Every set is a class, but

some classes are not sets. These are called "proper classes". Proper

classes can have members just like sets, but proper classes cannot be

members of classes. Hence, a class that is a member of a class must not

be a proper class, i.e., must be a set.

--

Dave Seaman

U.S. Court of Appeals to review three issues

concerning case of Mumia Abu-Jamal.

<http://www.mumia2000.org/>

Mar 23, 2006, 2:23:06 PM3/23/06

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Dave Seaman said,

> Hence, a class that is a member of a class must not

> be a proper class

You say "a class that is a member of a class must not be a proper

class". I don't know what it means for a class to be a member of a

class. Let me label the things in your statement "a class, P, that is a

member of a class, Q, must not be a proper class". The most reasonable

interpretation I can give this, treating P and Q as predicates, is "if

there is a set p such that (for all x, x is in p if P(x)), and Q(p),

then P is not a proper class". But this seems too trivial a statement

to be worth saying.

Is my interpretation correct?

Mar 23, 2006, 3:15:50 PM3/23/06

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Instead of complicating with various predicates, we can derive

definitions from just the membership relation:

Definitions:

x is a set iff ((x has at least one member or x is the empty set) and x

is a member of some y).

x is an urelement iff (x has no members and x is not the empty set).

x is a proper class iff (x has at least one member and x is not a

member of any y).

x is a class iff x is a set or x is a proper class.

(So all sets are classes, but not all classes are sets, since proper

classes are not sets.)

Another way of looking at it:

There are objects. There is a relation that we call the membership

relation. For any object, there are four possibilities:

(1) The object has members and the object itself is a member of some

object.

(2) The object has members but the object itself is not a member of an

object.

(3) The object has no members but the object itself is a member of some

object.

(4) The object has no members and the object itself is not a member of

an object.

It turns out that, in the axioms for most set theories, (4) never

occurs. Usually, in a set theory, there is no object that has no

members but is not itself a member of some object. So let's throw out

(4).

Any object satisfying (1) is a class and a set.

Any object satisfying (2) is a class but not a set (it is a proper

class).

An object satisfying (3) is either the empty set or an urelement (a

memberless object that is not the empty set). If the object is the

empty set, then it is a class and a set. If the object is an urelement,

then it is neither a class nor a set.

We could devise theories in the following ways

The following are different forms of set theory that are viable:

with just sets; no urelements; and no proper classes or

with just sets and urelements; and no proper classes or

with just sets and proper classes; and no urelements or

with sets, urelements, and proper classes.

The following don't seem to be forms of set theory in use:

with just urelements or

with just proper classes or

with just urelements and proper classes.

MoeBlee

Mar 23, 2006, 3:41:28 PM3/23/06

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> Is my interpretation correct?

No, I was speaking of NBG. As I explained, classes in NBG are not predicates.

Mar 23, 2006, 4:18:06 PM3/23/06

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Dave Seaman said:

> As I explained, classes in NBG are not predicates

Ay yes, I missed that. In NBG classes are reified so the statement

makes perfect sense. But "Vicious Circles" is working with ZF

(-foundation +anti-foundation) and the statement seems not to make any

sense in that context.

Mar 23, 2006, 7:13:34 PM3/23/06

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MoeBlee said:

> Instead of complicating with various predicates, we can derive

> definitions from just the membership relation

But in ZF (or ZF without Foundation as discussed in this book) there is

no such thing as a class, at least as a first class object. Talk about

classes is surely just a different way of talking about predicates. So

I always need to translate any kind of statement about classes back

into predicates - or at least know that there is such a translation.

> There are objects

In ZF there are only sets and in variations of ZF there may be

urelements. In NBG Set Theory classes are also objects, but that's not

what we're talking about here.

So I'm still confused.

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