Can classes be elements of classes?
Dan Piponi
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
Dave Seaman
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/>
Dan Piponi
> 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?
MoeBlee
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
Dave Seaman
> Is my interpretation correct?
No, I was speaking of NBG. As I explained, classes in NBG are not predicates.
Dan Piponi
> 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.
Dan Piponi
> 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.