[FOM] A little trouble with definition of "binary relation" in Wikipedia

[Rupert McCallum]

I believe there are such theories, but you don't need to use them to
define an ordered triple of proper classes. You could define the
ordered triple (X,Y,Z) to be the set of all a such that a=(1,x) where x
is in X, or a=(2,y) where y is in Y, or a=(3,z) where z is in Z. Then
the ordered triple of classes itself becomes a class. This works fine
in NBG.

--- Victor Makarov <viktormakarov at hotmail.com> wrote:

> The following definition of "binary relation" one can find in
> Wikipedia:
> 
> ( http://en.wikipedia.org/wiki/Binary_relation#Formal_definition  )
> 
> "A binary relation R is usually defined as an ordered triple (X, Y,
> G) where 
> X and Y are arbitrary sets (or classes), and G is a subset of the
> Cartesian 
> product X × Y."
> 
> But usually in set theories with classes (for example, NBG) an
> ordered 
> triple (X, Y, G) is definend as
> 
> the ordered pair (X, (Y, G)); and an ordered pair(a,b)  is defined as
> the 
> set { {a},  {a, b} }.
> 
> Because elements of sets must be sets, X, Y, G must be also sets (not
> proper 
> classes).
> 
> 
> My question is:
> 
> Are there set theories with classes, where proper classes can be
> elements of 
> other classes?
> 
> Thanks in advance,
> 
> Victor Makarov
> 
> 
> > _______________________________________________
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> 



 
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