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

[Thomas Forster]

Yes, there are such theories but you don't need them. (In particular you
don't need to know about Ackermann set theory!)  When you want an ordered
pair of two NBG-style proper classes, use their (Wiener-Kuratowski)
cartesian product instead: there is nothing sacred about the
Wiener-Kuratowski implementation of pairing and unpairing.

If you are unhappy about having a different implementation of 
pairing-and-unpairing for sets and for proper classes, then use the same 
(cartesian-product-using-W-K-pairs) for both!


      tf



On Sun, 28 Jan 2007, Victor Makarov 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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