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

Thomas Forster T.Forster at dpmms.cam.ac.uk
Mon Jan 29 04:29:39 EST 2007

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!


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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