# [FOM] A little trouble with definition of "binary relation" in Wikipedia (fwd)

eomodeo@units.it eomodeo at units.it
Thu Feb 1 04:04:16 EST 2007

```In the Tarski-Givant book "A formalization of set theories without
variables", pp.130-132, I found the following notion of 'ordered pair
for set-theoretical systems (excluding individuals) which admit proper
classes': the ordered pair whose first and second component are the
classes r and s, respectively, is
[r,s] = {{u}: u in r} + {{{v},0}: v in s},
where I'm indicating by '+' and '0' the class union operation and the
empty set.
Retrieving the components r,s from a class of this form is
unambiguous: the members of its sigleton elements form r, and from the
doubletons---via a but slightly more elaborate operation---we can get s.

--Eugenio Omodeo

Quoting "Robert M. Solovay" <solovay at math.berkeley.edu>:

> This doesn't work. The cartesian product of two factors [at least one
> of which is the null set] is null.
>
>          --Bob Solovay
>
> On Mon, 29 Jan 2007, Thomas Forster wrote:
>
>> 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)
>> 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!
>
>
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