🚀 go-pugleaf

In article <1992Aug25.120515.15090@husc3.harvard.edu> zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>Greetings,
[...]
>Je suis pr\^et \`a partir pour Paris jusqu'a 17 septembre; par
>cons\'equent, il me sera impossible de tenir mes propos provocants.
>Veuillez envoyer vos messages concernant mes defauts personnels \`a ma
>adresse not\'ee ci-dessous.
[...]

I cannot resist. Correcting MZ is such a pleasure.
It's "mon adresse" and not "ma adresse".
I confess however that the rest of his French looks better
than my English.

BTW, Mishka, si tu passes par l'inria, viens donc me faire
une petite visite au batiment 9 ou appelle moi a paname.
Je suis dans l'annuaire.


Since this can hardly justify a posting by itself here is something 
more interesting I hope.

Russel's paradoxical set is the set of normal sets, i.e. sets which
don't belong to themselves.
However, the set of all sets, let's call it S, is often considered 
paradoxical because of the following, with P(S) the powerset of S:

P(S) is included into S, then Card(P(S)) <= Card (S)
which contradicts a theorem saying that
Card(P(S)) > Card(S)

However, looking back to the proof for Card(P(S)) > Card(S)
it appears that the proof uses a set might quite be paradoxical !
The idea is to consider a function from S onto P(S).
If you take the identity the dubious set is indeed Russel's set !

Thus you might as well reject the proof rather than the concept
of set of all sets. However if you define sets with ZF it seems
that you must accept the proof. The separation "axiom" which is
in facta theorem of ZF, implies that the above dubious set be
a proper set. (Am I wrong ?)

The question is now, whether or not alternative axiomatizations
have been proposed that are consistent with the existence of the
set of all sets ?

Also, I don't know anything yet about categories. I have heard
though that the category of all the categories is not a pb.
Does category theory have similar paradoxes though ?

Do the problems with the set of all sets prevent the possibility
to properly define an Abstract Data Type "set" ? Set of anything
of course.

Finaly, for computer scientists, is it problematic to have a
universal type. I think of hierarchies in Object Oriented
languages, where you have a root like Object or Class.

Mikal Ziane.