The set of all sets

N.G. du Bois (n.dubois@wxs.nl)
Mon, 11 Oct 1999 17:32:49 +0200 (MET DST)

Hi to all,

I have a question.

In ZF set theory the powerset of a set S denoted as P(S) cannot be a
member of S. Because of that you cannot construct the set of all sets,
because it will lack its own powerset, which is a set and so ought to be
a member of the set of all sets.

Let's take a set S and construct its P(S). I can after that construct
P(P(S)) and after that P(P(P(S))) and so on.

Now let us take a Boolean logic where P(S) can perhaps be a member of S.
Let R be the set of all sets.
I can construct a P(R).
P(R) must be a member of R.
But then I get a new R' containing P(R),
And after that a P(R'), etcetera.
So I must conclude that R is not the set of all sets.

Now I'm wondering: in Fuzzy logic it should be possible for a P(S) to
have a degree of membersip in S. "The whole in the part", read B.
Kosko's book "Fuzzy thinking".

The question is: can I construct in fuzzy logic a (fuzzy) set of all
sets?

(Ofcourse this has to do with Russell's antinomy, but only the first
part.)

I am looking out for your suggestions,

Best regards,

Nico du Bois.

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