Re: Small Set theory:Revised.


zuhair schrieb:
From all of this dicussion, this Russell paradox is becoming clearer
to me. I think that this paradox is a deeper one that I originally
thought, now I am thinking that we can solve this paradox in one of
three aproaches.

1) Limit P. so we should have a kind of a type theory that allowes some
kinds of P and do not allow other kinds.

Maybe. Type theory is one approach against the Russel paradox.
But that's a completely different story than condensing ZFC into 4
axioms...


2) Apply a non binary logical concepts, I mean we lessen the
application of the law of excluded middle as regards this case.

Maybe. Again converting to become intuitionist is a valid but totally
different story...


3) "Existance" predicated. i.e. we treate "existance" as a predicate.

for example.

AxAy (((P[y]->yex)&(~P[y]->(~yex&~y=x))) <-> x Exist.)

What is the quantor Ax quantisizing about if not /existing/ sets x??
And if Ax runs over non-existing sets, does that mean that you want
((P[y]->yex)&(~P[y]->(~yex&~y=x)))
iff x exists whereas y may or may not exist?

If you start from ...
AxAy (((P[y]->yex)&(~P[y]->(~yex&~y=x))) <-> x Exist.)

Now if P[y]<->y=0

Ay (((y=0->0ex)&(~y=0->(~yex&~y=x))) <-> x Exist.)
or simply
Ay ((0ex&(~y=0->(~yex&~y=x))) <-> x Exist.)

if x = 0
.... and specialize for x=0 ...

Ay (((y=0->0e0)&(~y=0->(~ye0&~y=0))) <-> x Exist.)
rather
Ay ((0e0&(~y=0->(~ye0&~y=0))) <-> 0 Exist.)

Ay(false & true <-> x Exist)

Ay(false<-> x Exist)

Ay(true<->x do not Exist).
.... and obtain this ^^ (which is equivalent to "x does not exist")
you have shown that the empty set does not exist.
Congratulation.

.

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