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Mereo-relational Theory.
Posted:
Jul 31, 2012 2:48 AM
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Mereo-relational Theory: A theory in first order logic with identity
"=", part-hood "p" and Ordered pair
"()".
Axioms: Identity theory axioms +
(1) x p x
(2) x p y & y p z > x p z
(3) x p y & y p x > x=y
Define (pp): x pp y <> x p y & ~y p x
Define (O): x O y <> Exist z. z p x & z p y
(4) x pp y > Exist z. z p y & ~ z O x.
Define(atom): x is atom <> ~Exist y. y pp x
(5)Exist z. phi >
(Exist x. for all y. y O x <>
Exist m. phi(m) & y O m)
Define (whole of (phi)s): x is a whole of (phi)s iff whatever overlaps
with x do overlap with an object satisfying phi.
(6) Exist x. for all y. y p x > Exist z. z pp y
Define(void): x is void <> for y. y p x > Exist z. z pp y
(7) Every ordered pair of wholes of atoms is void
(8) Every ordered pair of wholes of void ordered pairs is an atom.
(9) Ordered pairs are identical iff they have exactly the same
projections.
(a,b)=(c,d) <> a=c & b=d
/ Theory definition finished.
I think this is consistent and it can prove Z! However the main
question is if it can prove ZF?
Zuhair
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