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Mereology > Set x Set > Mereology
Posted:
Jul 1, 2012 7:30 AM
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Language: first order logic
Primitives: identity "=", Label , Division "D"
D(x,y,z) is read as: x is divided into y and z
Axioms:
for all x,y,z. D(x,y,z) > D(x,z,y)
for all, x,y,z. D(x,y,z) > ~y=z & ~y=x & ~z=x
Define (part): y part of x <> Exist z. D(x,y,z)
for all x,y,z. y part of x & z part of y > z part of x
for all x,y.(for all z.z part of y >z part of x)&~y=x >y part of x
for all x. Exist y. y part of z
for all x,y. (for all z. z part of y <> z part of x) >x=y
for all x,y,z. D(x,y,z) > ~Exist u. u part of y & u part of z
for all x,y,z. x label y & x label z > y=z
Define (L): L(x) <> Exist y. x label y
for all x,y. L(x) & L(y) & ~x=y >~Exist z. z part of y & z part of x
Define(wholly labeled): x is wholly labeled <>
for all y. y part of x > Exist z. (z part of x or z=x) & L(z) &
Exist u. u part of y & u part of z
if phi is a formula in which x is not free.
(Exist t. L(t) & phi) > Exist x. x is wholly labelled &
for all y. (y part of x or y=x ) & L(y) <> L(y) & phi
Define(0): x=0 <> for all y. (y part of x or y=x) <>
~Exist z. z is wholly labeled & y part of z
Define (class): x is a class <> x=0 or x is wholly labeled
Define (e): y e x <> x is a class & Exist z. z part of x & z label y
Define (set): x is a set <> x is a class & Exist y. y label x
Define (Ur-element): x is a Ur-element <> ~ x is a class
Define (Ur-member): x is Ur-element & Exist y. y label x
Define (subclass): y subclass of x <> y=0 or
y is a class & x is a class & (y part of x or y=x)
/
Now is this system interpret-able in set theory (primitives of =,e), I
think the answer is towards the positive!
Take a set theory having axioms enough to define infinite sets, for
simplicity lets work with well founded sets. Now D(x,y,z) can be
defined as x= y U z & y disjoint of z & all of x,y,z are infinite.
Labels can be defined as x=L(y) <> x={{z,y}| z e y}, however this is
not flexible enough to define situations of circular labeling .
However I think it can be done with membership and identity. The
atomic approach of Mereology can be easily interpreted in set theory
where labels defined as singletons and division as above but with the
condition that it yields at least one finite set.
Anyhow I think that set theory can interpret all the above, but I
don't know which direction is simpler really.
Zuhair
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