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Re: Mereology > Set x Set > Mereology
Posted:
Jul 2, 2012 5:59 AM
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On Jul 1, 2:30 pm, Zuhair <zaljo...@gmail.com> wrote:
> 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
I think we can define x label y as x being a set of all iterated
singletons of y or the set of x and all iterated singletons of y.
An iterated singleton of x is defined recursively
x is 1_singleton of y <> x={y}
x is i+1_singlteon of y <> Exist z. z is i_singleton of y & x={z}
so x label y <> x={z| z is i_singleton of y & i e N} or x={z|(z is
i_singleton of y &i e N) or z=x}
where N={1,2,3,..}
Sets to be restricted to those that are infinite and such that non of
its elements is an iterative singleton of it.
I think this will take care of circular labeling, however it cannot
model set theory that refutes singletons.
I think that for every Mereology+ theory there is a set theory that
can interpret it, but there cannot be a single set theory that can
interpret ALL mereology+ theories.
Anyhow I think both approaches are somehow equivalent "Set theory" x
"Labeled Mereology", I personally prefer Mereology based approach
since the primitives of it are understood informally in a more
stricter sense.
Zuhair
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