
Re: Mereology > Set x Set > Mereology
Posted:
Jul 2, 2012 5:59 AM


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 (Urelement): x is a Urelement <> ~ x is a class > Define (Urmember): x is Urelement & 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 interpretable 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

