```Date: Jul 2, 2012 5:59 AM
Author: Zaljohar@gmail.com
Subject: Re: Mereology > Set x Set > Mereology

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.>> ZuhairI think we can define x label y as x being a set of all iteratedsingletons of y or the set of x and all iterated singletons of y.An iterated singleton of x is defined recursivelyx 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 isi_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 ofits elements is an iterative singleton of it.I think this will take care of circular labeling, however it cannotmodel set theory that refutes singletons.I think that for every Mereology+ theory there is a set theory thatcan interpret it, but there cannot be a single set theory that caninterpret ALL mereology+ theories.Anyhow I think both approaches are somehow equivalent "Set theory" x"Labeled Mereology", I personally prefer Mereology based approachsince the primitives of it are understood informally in a morestricter sense.Zuhair
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