|
|
Re: Mereology as the bases for relations in the finite world
Posted:
Jul 13, 2012 4:34 PM
|
|
On Jul 13, 11:32 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Jul 13, 7:07 pm, Zuhair <zaljo...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Jul 13, 11:00 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > > Primitives:
>
> > > Identity: =
> > > Division: D(x,y,z) read as x is divided to x and y
> > > Intervening: a>>x<<b read as x is intervening between a and b
> > > both are three place predicates
> > > glob, a one place predicate
>
> > > Axioms:
> > > I. D(x,y,z) > D(x,z,y)
> > > II. D(x,y,z) > ~y=z & ~x=y & ~x=z
> > > Define (part): y part of x <> Exist z. D(x,y,z)
> > > III. x part of y & y part of z > x part of z
> > > IV. Exist y. y part of x
> > > V. (for all x. x part of y > x part of z) & ~y=z > y part of z
> > > VI. D(x,y,z) > ~Exist m. m part of y & m part of z
> > > VII. (for all z. z part of y <> z part of x) > x=y
> > > Define: O(x,y) <> Exist z. z part of x & z part of y
> > > VIII. x is a glob & y is a glob > ~O(x,y)
> > > Define: x is a whole of phi's <>
> > > for all y. y part of x > Exist z. (z part of x or z=x) & phi(z) &
> > > O(y,z)
>
> > > IX. if phi,pi are formulae in which x is not free, then: ((Exist z.
> > > phi & pi) >
> > > (Exist x. x is a whole of phi's & for all y. (y part of x or y=x ) &
> > > phi(y) <> phi(y) & pi)) is an axiom
> > > X) a,b are globs >
> > > Exist x. x is a glob & a>>x<<b & ~ O(x,a) & ~O(x,b) & for all m,n.
> > > m>>x<<n > m=a & n=b
> > > Define(dedicated intervening): x is a dedicated intervening object <>
> > > Exist a,b. (a,b are globs & a>>x<<b & ~ O(x,a) & ~O(x,b) & for all
> > > m,n. m>>x<<n > m=a & n=b)
> > > Define (dedicated intervening glob): x is an intervening glob <> x is
> > > a glob & x is a dedicated intervening object.
> > > Define (relation): x is a relational object <> x is a glob & x is a
> > > whole of dedicated intervening objects
> > > Define (similar): x,y are wholes of dedicated intervening objects >
> > > x is similar to y <> for all n,m,v ( n,m are globs & v part
> > > of x & n>>v<<m & v is a dedicated intervening object > Exist v*. v*
> > > part of y & v* is a dedicated intervening object & n>>v*<<m)
> > > & for all n,m,v ( n,m are globs & v part of y & v is a dedicated
> > > intervening object & n>>v<<m >
> > > Exist v*. v* part of x & v* is a dedicated intervening object &
> > > n>>v*<<m)
> > > XI. x is a whole of dedicated intervening globs > Exist x*. x is a
> > > relational object & x similar to x*
> > > XII. Exist x. x is a glob.
> > > /
> > > The nice thing about that system is that it can assign representative
> > > globs to finite wholes of objects.
> > > Lets say that for a whole X of globs a "representational whole" of X
> > > is a whole of intervening globs between pairs of globs in X such that
> > > the whole of all globs intervened by them is X itself!
>
> > > Example: Let X be the whole of Glob1,Glob2,Glob3
> > > Let Glob(1,2) , Glob(2,3) be the intervening globs between
> > > Glob1,Glob2
> > > and between Glob2,Glob3 respectively.
> > > Now the whole Y of Glob(1,2),Glob(2,3) will be a representational
> > > whole of X
> > > Now if we have any finite whole of globs Glob1,....,Globn
> > > then we would have a representational whole of it having n-1 globs in
> > > it
> > > and a representational whole of that would having n-2 globs in it and
> > > so
> > > on if we continue... we'll eventually reach to a single glob. Now
> > > this
> > > single glob would stand as a representative glob of the original
> > > whole
> > > Glob1,...,Globn Of course we can have many such representative globs
> > > for a single
> > > whole of globs, but the main point is that for finite wholes we can
> > > reach at globs that
> > > represent them.
> > > Unfortunately no comparable matter can be applicable to infinite
> > > wholes.
>
> > > Using those represetive globs of wholes of globs one can interpret
> > > finite sets!
>
> > > Zuhair
>
> > One can actually extend this system,
>
> > Define (identity intervening globs): x is an identity intervening glob
> > <>
> > x is a glob & Exist a. a>>x<<a.
>
> > For all i,j=1,2,3,....
> > for all x. x is an identity intervening glob >
> > Exist a1,..,ai ,b1...,bj for all a,b. a>>x<<b > (a=a1 or a=a2
> > or ...a=ai) & (b=b1 or...b=bj) is an axiom.
>
> > Axiom. if phi is a formula then:
> > Exist x,y. phi(x) & phi(y) & x,y are wholes of globs & ~x=y > Exist x.
> > (x is an identity intervening glob & Exist a. a is a whole of globs &
> > phi(a) & a>>x<<a & for all m,n. m>>x<<n & ~m=a & ~n=a > ~m=n) is an
> > axiom.
>
> > Now suppose we defined natural numbers as equivalence wholes of finite
> > wholes of globs. Now the fussion of all those will be the whole of all
> > natural numbers. lets call it N. Now the whole of all identity
> > interevening globs between subwholes of N with themselves will stand
> > in a comparable way to the set of all subsets of N.
>
> > In this way we can define reals as well.
>
> > Such theory is strong enough to interpret Z perhaps or even stronger.
>
> > Zuhair
>
> Just an example of how strong this methodology could be:
>
> A number can be defined as the whole of all identity intervening globs
> between wholes that are bijective to some specified whole
>
> Example: the whole of all identity intervening globs between wholes of
> globs bijective to a whole of globs having ONE glob is number 1.
>
> A finite number is a number that have immediate predecessor where
> every predecessor of it other than number 1 do have an immediate
> predecessor.
>
> A whole that may be taken to represent the "set" of all finite numbers
> is the whole of all identity intervening globs between numbers.
...between "finite" numbers.
>
> Also a whole that may be taken to represent the "set" of all subsets
> of the set of all finite numbers is the whole of all identity
> intervening globs between subwholes of the set of all finite numbers.
> This is like Power(Omega)
>
> We can also ascend to have various iterative powers of Omega,
> PP(Omega), PPP(Omega), ....
>
> The fusion of all those powers is the Omega_th power of Omega, and we
> can also have iterative powers of it, and so on
>
> The set of all sets can also be represented here as the whole of all
> identity intervening globs between whole of identity intervening
> globs. A nice result that is shown immediately is that the set of all
> sets is the union of all sets.
>
> Anyhow I'm not sure yet it this methodology can withstand the wind.
>
> Zuhair
|
|
