The definition of "set' and " membership "

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Guest

Posted: Sat Apr 26, 2008 7:00 pm

Hi all,

Actually using Mereology we can DEFINE 'set' and 'set membership' in
the following manner.

We introduce a two place primitive relation symbole N
which stands for " is the name of " , to the primitives of identity
and part-hood that is used in Mereology, and then define epislon
membership in the following manner:

Define: x e y <-> Ez ( x N z & y atomic part of z )

were x N z is read as x is the name of z.

Define: x is a set <-> Ez ( x N z )

So sets are names of heaps.

It is not difficult to introduce axioms that will lead to all ZF
axioms.

For example extensionality will be

Axiom: x N y -> [~Ez(~z=x & z N y) & ~Ez(~z=y & x N z )]

i.e N is bijective from names to heaps.

Axiom: x N y -> x is atomic

To build the hierarchy of sets.

If we want to element the ur-elements we axiomatize the following:

Axiom: x is atomic -> x is a set

To produce ZFC we axiomatize the following

Size limitation: x subnumerous to V <-> x is a set.

were V is the heap of all atoms.

Power:x is a set -> Power(x) is a set

Infinity: the heap of all finite ordinals is a set.

This will produce all axioms of null,
pairing,union,separation,replacement and global choice.

We can also use ackermann's approach.

Note: the variable present above are heaps.

Guest

Posted: Sun Apr 27, 2008 5:15 am

On Apr 27, 7:29 am, J Jones <jonescard...@aol.com> wrote:

Quote:

Zaljo...@gmail.com wrote:
Hi all,

Actually using Mereology we can DEFINE 'set' and 'set membership' in
the following manner.

We introduce a two place primitive relation symbole N
which stands for " is the name of " , to the primitives of identity
and part-hood that is used in Mereology, and then define epislon
membership in the following manner:

Define: x e y <-> Ez ( x N z & y atomic part of z )

were x N z is read as x is the name of z.

Define: x is a set <-> Ez ( x N z )

So sets are names of heaps.

It is not difficult to introduce axioms that will lead to all ZF
axioms.

For example extensionality will be

Axiom: x N y -> [~Ez(~z=x & z N y) & ~Ez(~z=y & x N z )]

i.e N is bijective from names to heaps.

Axiom: x N y -> x is atomic

To build the hierarchy of sets.

If we want to element the ur-elements we axiomatize the following:

Axiom: x is atomic -> x is a set

To produce ZFC we axiomatize the following

Size limitation: x subnumerous to V <-> x is a set.

were V is the heap of all atoms.

Power:x is a set -> Power(x) is a set

Infinity: the heap of all finite ordinals is a set.

This will produce all axioms of null,
pairing,union,separation,replacement and global choice.

We can also use ackermann's approach.

Note: the variable present above are heaps.

Are you nicking any of my stuff on heaps and sets? Didn't I say that a
set is defined by its name? Where's your recognition of that?- Hide quoted text -

- Show quoted text -

I don't know any of your stuff actually.
I took this idea from Randall holms book : elementary set theory with
universal set.
but of course his idea is different from the one here.

Can you present me with a link to your stuff on that.

Second I didn't say a set is defined by its name, I said a set is a
name of a heap.
So if we say that x is a set, then this mean that there exist a heap y
such that x is the name of y. and if we say that there exist a heap y
such that x is the name of y then this mean that x is a set.

y in x is identical to saying that: there exist a heap z such that y
is atom of z
and x is the name of z.

Zuhair

Marshall

Posted: Sun Apr 27, 2008 5:32 am

Guest

On Apr 27, 7:29 am, J Jones <jonescard...@aol.com> wrote:

Quote:

Are you nicking any of my stuff on heaps and sets? Didn't I say that a
set is defined by its name? Where's your recognition of that?

Nominalism is many centuries old. Although it's not in
use in ZFC, there are many programming languages
that use nominal type systems. In fact that's the most
common way of doing things in programming languages.

Marshall

John Jones

Posted: Sun Apr 27, 2008 9:29 am

Joined: 26 Oct 2004 Posts: 4263

Zaljohar@gmail.com wrote:

Quote:

Are you nicking any of my stuff on heaps and sets? Didn't I say that a
set is defined by its name? Where's your recognition of that?

george

Posted: Sun Apr 27, 2008 3:39 pm

Guest

On Apr 27, 1:00 am, Zaljo...@gmail.com wrote:

Quote:

Hi all,

Actually using Mereology we can DEFINE 'set' and 'set membership' in
the following manner.

No, we can't, because you haven't given any axioms for
"mereology".

Quote:

We introduce a two place primitive relation symbole N
which stands for " is the name of " , to the primitives of identity
and part-hood that is used in Mereology,

Except that YOU haven't demonstrated any capacity to use
these primitives.

Quote:

and then define epislon

You certainly don't need to introduce "name of" to define epsilon
in terms of subsethood. If you have some other notion of parthood
then YOU NEED TO POST SOME AXIOMS.

Guest

Posted: Mon Apr 28, 2008 12:50 am

On Apr 27, 6:39 pm, george <gree...@cs.unc.edu> wrote:

Quote:

On Apr 27, 1:00 am, Zaljo...@gmail.com wrote:

Hi all,

Actually using Mereology we can DEFINE 'set' and 'set membership' in
the following manner.

No, we can't, because you haven't given any axioms for
"mereology".

We introduce a two place primitive relation symbole N
which stands for " is the name of " , to the primitives of identity
and part-hood that is used in Mereology,

Except that YOU haven't demonstrated any capacity to use
these primitives.

and then define epislon

You certainly don't need to introduce "name of" to define epsilon
in terms of subsethood. If you have some other notion of parthood
then YOU NEED TO POST SOME AXIOMS.

Yea, I already posted that before in a separat topic.

I will post them again here, they are only four axiom schemes:

Symbols used:

\p : is a part of
\pp : is a proper part of
\*p : is atomic part of

Definitions:

y \pp x iff ( y \p x and y neq x ).

y is atomic iff Ez ( z \pp y and Au ( u neq z -> u=y ) ).

y \*p x iff ( y \p x and y is atomic )

Axioms:

M1: Transitive: (x \p y and y \p z) -> x \p z

M2: Accumulation: Az (z \*p y -> z \*p x) -> y \p x

M3: Extensionality: Az (z \*p x iff z \*p y) -> x=y

M4: Comprehension: if Phi is a formula in which x is not free,
then all closures of

Ay( Phi(y) -> y is atomic) -> Ex Ay ( y \*p x iff Phi(y) )

are axioms.

Theorems:

T1: Ax (x \p x)

Proof: M2

T2: Ex Ay not( y \*p x ).

Proof: Let Phi(y) iff y neq y

Then substitute y neq y in M4 we get:

Ay( y neq y -> y is atomic) -> Ex Ay ( y \*p x iff y neq y )

The left hand is trivially true, thus the right hand is true.

Theorem proved.

T3: E!x Ay not( y \*p x )

Proof: M3

Define: x= [ ] iff Ay not ( y \*p x )

[ ] is what Mereologists call as 'bottom'

This leads to T4:

T4: ~x=[ ] -> Ey ( y \*p x )

T5: Az (z \*p y -> z \*p x) <-> y \p x

Proof: M1 and M2.

T6: Ex Ay ( y \p x )

Proof: Let Phi(y) iff y is atomic

from comprehension we get

Ex Ay ( y is atomic -> y \*p x )

from extensionality we proove that E!x Ay ( y is atomic -> y \*p x )

Define x= V iff Ay ( y is atomic -> y \*p x )

So V is the heap of all atoms

From T5 V is the heap of all heaps.

To prove that, we say heaps are either [ ] or non empty

if x=[ ] then x \p V . M2
if x neq [ ] then there must exist an atom in x, Now all atoms in x
would be in V ( definition of V ) so x \p V. Theorem proved.

So V is the heap of all heaps.

Zuhair

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