From Newsgroup: sci.logic

On 07/08/2015 01:38 AM, Zuhair wrote:

Language: second order logic with predication limited to be over objects only.

Primitives:
Part-hood symbolized by P
fulfillment instance symbolized by <,> (an object, predicate two place partial function symbol), so y=<x,Q> is read as y is the fulfillment instance of x in Q, or alternatively the fulfillment instance of Q by x.

Axioms:
[1] Mereology: General extensional atomic mereology without bottom

[2] Fulfillment: R(x) ^ Q(y) ^ a=<x,R> ^ b=<y,Q> -> [a=b <-> x=y]

[3] Instances: y=<x,Q> -> y is an atom.

Define: x=eQ <-> for all y ((y is an atom ^ y P x) <-> Exist z(y=<z,Q>))

"eQ" is read as: the extension of Q.

Define: x is a set <-> Exist Q (x=eQ ^ for all y (Q(y) -> Exist z (z=<y,Q>)))

Define: y E x <-> Exist Q (x=eQ ^ Exist z (z=<y,Q>) ^ Q(y))

"E" is read as "is a member of"

[4] Extensionality: [for all x (x E eQ <-> x E eR)] -> [Q<->R]

[5] Rejection: [Not exist x (Q(x))] <-> Exist y,z (y=<z,Q> & not Q(z))

[6] Acceptance: if phi is a formula only using E and = as predicates, then [(for all x (Q(x) <-> phi)) ^ y is a set ^ Q(y) -> Exist z (z=<y,Q)]
is an axiom.

/Theory definition finished.

A weaker form of Extensionality is:

eQ is a set ^ for all y (y E eQ <-> y E eR) -> [Q<->R]

A nice feature of this theory is that it is compatible with the following aesthetic principle.

"Every part of an extension of a predicate is an extension of a predicate".

This will render the empty set a single atom, and prove Lewis's aesthetic principle of parts of classes being identical to subclasses, since all classes after Lewis are non empty. Here this account, unlike Lewis's, enrolls the empty set among classes and thus views it as a pure object.

Now this theory does interpret Z and I'd think ZF also over the realm of hereditarily set extensions of it. And those are definable here as:

Define: x is hereditarily a set <->
x is a set ^ for all y (Exist Q,z(z=<y,Q> ^ z P TC(x)) -> y is a set)

(TC standing for transitive closure is definable using the unrestricted composition principle of Mereology, this is the mereological intersection of all mereological collections having x as a part of them and for each fulfillment instance that is part of them every atom of the first argument of that fulfillment instance is a part of them also)

ALL the above axioms are emanations of Mereologicism, none of them emanates from intuitive contemplation of sets on the abstract level (i.e. taking set membership as a primitive concept), so none of them is a set theory rule.

This means that Set theory is explained in pure Mereologicism!

Mereologicism seems to be a more 'concrete' than the abstract Set Theory, so one can say that the later is founded in the former.

So it seems that Mathematics would ultimately be founded in Mereo-Logicism.

Zuhair

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From Newsgroup: sci.logic

On 09/28/2026 11:37 AM, Ross Finlayson wrote:

On 07/08/2015 01:38 AM, Zuhair wrote:

Language: second order logic with predication limited to be over
objects only.

Primitives:
Part-hood symbolized by P
fulfillment instance symbolized by <,> (an object, predicate two place
partial function symbol), so y=<x,Q> is read as y is the fulfillment
instance of x in Q, or alternatively the fulfillment instance of Q by x.

Axioms:
[1] Mereology: General extensional atomic mereology without bottom

[2] Fulfillment: R(x) ^ Q(y) ^ a=<x,R> ^ b=<y,Q> -> [a=b <-> x=y]

[3] Instances: y=<x,Q> -> y is an atom.

Define: x=eQ <-> for all y ((y is an atom ^ y P x) <-> Exist z(y=<z,Q>))

"eQ" is read as: the extension of Q.

Define: x is a set <-> Exist Q (x=eQ ^ for all y (Q(y) -> Exist z
(z=<y,Q>)))

Define: y E x <-> Exist Q (x=eQ ^ Exist z (z=<y,Q>) ^ Q(y))

"E" is read as "is a member of"

[4] Extensionality: [for all x (x E eQ <-> x E eR)] -> [Q<->R]

[5] Rejection: [Not exist x (Q(x))] <-> Exist y,z (y=<z,Q> & not Q(z))

[6] Acceptance: if phi is a formula only using E and = as predicates,
then
[(for all x (Q(x) <-> phi)) ^ y is a set ^ Q(y) -> Exist z (z=<y,Q)]
is an axiom.

/Theory definition finished.

A weaker form of Extensionality is:

eQ is a set ^ for all y (y E eQ <-> y E eR) -> [Q<->R]

A nice feature of this theory is that it is compatible with the
following aesthetic principle.

"Every part of an extension of a predicate is an extension of a
predicate".

This will render the empty set a single atom, and prove Lewis's
aesthetic principle of parts of classes being identical to subclasses,
since all classes after Lewis are non empty. Here this account, unlike
Lewis's, enrolls the empty set among classes and thus views it as a
pure object.

Now this theory does interpret Z and I'd think ZF also over the realm of
hereditarily set extensions of it. And those are definable here as:

Define: x is hereditarily a set <->
x is a set ^ for all y (Exist Q,z(z=<y,Q> ^ z P TC(x)) -> y is a set)

(TC standing for transitive closure is definable using the
unrestricted composition principle of Mereology, this is the
mereological intersection of all mereological collections having x as
a part of them and for each fulfillment instance that is part of them
every atom of the first argument of that fulfillment instance is a
part of them also)

ALL the above axioms are emanations of Mereologicism, none of them
emanates from intuitive contemplation of sets on the abstract level
(i.e. taking set membership as a primitive concept), so none of them
is a set theory rule.

This means that Set theory is explained in pure Mereologicism!

Mereologicism seems to be a more 'concrete' than the abstract Set
Theory, so one can say that the later is founded in the former.

So it seems that Mathematics would ultimately be founded in
Mereo-Logicism.

Zuhair

The accounts of heno-theories make for that there are a wide
variety of "theories-of-one-relation", each with the neat accounts
of whether and how they model fixed-point and the extra-ordinary
for each other, or not.


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