Mereological Basis of Set Theory

To General Extensional Mereology [$\sf GEM$][1] add primitives of non-directional linking, which is a ternary relation symbol, and the primitive binary relation "is connected to". The first denoted by $ a \overset {l} {-} b $ (read as: $l$ links $a$ and $b$), and the second by $\mathcal C$. Add a unary partial function $\mathcal R$, called the *Revealing* function, with the intention to be revealing of a true size comparison. Add:

**Connectedness** axioms:

**C1-Hyper-reflexive:** $\mathcal O(x,y) \implies x \mathcal C y$

**C2-Symmetric:** $x \mathcal C y \implies y \mathcal C x$

***Define:*** $x \ p^* \ y \iff x \ p \ y \land atom(x)$

**C3-Partitive:** $\neg atom(x) \land \neg atom(y) \implies \\ [x \mathcal C y \iff \exists a \exists b: a \ pp \ x \land b \ pp \ y \land a \mathcal C b]$

***Define:*** $\begin{align} \operatorname {Unit}(x) \iff & \forall y \forall z (\operatorname {fusion}(y,z)=x \implies y \mathcal C z) \, \land \\& \forall k (\neg \mathcal O(k,x) \implies \neg k \mathcal C x)\end{align} $

***Define:*** $\operatorname {Atomic}(x) \iff \forall y\, (y \ p \ x \implies \exists z: z \ p^* \ y)$

**Linking** axioms:

**L1-Symmetric:** $ a \overset {l} {-} b \implies b \overset{l} {-} a$

**L2-Projective:** $   a\overset {l}{-}b \land c\overset {l}{-}d \implies  (a=c \land b=d) \lor (a=d \land b=c)$

**L3-Atomic:** $a\overset {l}{-}b \implies  a,b,l \text { are atoms } $

**L4-Distinctive:** $ a \overset{l} - b \implies l \neq a, l \neq b, a \neq b$

***Define:*** $l \equiv k \iff \exists a \exists b: a \overset{l}-b \land a \overset{k}-b$

**Parsimony** axioms:

**P1-Internal:** $l \equiv k, l \ p \ X , k \ p \ X, \operatorname {Unit}(X) \implies l=k$

**P2- External:** $\forall \ \operatorname {Units} \ X,Y: \\\forall l \ p^* \ X \exists k \ p^* \ Y: l \equiv k \land \\\forall k \ p^* \ Y \exists l \ p^* \ X: l \equiv k \\ \implies X=Y$

***Define:*** $ Cloud(X)=W \iff  \operatorname {Unit}(W) \land \operatorname {Atomic}(X) \land \\ \forall l \ p^* \ W \exists a \ p \ X  \exists b \ p \ X: a \overset {l}- b    \\ \land \forall a \ p^* \ X \, \forall b \ p^* \ X [a \neq b \implies \exists l \ p \ W: a \overset{l} -  b]  $

***Define:*** $x \text { is a } Cloud   \iff \exists y: x=Cloud(y)$

**Non-links:** $\exists k: atom(k) \land k: \neg \exists a \exists b: a \overset{k} - b$ 

**Gunk**: $\exists G: \neg \exists a: atom(a) \land a \ p \ G$

***Define*** $\frak G$ as the fusion of all Gunk (atomless) material.

***Define*** $\varnothing$ as the complement of the fusion of all Clouds. 

**Dichotomy:** $\operatorname {Unit}(x) \iff x=\varnothing \lor x \text { is a }Cloud$

***Define (Set-hood):***  $Set(X) \iff \operatorname {Unit}(X) $

***Define (Set Membership):***  

$Y \in X \iff \operatorname {Unit} (X) \land \exists Z: X=Cloud(Z) \land Y \ p \ Z \land \operatorname {Unit} (Y)$

This way the empty set becomes $\varnothing$ itself.

We call the above system as **Linked-Connective-Mereology**.

Now to make a system that can interpret $\sf ZFC$, all  we need to add is just one **size criterion**. We can coin:

 **Axiom Schema of Size:** *If Units of some object can be put in one-to-one relationship with parts of a Unit, then the fusion of those Units has a Cloud.* That is, for every function $F$:

$$ K \text { is fusion of } \operatorname {Units}  \land  \operatorname {Unit}(L) \land \\\forall a \forall b \, [F(a)= F(b) \implies b=d ] \land \\ \forall x \, [ x \ p \ K \land \operatorname {Unit}(x) \implies \exists y: y \ p \ L \land F(x)=y] \\ \implies \\ \exists W: W=Cloud(K) $$

**Axiom of Infinity:** $\mathcal R$ is one-to-one from the naturals to parts of $\frak G$. That is, 

$$ \forall a \forall b \, [\mathcal R(a)= \mathcal R(b) \implies b=d ] \land \\ \forall n \, [ \operatorname {Natural}(n) \implies \exists g: g \ p \ {\frak G} \land \mathcal R(n)=g]$$

Where "$\operatorname {Natural}$" is defined in the usual sense.

So, this theory, in some sense, reduces the standard line of set theory to a size notion over the rest of its axioms.

Note: $p$ stands for parthood relation, $pp$ for proper parthood relation, $\mathcal O$ for Overlap relation, $\operatorname {fusion}$ and $atom$ stand for Mereological fusion and Mereological atom.

  [1]: https://plato.stanford.edu/ENTRIES/mereology/#UnrCom

ANOTHER VERSION ( The more fundamental one)

   To Atomic General Extensional Mereology [$\sf  AGEM$][1] add primitives of non-directional linking, which is a ternary relation symbol, and the primitive binary relation "is connected to". The first denoted by $ a \overset {l} {-} b $ (read as: $l$ links $a$ and $b$), and the second by $\mathcal C$. Add the following axioms:

**Connectedness** axioms:

**C1-Hyper-reflexive:** $\mathcal O(x,y) \implies x \mathcal C y$

**C2-Symmetric:** $x \mathcal C y \implies y \mathcal C x$

***Define:*** $x \ p^* \ y \iff x \ p \ y \land atom(x)$

**C3-Atomic:** $x \mathcal C y \iff \exists a \exists b: a \ p^* \ x \land b \ p^* \ y \land a \mathcal C b$

***Define:*** $\begin{align} \operatorname {Unit}(x) \iff & \forall y \forall z (\operatorname {fusion}(y,z)=x \implies y \mathcal C z) \, \land \\& \forall k (\neg \mathcal O(k,x) \implies \neg k \mathcal C x)\end{align} $

**Linking** axioms:

**L1-Symmetric:** $ a \overset {l} {-} b \implies b \overset{l} {-} a$

**L2-Projective:** $   a\overset {l}{-}b \land c\overset {l}{-}d \implies  (a=c \land b=d) \lor (a=d \land b=c)$

**L3-Atomic:** $a\overset {l}{-}b \implies  a,b,l \text { are atoms } $

**L4-Distinctive:** $ a \overset{l} - b \implies l \neq a, l \neq b, a \neq b$

***Define:*** $l \equiv k \iff \exists a \exists b: a \overset{l}-b \land a \overset{k}-b$

**Parsimony** axioms:

**P1-Internal:** $l \equiv k, l \ p \ X , k \ p \ X, \operatorname {Unit}(X) \implies l=k$

**P2- External:** $\forall \ \operatorname {Units} \ X,Y: \\\forall l \ p^* \ X \exists k \ p^* \ Y: l \equiv k \land \\\forall k \ p^* \ Y \exists l \ p^* \ X: l \equiv k \\ \implies X=Y$

***Define:*** $ Cloud(X)=W \iff  \operatorname {Unit}(W) \land \\ \forall l \ p^* \ W \exists a \ p \ X  \exists b \ p \ X: a \overset {l}- b    \\ \land \forall a \ p^* \ X \, \forall b \ p^* \ X [a \neq b \implies \exists l \ p \ W: a \overset{l} -  b]  $

***Define (Set-hood):***  $\begin{align}Set(X) \iff \exists Y: & (Y \text { is fusion of Clouds } \lor Y=\varnothing) \land \\ & X=Cloud(Y) \end{align} $

***Define (Set Membership):*** : $Y \in X \iff \\Set(X) \land \exists Z: X=Cloud(Z) \land Y \ p \ Z \land \operatorname {Unit} (Y)$

**Non-links:** $\exists x: atom(x) \land \neg \exists a \exists b: a \overset{x} - b$ 

$\varnothing$ is the complement of the fusion of all Clouds. 

The empty set is $Cloud(\varnothing)$

A class is more prudently defined as a fusion of Units. Membership in a class is defined as being a Unit that is a part of that class.

So with the exception of singleton classes of sets, which are equal to their sole elements which are sets, all other classes are not sets. Also there is no empty class.

This depiction of set/class and their membership, accommodates virtually all kinds of set/class theories.

One nice interpretation is to stipulate that the totality of all non-linking atoms is a Unit, and that all otherwise Units are Clouds; by then we can redefine sets and their membership as:

***Define (Set-hood):***  $Set(X) \iff \operatorname {Unit}(X) $

***Define (Set Membership):*** : $Y \in X \iff \\\operatorname {Unit} (X) \land \exists Z: X=Cloud(Z) \land Y \ p \ Z \land \operatorname {Unit} (Y)$

This way the empty set becomes $\varnothing$ itself.

Now to make a system that can interpret $\sf ZFC$,  we need to add **size criterions**. We can coin an **Axiom of Size**:

If Units of some object can be put in one-to-one relationship with parts of a Unit, then the fusion of those Units has a Cloud.

If we add another size criterion that of **Infinity**,  i.e, there is a Unit composed of infinitely many atoms. Then we get to interpret $\sf ZFC$.

So, in some sense this theory does reduce the standard Set Theory to this kind of **Linked-Connective-Mereology + Size criterions**. 

This in some sense supports David Lewis's notion of [Mathematics is Megethology][2]. In nutshell, that mathematics reduces to Set Theory which with the aid of some Mereology would be reducible to some size notions. However, here we delve more into the nature of the Singleton Function of Lewis, which is the Cloud function here.

Note: $p$ stands for parthood relation, $\mathcal O$ for Overlap relation, $\operatorname {fusion}$ and $atom$ stand  for Mereological fusion and Mereological atom.

  [1]: https://plato.stanford.edu/ENTRIES/mereology/#UnrCom

  [2]: https://andrewmbailey.com/dkl/Mathematics_is_Megethology.pdf