Lewis's approach to Mereological foundation of set theory, is very interesting by itself. The following shows that it can indeed provide an interpretation for Ackermann's set theory!

In brief let our base theory be Atomic General Extensional Mereology "AGEM". Add to it a binary relation symbol $\mathcal L$ to stand for "is a label of", on top of part-hood $\mathcal P$ and equality $=$, binary relations.

If we add axiom of "Bottom" to the underlying Mereology (which Lewis refuse based on non-technical grounds), then technically we can axiomatize the "bottom" atom $\emptyset$ to be a non-labeling atom (so it would stand for the empty class $\emptyset$, see below).

Of course adding Bottom would call us to re-define atom as an object whose only parts are itself and bottom; i.e. an object that do not have a proper part other than bottom.

We set two important axioms about labeling:

  1. Labeling is a bijective partial function.
  2. Labels are non-overlapping

Formally those are:

$\forall a,b,x,y \ (x \ \mathcal L \ a \wedge y \ \mathcal L \ b \to [x=y \leftrightarrow a=b] \wedge \neg \ x \ \mathcal O \ y) $

Where $\mathcal O$ means "overlaps", i.e. Share a common part other than bottom.

A class is defined as in Lewis as a Mereological totality of labels.

$class(x) \iff \forall y (y \ \mathcal O \ x \to \exists z (label(z) \wedge z \ \mathcal P \ x \wedge y \ \mathcal O \ z))$

$label(z)$ means "$z$ is a label", i.e. $\exists k (z \ \mathcal L \ k)$

Class Membership is defined as:

$y \in x \iff class(x) \wedge \exists z (z \ \mathcal P \ x \wedge z \ \mathcal L \ y)$

In English: $y$ is a member of $x$ if and only $x$ is a class and there is a part of $x$ that labels $y$.

Now a set is a defined as a class that has a label. Formally:

$set(x) \iff class(x) \wedge \exists l (l \ \mathcal L \ x)$

A class is said to be $nice$ if and only if all of its labeling parts are atoms; i.e. its a mereological totality of labeling atoms. Formally:

$nice(x) \iff class(x) \wedge \forall y \ \mathcal P \ x [label(y) \to atom(y)]$

A nice set is a nice class that is labeled by an atom.

Now we only need ONE scheme to interpret all axioms of Ackermann's set theory that is:

If $\psi$ is a formula in which all and only symbols $``y,x_1,..,x_n"$ occur free, that can only use symbols of $``= , \in , class"$, as predicate symbols, then: $$\forall \text {nice sets } x_1,..,x_n \\ [\forall y (\psi \to nice(y)) \to \forall y (\psi \to \text{ nice set} (y))]$$; is an axiom.

This would interpret Ackermann's set theory over the realm of "pure" nice sets, i.e. nice sets that are $\in$-hereditarily nice.

I see this interpretation of Ackermann's set theory in Lewis like approach to Mereological foundation of set theory, very interesting.

To be noted is that this theory proves the consistency of Ackermann's set theory, so it is stronger than ZFC.

Question: Is there is clear inconsistency with this theory?

Question: had there been prior work on that specific line of approach?


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