[FOM] Model theory for mereology, without sets?

[Allen Patterson Hazen]

David Nicolas asks about the possibility of developing model theory for
mereology in a mereological framework, without appeal to set theory. 
Without having given it much thought, one problem suggests itself.  Model
theory refers to assorted "finitely structured" entities: thus, for
example, formulas get satisfied (or not, as the case may be) by SEQUENCES
of objects from the domain.  (Usually more convenient to ignore questions
about the length of the sequence and use infinite ones, but Tarski noted
that finite sequences would do.)

Now, such entities are provided by very elementary set theory, but can be
problematic in frameworks that (for whatever philosophical or other
motivation) avoid sets.  No general way of defining even ordered pairs is
available in pure (First-Order) mereology.  The appendix to David Lewis's
"Parts of Classes"  finesses this, but at cost: it uses essentially
SECOND-order mereology (plural quantification on top of mereology), and it
postulates lots and lots and lots of mereological atoms.

Given atomicity, maybe you can do more.  Syntax is naturally atomic:
formulas, etc, are naturally construed as sequences of discrete symbols
which can be treated as atoms.  Quine and Goodman's "Steps toward a
constructive nominalism" (JSL late 1940s) makes some progress toward
developing logical syntax using mereological concepts and assuming symbols
can be treated as atoms.  It does not, however, attempt model theory.

Allen Hazen
Philosophy Department
University of Melbourne