[FOM] Nonempty Finite Interval Mereology

[A.P. Hazen]

Harvey Friedman writes--
>The kind of mereology being discussed currently on the FOM can be reasonably
>identified, model theoretically, with the partial order of all nonempty
>finite open intervals in the real line under inclusion.

----Mereology usually considers a bit more than the intervals: basic 
axiom of mereology says that  for any non-empty set of "individuals" 
there is an individual which is their fusion.  So, taking nonempty 
finite open intervals as  "individuals" we ought to have also unions 
of individuals (or rather interiors of unions of individuals: regular 
open sets).  The axiomatics of this structure have been studied at 
least since Tarski (whose "Foundations of the Geometry of Solids," 
ch. 2 of his "Logic, Semantics, Metamathematics," takes the set of 
regular open sets in a Euclidean space as a model for "solids," 
interpreting the mereological notion "part  of" by containment).  I 
could be wrong, but my impression is that the real line is a rich 
enough structure to provide counterexamples to any sentence of 
first-order mereology (= first-order language with variables 
construed asranging over "individuals" and "part of" or something 
interdefinable with it as vocabulary) that fails in ANY atomless 
mereology.  (So the theory is the theory of a complete Boolean 
Algebra with thebottom element left off.)
    Cf. also Tarski's 1935 "Foundations of  Boolean Algebra," (= ch 11 
of "LSM") and the finalfootnote added to it in the book.
---
Allen  Hazen
Philosophy Department
University of Melbourne