[FOM] Nonempty Finite Interval Mereology
a.hazen at philosophy.unimelb.edu.au
Sat Nov 19 01:46:16 EST 2005
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.
University of Melbourne
More information about the FOM