[FOM] Nonempty Finite Interval Mereology

[Harvey Friedman]

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.

One can also consider the quasi order of all intervals of finite nonzero
length in the real line under the relation of almost inclusion (say,
inclusion among their interiors).

I claim that the above poset has a nice complete finite set of axioms. It
doesn't make any difference whether equality is used since it is defined as
A <= B and B <= A. 

I also claim that the above quasi order has a nice complete finite set of
axioms - the same as for the above partial order if we do not use equality.
We can also use equality, and get a nice complete finite set of axioms. But
here equality is not definable.

I won't go further into this on the FOM if someone can provide a reference
for these results. Are they known?

Harvey Friedman