FOM: interpreting (R,+,x) in (R^2,E); interpretability in general

[Stephen G Simpson]

 > From: Harvey Friedman <friedman at math.ohio-state.edu>
 > Subject: FOM: More Axiomatization of Geometry
 > Date: Mon, 1 Feb 1999 04:52:48 +0100
 > 
   ...
 > 
 > THEOREM 2. (R,0,1,+,x) is interpretable in (R^2,0,1,i,E) and vice
 > versa.

The presence of 0,1,i raises some additional questions.  What happens
if we drop i?  1,i?  0,1,i?  My feeling is that (R,0,1,+,x) should be
interpretable in (R^2,E) in *some* appropriate sense, but we need a
broader notion of "interpretable".  What is the right notion of
interpretability here?

This rang a bell with me because I recently had occasion to teach
predicate calculus to some advanced math students.  I tried to do
things at a somewhat higher or more general level than usual.  I
included such topics as many-sorted predicate calculus, conservative
extensions, extension by definitions, etc, and I ran into the question
of how to define interpretability in an appropriately general way.
This seems important.

-- Steve