FOM: interpreting (R,+,x) in (R^2,E); interpretability in general
Stephen G Simpson
simpson at math.psu.edu
Thu Feb 11 11:38:03 EST 1999
> 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
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