FOM: axioms of infinity
Stephen G Simpson
simpson at math.psu.edu
Mon Jul 17 17:04:10 EDT 2000
Alasdair Urquhart Wed, 28 Jun 2000 16:10:54 -0400, referring to my
message of Wed, 28 Jun 2000 13:16:53 -0400 (EDT), writes:
> [...] I haven't been able to find any kind of result such as that
> conjectured by Steve to the effect that "any finitely axiomatizable
> theory with only infinite models must interpret one of a small
> finite number of such theories." Somehow, this picture seems too
> optimistic to me, although it does not appear to be completely
> ruled out by Trachtenbrot's result. Does anyone know anything
> about this?
This discussion involving Baldwin (21 Jun 2000), me (28 Jun 2000),
Urquhart (28 Jun 2000), and Hazen (6 Jul 2000) is getting interesting.
Like Urquhart, I feel that the above "conjecture" (actually I never
officially conjectured it!) is probably false, but I can't seem to
The best I can do right now is the following.
Define an *axiom of infinity* to be a consistent sentence of
first-order predicate calculus which has no finite model. Let AxInf
be the set of axioms of infinity. It follows from Trakhtenbrot's
Theorem (or perhaps, a refinement of it) that AxInf is productive in
the sense of Post. This implies that, given (an r.e. index of) an
r.e. subset S of AxInf, we can effectively find a member of AxInf
which is not logically equivalent to any member of S.
More information about the FOM