FOM: axioms of infinity

Stephen G Simpson simpson at
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
refute it.

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.

-- Steve

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