FOM: axioms of infinity

[V. Yu. Shavrukov]

This is to point out that the conjecture

>any finitely axiomatizable theory with only infinite models must
>interpret one of a small finite number of such theories

appearing in a posting of Stephen Simpson is untrue under the interpretation
of "to interpret" as the usual relative interpretability as found in e.g.
Shoenfield's book, as well as under a few (but probably not all)
generalizations thereof.

A quote from Simpson:

>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.  [...]

(1) Any sentence interpreting an axiom of infinity is either inconsistent
or is itself an axiom of infinity.

(2) Suppose the conjecture were true with S the finite small set
of interpretability-minimal axioms of infinity.
Then in view of (1)

{sentences interpreting some element of S} =
         = {axioms of infinity plus inconsistent sentences}.

(3) By Trakhtenbrot, the r.h.s. of (2) cannot be r.e. as it
complements the set {sentences true in some finite model},
which is a member of an inseparable r.e. pair.

(4) The l.h.s. of (2) on the other hand is r.e. because
all you need to interpret a single sentence is a translation
and a finite number of proofs.

Contradiction.


regards,
V.Shavrukov