FOM: axioms of infinity
V. Yu. Shavrukov
vys1 at mcs.le.ac.uk
Thu Jul 20 04:13:58 EDT 2000
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
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