# FOM: axioms of infinity

John Baldwin jbaldwin at math.uic.edu
Thu Jul 20 14:58:11 EDT 2000

```Here a few more comments on the axiom of infinity issue.

Steve Simpson raised the question of whether there `up to
interpretability'
only a finite number of axioms of infinity.  This formalizes a bit
the informal remark I attributed to Tait in my earlier note; that
attribution was based on conversations of 25 years ago.

Thanks to Shavrukov for a clear answer to Steve's question.  I haven't
replied earlier because I can't really get a clear question formulated.
Here are a few reflections (and some examples).

1.  The relevance of Hanf/Peretyatikin:  I may be
misunderstanding something here but I believe Peretyatkin
has proved that for any recursively axiomatized
aleph_1 -categorical theory there is a finitely axiomatized
aleph_1 - categorical theory with the same turing degree
(and indeed further similarities.)  Depending upon
what notion of interpretation Steve has in mind, this
seems to me to mean the classification problem is the same for
recursively axiomatized theories as for finitely
axiomatized one.

What is the effect of a switch from finite to recursive on the
philosophical
significance of the question?

2.  Note that if we consider languages with one n-ary function f_n and
n unary function  p_i so that
f_n(p_1(x),  ... p_n(x)) =x
p_i(f_n(x_1,    ... x_n) = x_i

we have a family of different  (i.e. an explicit set of weaker theories
under interpretability ) but which all depend on the same (pairing
function)
idea.  So by a weaker notion of interpretability I mean one which counts
these as all the same.

(One could also tighten by insisting on a fixed finite language.)

3.  I agree that there is a progressive weakening of the general interest
of the question as one progresses from an arbitrary sentence to one
that is complete, to categoricity in power.  A point I tried to make
in my first post was that when the question became - categorical in
all powers, then a) there is now no sentence b) the area developed
into a new subject uniting permuation group theory and model theory
without any real claim to either side.

I think the general view of model theorists has been that the real
importance of this question has been as a draw to develop structure
theorems which would have otherwise not been thought about.

4.  There are several variants on these problems existing in the
model theory literature?

Must every totally categorical sentence with only infinite models
have the strict order property (i.e. linearly order an infinite
set of n-tuples)?  (I think this is due to McPherson.)

Is there a simple aleph-0 categorical finitely axiomatizable theory.
(Kim-Pillay)? This is open even for stable theories.  I recently refuted
the only live possibility for a counterexample to the stable case
by finding
that that theory was the almost sure theory for a probability measure on
finite
models.  (www.math.uic.edu/~jbaldwin   Probability and the Finite
Model Property).

```