FOM: Axioms of Infinity (& Help Request)

[John Baldwin]

I want to consider Hazen's remarks on `second order interpretability'
as a possible answer to my question about stronger notions of
interpretablity that might reflect sentences with the same underlying
principle forcing infinite models.

In the meantime, I will connect below some of Hazen's remarks with
the model theoretic literature.

On Sun, 23 Jul 2000, Allen Hazen wrote:

> 	Consider the following two First Order AoI:
> 	(1) R is a discrete linear order with bottom but no top
> 	(2) S is like the successor relation in arithmetic: Every object is
> S-related to a unique object, there is an object to which nothing is
> S-related (i.e. zero), and two objects S-related to the same object are
> identical.

Hazen points out that 1) is interpretable in 2) but suggests the converse
is false via the Gaifman locality theorem.

A simple way to see that 2) is not interpretable into 1 is:

a) 2) is unstable
b) instability is preserved by interpretation.
c) 1) is aleph-1 categorical and hence omega stable.
d) so 2 is not interpretable into 1).


Hazen continues:

 
> 	Now the HELP REQUEST.
> 
> 	Models of (2) have the feature that no object in their domain is
> related by the primitive relation of the theory (in either direction) to
> more than finitely many others (in fact, to more than two).  In contrast,
> an object in the domain of a model of (1) is R-related to infinitely many
> others.  Most foundationally interesting theories are, in this regard, more
> like (1): every natural number forms sums with all other natural numbers
> (so every object in a model of typical arithmetic theories is related to
> infinitely many objects by the ternary addition relation), every set in
> models of standard set theories belongs to infinitely many others, etc.  I
> think the property of (2) just isolated (I propose to call it "Local
> Finiteness," or, if that term has a different established meaning, "Finite
> Neighborhoodedness") ought to be of some interest.  It doesn't seem to make
> First Order theories trivial: I think I've found an undecidable example
> that "interprets" (though not in the standard sense) Pi-0-1 arithmetic.
> Can someone give me references to papers discussing this property?
> 
You are slicing the world in a different way than is standard in model
theory so the translation is not precise.  But here are several
comments.

As I said above 1) is (very) stable  2) is quintessentially unstable.

Among the very nice stable structures, the best are strongly minimal
sets.  Every definable set is finite or cofinite. (Like 1).

On strongly minimal sets algebraic closure defines a notion of algebraic
dependence (like vector spaces).  In a depends on b,c implies a depends
on b or c, the relation is said to be trivial.  (like 1) again.

algebraic closure is not well behaved in 2).


References: the many books on stability theory.

A connection between Gaifman's work and strongly minimal sets is
in the thesis last year of Shawn Hedman, L^n axiomatizability of
strongly minimal sets.
He is currently at: hedman at math.umd.edu

A further relevant reference:

Finite axiomatizability and theories with trival algebraic closure,
Dugald Macpherson, Notre Dame Journal of Formal Logic,
1991, pages 188-190

He proves every finitely axiomatized complete theory with trivial
algebraic closure (see my last reply to Hazen) has the strict order
property.

roughly: strict order property means some phi(xbar, ybar) defines
a partial ordering on n-tuples with a infinite totally ordered subset.



John Baldwin
Math Stat Computer Science
University of Illinois at Chicago