[FOM] Tight and Loose Axiomatizations; Axiomatic Dimension

[JoeShipman@aol.com]

Let T be a complete, recursively axiomatizable  theory which is not finitely 
axiomatizable, and has an axiom of infinity (some  sentence in T has no finite 
models).

Let A={A1,A2,...} be a (not  necessarily recursive) axiomatization of T.

Say A is "tight" if each  axiom in it is independent of all the others. If 
such an A exists, say T is  "tightly axiomatizable".  If A is also recursive, 
then say T is "tightly  recursively axiomatizable".

Say A is "loose" if any infinite subset of A  implies A. 
(The reason we assumed T has an axiom of infinity is to rule out  trivial 
"loose" axiomatizations which simply exclude larger and larger finite  models).

Say A is "essentially loose" if it has a finite subset  {B1,...,Bn} and an 
infinite subset {C1,C2,...} such that  {B1&...&Bn&C1,B1&...&Bn&C2,...} is  loose.

Example: Let T1 be the theory of Algebraically Closed Fields of  
Characteristic 0. T1 is tightly recursively axiomatizable. If AF is the  conjunction of 
the field axioms, Cn is the sentence saying 1 added to itself n  times is 0, and 
Dn is the sentence saying all polynomials of degree n have  roots, then {AF, 
~C2, D2, ~C3, D3, ~C5, D5, ~C7, D7, ~C11, D11, ...} is tight  (Shipman 2005).

Example: Let T2 be the theory of Real Closed Fields in  the language of 
fields without "<". There is an axiomatization which includes  the sentences "-1 is 
not a sum of n squares" for each n, the sentences "Every  polynomial of 
degree 2n+1 has a root" for each n, and the sentence AF. This  axiomatization is 
not tight, nor is any subset tight, nor is it loose, nor is it  essentially 
loose. Note that T2 does have an axiom of infinity, since "AF &  'Every polynomial 
of degree n has a root' " is an axiom of infinity for all  n>1.

Example: Let T3 be the Theory of Real Closed Fields in the  language of 
fields with "<". We can loosely axiomatize this {AOF2&D3,  AOF2&D5, AOF2&D7, 
AOF2&D9, ....} where AOF2 is the conjunction of  the axioms for ordered fields with 
the sentence that every positive element has  a square root, and Dn for odd n 
is the sentence that every degree-n polynomial  has a root.

The key difference between T2 and T3 is that in T2 we had two  distinct 
phenomena ("realness" and "completeness") which each required  infinitely many 
axioms to represent, while in T3 the addition of the order  relation symbol allows 
"realness" to be finitely axiomatized.

Question:  Is every axiomatization of T3 essentially loose? If not, can you 
come up with a  nontrivial example of a theory for which every axiomatization 
is essentially  loose? (Remember that we assume T has an axiom of infinity to 
avoid trivial  examples.)

There seems to be a notion of dimension underlying this -- in  some sense, T1 
has dimension infinity, T2 has dimension 2, and T3 has dimension  1. 

We could formalize this by saying an axiomatization A has dimension n  if it 
contains disjoint subsets F U B1 U B2 ... U Bn, where F is finite and the  Bi 
are infinite, such that any subset of A which includes F and includes  
infinitely many elements of each Bi implies A. A theory T has dimension n if it  has 
an axiomatization of dimension n but no axiomatization of dimension n+1, and  
dimension infinity if it has axiomatizations of all finite dimensions  >0.

This suggests that a finitely axiomatizable theory has dimension  0; it seems 
a bit odd that a theory of dimension infinity can be given  axiomatizations 
of all degrees >0 but not of degree 0, but maybe no odder  than the fact we can 
map omega^omega into omega^1 but not into a finite  set.

Is this is a well-known model-theoretic concept in a new  guise?

-- Joe Shipman