[FOM] Questions on axiomatizations of complete theories

[joeshipman@aol.com]

In the following, T always refers to a complete theory which is 
recursively axiomatizable (hence decidable) but not finitely 
axiomatizable.

Say that an axiomatization A = {A1, A2, ...} of T is "tight" if every 
axiom in it is logically independent of the rest.

Say that A = {A1, A2, ...} is "loose" if any infinite subset of A 
implies all the axioms in A.

These are two extremes. Every axiomatizable theory has a "loose" 
axiomatization: convert {A1, A2, A3, ... } to {A1, A1&A2, A1&A2&A3, 
...}.

Can anyone find nontrivial examples of T (preferably theories that are 
already considered interesting) such that

1) No recursive axiomatization of T is tight?

2) Every recursive axiomatization of T contains a recursive "loose" 
subset?

-- JS