[FOM] Logically Full Structures and Theories

[Harvey Friedman]

It appears that a nice research direction MAY have emerged out of my 
efforts to reformulate my

"explicit upward Lowenheim Skolem theorem"

in order that Baldwin MIGHT think that it was more model theoretic.

Let M be a relational structure. We say that M is *logically full* if 
and only if every consistent sentence in first order predicate 
calculus with equality has a model that is M definable.

We elaborate: a model is M definable iff its domain and relations and 
functions are all M definable. Thus the domain can be a proper subset 
of the domain of M, and equality must remain equality - although one 
might wish to consider alternatives.

We call a set T of sentences logically full if and only if all models 
M of T are logically full.

The idea is to develop some necessary and some sufficient conditions 
for logical fullness of structures and of theories.

The ring of integers is logically full. The field of rationals is 
logically full. The field of reals and the field of complexes are not 
logically full.

Note that PA plus the true Pi-0-1 sentences is logically full. PA can 
be replaced by weak fragments of PA here.

Is there a particularly mathematically natural theory that is logically full?

Harvey Friedman