[FOM] Logically Full Structures and Theories

[John T. Baldwin]

I copy below Harvey's note on logically full theories.  


The notion of logically full does seem to me to  be of model theoretic 
interest.  A closely related notion
is studied in 5.5 of Hodges, Model theory entitled: Theories that 
interpret anything
See also appendix A3 of the same book which shows the class of nil-2 
groups is (if I am reading right)
logically full..  

Certainly a characterization of this class would be interesting.

I can't resist remarking that while Harvey came at this from 
considerations involving the set theoretic structure of the universe of
models (which I hold to be non-model theoretic), this aspect is missing 
from the new project which is classical model theory.

Harvey Friedman wrote:

> 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
>
>
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