# Correction: [FOM] Logically Full Structures and Theories

John T. Baldwin jbaldwin at uic.edu
Sat Jul 5 09:33:21 EDT 2003

```I correct one point in my previous note.

John T. Baldwin wrote:

> 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..

correction: Mekler's theorem (see A3 of Hodges) asserts every structure
is interpretable in SOME nil-2 group.  This is considerably weaker than
asserting
nil-2 groups are logically full since
Harvey's notion of a theory T  being logical full  requires every
sentence  interpretable in every model of T.

A group is nil-2 if both the commutator subgroup G' of G is Abelian and
G/G is abelian.  An alternative defnition is that the
upper central series has length 2.   On the one hand these are the
non-abelian groups that are closed to be Abelian; on the
other hand they can represent a wide range of phenomena.  It is the
second aspect that movtivated Mekler's theorem.  In

> 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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>
>
>
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