Correction: [FOM] Logically Full Structures and Theories

[Harvey Friedman]

Reply to Baldwin 8:33AM 7/5/03.

>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

Here is a conjecture. I normally try to prove or refute my 
conjectures before I disseminate them, but I am just too pressed for 
time, and have to compromise from time to time.

Let us call a (commutative) ring (with unit) Diophantine Correct, or 
DC, if and only if every every polynomial with integer coefficients 
that has a zero in the ring has a zero in the actual integers.

The DC rings are obviously just the models of an extremely simple 
theory - though the set of axioms is complete Pi-0-1.

CONJECTURE. The DC axioms are complete. I.e., every consistent 
sentence has a model definable in every DC ring.

If the the conjecture is too strong, there are some obvious 
weakenings. E.g., use ORDERED rings, etc.

Are the DC rings interesting?