TITLE: Are computable models sufficient?
QUESTION [7 upvotes]: What I mean is this. By downward Lowenheim-Skolem theorem, first-order formula Q is a always true iff it is true in every countable structure. But is there some first-order formula Q which is true in every computable structure and false in some non-computable structure? My feeling is that of course the answer should be "yes", but I can't construct an example. I feel also that the questions of the sort has been widely studied. (For example, maybe some study of conditions under which first-order theory has computable model or hasn't). Do you know anything about that?
Thanks in advance.

REPLY [5 votes]: Carl's answer is correct. 
There is also a to more direct way to achieve the same thing. The Tennenbaum's theorem holds for much weaker theories, e.g. $I\Delta_0$ (even for far weaker theories like $IOpen$ plus some number theoretic principles, but not for $IOpen$, a result due to Shepherdson). 
$I\Delta_0+exp$ is finitely axiomatizable, see Haim Gaifman and Constantine Dimitracopoulos, "Fragments of Peano's Arithmetic and the MRDP Theorem". It is also a sub-theory of $PA$.
For more on Tennenbaum's theorem and weak arithmetics, have a look at this paper: 
Shahram Mohsenipour, "Hierarchies of Subsystems of Weak Arithmetic", to appear 
in "Set theory, Arithmetic, Philosophy: Essays in Memory of Stanley Tennenbaum" (edited by J. Kennedy and R. Kossak), Cambridge University Press.