Does Godel's incompleteness theorem admit a converse? Let me set up a strawman:
One might entertain the following criticism of Godel's incompleteness theorem:
why did we ever expect completeness for the theory of PA or ZF in the first place?
Sure, one can devise complete theories semantically (taking all the statements that hold 
in some fixed model), but one has usually discovered something special (e.g. elimination of
quantifiers) when a naturally framed theory just turns out complete.
Now perhaps one could defend Godel's theorem as follows:
By Godel, the theory of the standard natural numbers has no recursive axiomization, but equally remarkably PA has no recursive non-standard models (Tennenbaum's theorem).  That means that the incompleteness of arithmetic has a deeper character than, say, the incompleteness of group theory -- there exhibiting groups with distinct first-order properties easily suffices.
My question:
Does there exist any sort of converse to Godel's incompleteness theorem.  A converse might say that when one has incompleteness and also some reasonable side condition (I'm suggesting but not committed to "there exists only one recursive model"), then there must exist some self-reference mechanism causing the incompleteness? Or stronger perhaps, the theory must offer an interpretation of some sufficiently strong theory of arithmetic?
 A: I will attempt to answer the first question, which I think is more philosophy than math. Consider the integers. We have strong intuitions about the integers being a very definite set of things, which we all have access to. Furthermore, if we accept a naive second order axiomization, the integers are the only thing this axiomization describes. This should survive formalization: after all we only want to describe this one model that Peano arithmetic seems to describe perfectly.
Obviously what I said above is only intuition, and is in fact hopeless. But I am willing to use second order theories to save arithmetical consistency. Of course, this has a price: describing models is hard, logic on second order theories loses a lot of nice features, but if you are willing to sacrifice an entire branch of mathematics in your worldview, you can somewhat avoid having to consider weird models of the integers.
A: A theory containing the axiom of arithmetics without the multiplication operator is complete. In some sense, a recursive theory containing the axiom of arithmetics (with the multiplication operator) may be considered as the minimum that is required for incompleteness. You ask whether any incomplete theory with one recursive model has to have self-reference property. Well it seems that we can embed this theory into the recursive theory of PA. Recursive theory that contains PA with PM as its inference rule is the minimum required for achieving incompleteness. 
