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## 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?

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The phrase 'causes the incompleteness' is problematic. One way to show that ZFC is incomplete is to use forcing to show that ZFC does not prove CH nor its negation. This is much closer to incompleteness for the theory of groups than Gödel's argument. Furthermore, it is hard to imagine how the independence of CH might be caused by some syntactic self-reference mechanism. – François G. Dorais Oct 22 2011 at 12:48
On the other hand, if you restrict to $\Pi_1$ statements, then I believe the answer is yes since you can encode $\Pi_1$ facts into Con($T$). See mathoverflow.net/questions/67214 This is probably the limit since Con($T$) is a $\Pi_1$ statement for any axiomatizable theory $T$. – François G. Dorais Oct 22 2011 at 12:53
For every undecidable set $X$, there are theories whose decision problem is of the same Turing degree as $X$. Hence one cannot expect that there is a single undecidable problem that lies behind all undecidable theories. – David Harris Oct 22 2011 at 14:13
I like your question, but the motivation seems far-fetched to me. Just to recall the obvious: we believed (or we hoped) that PA and ZF were complete, because Euclidean Geometry is complete (once corrected for the gaps in Euclid's text relative to the real), and was felt to be complete well before it was formally proved by Tarski, and that Eucildean Geometry has been a strong ideal of what a mathematical theory should be. When we felt strong enough to do with mathematics at large what Euclid did with geometry (that is, after Cantor and Frege), we tried... and we failed. We are still crying... – Joël Oct 22 2011 at 15:11
Godel's second incompleteness theorem states that a theory $T$ does not prove its consistency assuming that (1) $T$ is consistent, (2) (the set of Godel codes of) $T$ is recursively enumerable, (3) $T$ contains a large part of the Peano axiom system. Now (1) is obviously needed and (2), (3) are necessary even to formulate the claim, i.e., Con(T), so in this sense, Godel's second incompleteness theorem has a converse. – Péter Komjáth Oct 22 2011 at 20:02