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I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being negative results about decidability and established by diagonal arguments (and in the 1930's), so they must somehow be two ways to view the same matters. And I thought that Turing used a universal Turing machine to show that the halting problem is unsolvable. (See also this math.SE question.)

But now that (teaching a course in computability) I look closer into these matters, I am rather bewildered by what I find. So I would like some help with straightening out my thoughts. I realise that on one hand Gödel's diagonal argument is very subtle: it needs a lot of work to construct an arithmetic statement that can be interpreted as saying something about it's own derivability. On the other hand the proof of the undecidability of the halting problem I found here is extremely simple, and doesn't even explicitly mention Turing machines, let alone the existence of universal Turing machines.

A practical question about universal Turing machines is whether it is of any importance that the alphabet of a universal Turing machine be the same as that of the Turing machines that it simulates. I thought that would be necessary in order to concoct a proper diagonal argument (having the machine simulate itself), but I haven't found any attention to this question in the bewildering collection of descriptions of universal machines that I found on the net. If not for the halting problem, are universal Turing machines useful in any diagonal argument?

Finally I am confused by this further section of the same WP article, which says that a weaker form of Gödel's incompleteness follows from the halting problem: "a complete, consistent and sound axiomatisation of all statements about natural numbers is unachievable" where "sound" is supposed to be the weakening. I know a theory is consistent if one cannot derive a contradiction, and a complete theory about natural numbers would seem to mean that all true statements about natural numbers can be derived in it; I know Gödel says such a theory does not exist, but I fail to see how such a hypothetical beast could possibly fail to be sound, i.e., also derive statements which are false for the natural numbers: the negation of such a statement would be true, and therefore by completeness also derivable, which would contradict consistency.

I would appreciate any clarification on one of these points.

You have one conceptual problem: algorithmic decidability (Halting problem) and derivability resp. provability (logics) are two very different concepts; you seem to use "decidability" for both. – Raphael – 2012-03-15T18:10:47.357

yes indeed the two proofs are conceptually extremely similar and in fact one way to look at it is that Godel constructed a sort of turing-complete logic in arithmetic. there are many books that point out this conceptual equivalence. eg Godel Escher Bach by hofstadter or Emperors New Mind by penrose.... – vzn – 2012-10-11T16:19:51.807

1@Raphael: I am very well aware that there is a large conceptual difference between the statements of incompleteness theorem and of the undecidability of the halting problem. However the negative form of incompleteness: a sufficiently powerful formal system cannot be both consistent and complete, does translate into an indecidability statement: since the set of theorems deducible in a formal system is semi-decidable by construction, completeness would make the set of non-theorems semi-decidable as well (as negations of theorems, assuming consistency, or else as the empty set), hence decidable. – Marc van Leeuwen – 2012-03-17T18:04:56.457

I see, thanks for clarifying. – Raphael – 2012-03-18T00:18:35.127