Relationship Between Incompleteness and The Halting Problem

Yesterday, while googling around for information on hyperoperations, I came across Scott Aaronson’s essay, “Who can name the bigger number?” You should go read it. I’ll wait.

On the halting problem:

The proof is a beautiful example of self-reference. It formalizes an old argument about why you can never have perfect introspection: because if you could, then you could determine what you were going to do ten seconds from now, and then do something else. Turing imagined that there was a special machine that could solve the Halting Problem. Then he showed how we could have this machine analyze itself, in such a way that it has to halt if it runs forever, and run forever if it halts. Like a hound that finally catches its tail and devours itself, the mythical machine vanishes in a fury of contradiction.

What if we could solve the halting problem? On hypercomputation:

Suppose we could endow a Turing machine with a magical ability to solve the Halting Problem. What would we get? We’d get a ‘super Turing machine’: one with abilities beyond those of any ordinary machine. But now, how hard is it to decide whether a super machine halts? Hmm. It turns out that not even super machines can solve this ‘super Halting Problem’, for the same reason that ordinary machines can’t solve the ordinary Halting Problem. To solve the Halting Problem for super machines, we’d need an even more powerful machine: a ‘super duper machine.’ And to solve the Halting Problem for super duper machines, we’d need a ‘super duper pooper machine.’ And so on endlessly.

Wikipedia’s hypercomputation page is silent on the super Halting Problem. Is this the work of the hypercomputation Illuminati, trying to suppress an inconvenient truth?

Then it occurred to me: the structure of the super Halting Problem is virtually identical to the “Contracrostipunctus” dialogue in [Gödel, Escher, Bach] (very recommended, buy a copy), in which the Tortoise tells Achilles about his collection of phonograph-destroying records. The super Halting Problem can be thought of as a record that cannot be played on a super Turing machine, which makes one wonder: what is the relationship between the incompleteness theorems and the halting problem?

My understanding of it is this: Gödel’s incompleteness theorems state that for any sufficiently powerful formal system, there are assertions that cannot be proven either true or false. Concretely, perhaps all the recent work on the twin prime conjecture has been in vain and it is unprovable. One wonders, then: given some proposition, can you decide in advance whether or not it is provable? If we could decide in advance, we could avoid wasting a whole lot of time trying to solve something that cannot be solved.

At this point, we need to step back and consider just how one goes about proving something. You start with a proposition, carry out some operations, and then voila!, it is proved. The difference between a provable statement and an unprovable statement is the number of steps we have to carry out in order to prove it. If a proposition is provable, it can be proven in a finite number of steps. If not, you can use as many steps as you like and you will never prove it.

This is the halting problem!

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