Hi Abram, welcome.

On 24 Nov 2008, at 17:52, Abram Demski wrote (in part):

> The little puzzle is this: Godel's theorem tells us that any
> sufficiently strong logic does not have a complete set of deduction
> rules; the axioms will fail to capture all truths about the logical
> entities we're trying to talk about. But if these entities cannot be
> completely axiomized, then in what sense are they well-defined? How is
> logic logical, if it is doomed to be incompletely specified? One way
> out here is to say that numbers (which happen to be the logical
> entities that Godel showed were doomed to incompleteness, though of
> course the incompleteness theorem has since been generalized to other
> domains) really are incompletely specified: the axioms are incomplete
> in that they fail to prove every sentence about numbers either true or
> false, but they are complete in that the ones they miss are in some
> sense actually not specified by our notion of number. I don't like
> this answer, because it is equivalent to saying that the halting
> problem really has no answer in the cases where it is undecidable.



I am not sure I follow you here. All what Godel's incompleteness  
proves is that no machine, or no axiomatisable theory can solve all  
halting problems.
The undecidability is always relative to such or such theory or  
machine prover. For self-modifying theorem prover, the undecidable  
sentence can evolve.  (extensionaly, and yet remain the same  
intensionally)

For such machine the self-stopping problem become "absolutely-yet- 
relatively-to-them" undecidable.
Actually I am very happy with this, because , assuming comp, this  
could explain why humans fight on this question since so long. And we  
can bet it is not finished!

Tarski 's theorem is even more "religious", in the computationalist  
setting. It means that the concept of truth (about a machine) acts  
already like a "god" for that machine. No (sound) machine can givee a  
name to its own proof predicate.

See my paper on the arithmetical interpretation of Plotinus to know  
more. But the main reason of that paper is the failing of Aristotle  
materialism to address the mind-body problem. This is what we talk in  
the MGA thread in case you want catch the train. You can see my url  
for the papers if interested in the foundation of physics and mind.

Best,

Bruno


http://iridia.ulb.ac.be/~marchal/




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