Hi Bruno,
> 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)
I agree with everything you say there, so I'm not sure where you
aren't following me. I definitely agree with the idea of a
self-modifying theorem prover that becomes stronger over time-- I
think it is the right model. What I am saying in the paragraph you
quoted is that one "way out" is to claim that when we add axioms to
strengthen our system, we can choose either the axiom or the negation
arbitrarily, since either is consistent with the system so far. I've
argued with people who explicitly claim this. My opinion is that there
is only one correct choice for each addition. In the case of the
halting problem, we want to reflect the actual truth about halting; in
the (equivalent) domain of undecidable numerical statements, we still
want the actual truth.
Also, I should mention that the arithmetical hierarchy shows that some
problems are "more undecidable" than halting: if we had a halting
oracle, we would still be unable to decide those problems.
Schmidhuber's super-omegas are a perfect example.
http://www.idsia.ch/~juergen/kolmogorov.html
But you probably knew that already.
>
> 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!
This argument is given in longer form elsewhere? Perhaps that paper
you mention later on?
>
> 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.
To extend the model of the self-modifying theorem prover... the
scenario I use when thinking about truth is a population of entities
which, among other things, need to reason properly about each other.
(I could perhaps reduce this to one entity that, among other things,
needs to reason about itself; but that is needlessly recursive.) The
logic that these entities use evolves over time. In any given
generation, the entity who can represent the truth-predicate of the
most other entities will dominate. The question, then, is: what logic
will the population eventually converge to?
I think fair arguments could be given for the fixed-point or revision
theories in this scenario, but like I said, both create reference
gaps... so some creature could dominate over these by inventing a
predicate to fill the gap. That creature will then have its own
reference gaps, and yet more gap-filling predicates will be created.
My current thinking is that each gap-filling predicate will correspond
to an ordinal number, so that the maximal logic will be one with a
gap-filling predicate for each ordinal number. No gap will be left,
because if there was such a gap, it would correspond to an ordinal
number larger than all ordinal numbers, which is impossible. Of
course, this gives rise to an outlandish number of truth-values (one
for each ordinal number), when normally any more than 2 is considered
questionable.
--Abram Demski
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Received on Mon Nov 24 2008 - 15:52:44 PST