On 24 Nov 2008, at 21:52, Abram Demski wrote:
>
> 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.
I agree, for a category of pure machine, not yet confronted to some
"bigger or older" universal machine.
> 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.
Well some of them, like you, equivalent or bigger machine, will never
know, unless you are infinitely patient.
Insolubility here is hard, but it makes the mathematical world
necessarily ever surprising.
You can hope for a theory of everything, which preserves the mystery,
which makes it even more mysterious.
>
>
> 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.
That is why the computationalist hypothesis is non trivial. It gives a
prominent role to Sigma_1 completeness. The oracle can still play an
important role "from inside", but even this is not sure. (and then
comp gives importance to sigma_1 completeness relatively to an oracle).
>
>
> 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?
I have not publish this. I explain it informally in my french thesis
long version. But it is obvious I think, especially assuming comp. It
is implicit in my Plotinus paper (still on my first page of my url I
think).
Finite machine are limited. But finite machine which believes in the
induction axioms can know that they are limited, and can build
theories explaining those limitation. Formally this gives autonomous
progression. The correct one have to converge to some "non convergence
possible" in the horizon ...
>
>
>>
>> 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.
Relatively to some universal machine. I see subtle difficulties I
don't want to bore you with.
> In any given
> generation, the entity who can represent the truth-predicate of the
> most other entities will dominate.
Why?
> The question, then, is: what logic
> will the population eventually converge to?
If the entities believes in classical logic, and if they believe in
induction, they will converge toward the self-reference logic of
Solovay (G and G*, or GL and GLS nowadays).
>
>
> 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.
I think so.
>
>
> 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.
Not really, because those truth value are, imo, not really truth
value, but they quantify a ladder toward infinite credibility,
assurance or something. Perhaps security.
Still, you can do this constructively, and provably so, only up to a
constructive ordinal. In that case, your entire population will
develop a "last" large reference gap. Or ... your population will
"live" forever with "little gaps", and will converge toward the first
non constructive ordinal, but nobody, well no machines, will ever know
that.
Yet, from their first person point of view, it is different. From
their first person point of view they fill all gaps immediately.
Perfectly sound machines will identify truth and proof, consistently
so, and fill the gap. Only seen from outside, for not too much complex
machine, *we* can see they are "wrong" ... from outside, and correct
from inside.
This entails that "conscious machine" will have hard time to believe
that they are machine.
It is a bit like Skolem paradox. it depends how we will interpret the
ordinal. this can lead to very difficult questions, and the
hierarchies you mentionned can be useful. To be sure this could be too
technical for the list, where we discuss very basic fundamental
question, and no so many people knows the technic in logic.
Actually, we are discussing an argument that IF we are machine
(whatever we are, except the physical universe), then the physical
universe is not entirely computable. The physical universe cannot be
the output of a program, nor really directly generated by a program.
It has to emerge statistically from all possible computation going
through possible observers. It is generated, but only implicitly, by
the running of a universal dovetailer. A program whose existence
relies on Church thesis, which generates and executes all programs by
zigazaguing on all their executions. It has to zigzag because we are
not able, and never will be, to predict in advance which programs
stops or doesn't stop.
My general feeling is that incompleteness is the guaranty of
seriousness of the mechanist hypothesis in the cognitive science, and
even in the physical science. The lesson of Gödel's first
incompleteness theorem is a lesson in modesty (provably so assuming
comp). The lesson of Gödel's second incompleteness theorem is a lesson
of wonder: all self-introspecting universal machine can learn the
modesty lesson!
Incompleteness is also what makes Church thesis consistent (see the
first footnote of the Plotinus paper). This has no price, because it
makes computability the first absolute mathematical epistemological
notion. We don't have this for the notion of provability, nor of
knowledge, still less of provability. But we have modal logic which
are invariant for their interpretation thanks to Gödel, Löb and
Solovay. Do you know the provability logics?
Bruno Marchal
http://iridia.ulb.ac.be/~marchal/
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Received on Mon Nov 24 2008 - 17:03:57 PST