Russell Standish wrote:
> You are right when it comes to the combination of two independent
> systems A and B. What the original poster's idea was a
> self-simulating, or self-aware system. In this case, consider the liar
> type paradox:
>
> I cannot prove this statement
>
> Whilst I cannot prove this statement, I do know it is true, simply
> because if I could prove the statement it would be false.
But doesn't that depend on having adopted rules of inference and values, i.e. "the sentence is either true or false". Why shouldn't we suppose that self-referential sentences can have other values or that your informal reasoning above is a proof and hence there is contradiction.
ISTM that all this weirdness comes about because we have tried to formalize ordinary reasoning into LOGIC, which works wonderfully for many axiom sets and rules of inference, but not for all. That's why mathematicians generally get along just fine without worrying about completeness or the provability of consistency.
Brent Meeker
>
> To know that it is true, I am using self-reference about my own proof
> capabilities.
>
> I don't think anyone yet has managed a self aware formal system,
> although self-reproducing systems have been known since the 1950s, and
> are popularly encountered in the form of computer viruses. There has
> to be some relationship between a self-reproducing system and a
> self-aware system...
I think it would be almost trivial to create and AI system that would recognize the equivalent of "I cannot prove this statement" is true or mu. It would just be a trick of formal logic and would have very little to do with self-awareness.
Brent Meeker
>
> Cheers
>
> On Thu, May 24, 2007 at 09:45:45PM -0400, Jesse Mazer wrote:
>> I definitely don't think the two systems could be complete, since (handwavey
>> argument follows) if you have two theorem-proving algorithms A and B, it's
>> trivial to just create a new algorithm that prints out the theorems that
>> either A or B could print out, and incompleteness should apply to this too.
>>
>> Jesse
>>
>>
>>> From: Russell Standish <lists.domain.name.hidden>
>>> Reply-To: everything-list.domain.name.hidden
>>> To: everything-list.domain.name.hidden
>>> Subject: Re: Overcoming Incompleteness
>>> Date: Thu, 24 May 2007 23:59:23 +1000
>>>
>>>
>>> Sounds plausible that self-aware systems can manage this. I'd like to
>>> see this done as a formal system though, as I have a natural mistrust
>>> of handwaving arguments!
>>>
>>> On Thu, May 24, 2007 at 10:32:29AM -0700, Mohsen Ravanbakhsh wrote:
>>>> Thanks for your patience! , I know that my arguments are somehow
>>>> raw and immature in your view, but I'm just at the beginning.
>>>>
>>>> *S1 can simulate S2, but S1 has no reason to believe whatever S2 says.
>>>> There is no problem.
>>>> **Hofstadter "strange loop" are more related to arithmetical
>>>> self-reference or general fixed point of recursive operator*
>>>>
>>>> OK then it, becomes my own idea!
>>>> Suppose S1 and S2 are the same systems, and both KNOW that the other one
>>> is
>>>> a similar system. Then both have the reason to believe in each others
>>>> statements, with the improvement that the new system is COMPLETE. We've
>>> not
>>>> exploited any more powerful system to overcome the incompleteness in our
>>>> system.
>>>> I think this is a great achievement!
>>>> It's actually like this: YOU believe in ME. THEY give
>>>> you a godelian statement (You theoretically can not prove this
>>>> statement) you give it to ME and then see that I can neither prove it
>>>> nor disprove it, so you tell
>>>> THEM that their statement is true.
>>>> But the wonder is in what we do just by ourselves. We have a THEORY OF
>>> MIND.
>>>> You actually do not need to ask me about the truth of that statement,
>>> you
>>>> just simulate me and that's why I can see the a godelian statement is at
>>>> last
>>>> true. But in the logical sense ONE system wont be able to overcome the
>>>> incompleteness,
>>>> so I might conclude:
>>>> I'M NOT ONE LOGICAL SYSTEM!
>>>> This is how we might rich a theory of self. A loopy(!) and multi(!)
>>> self.
>>>>
>>>>
>>>> *
>>>>
>>>> *Mohsen Ravanbakhsh
>>>>
>>> --
>>>
>>> ----------------------------------------------------------------------------
>>> A/Prof Russell Standish Phone 0425 253119 (mobile)
>>> Mathematics
>>> UNSW SYDNEY 2052 hpcoder.domain.name.hidden
>>> Australia http://www.hpcoders.com.au
>>> ----------------------------------------------------------------------------
>>>
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>>
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Received on Fri May 25 2007 - 00:31:31 PDT