Thank you, Joseph, for your answer. So it seems that both answers may be correct, right?

Hello,

As far as I understand, both possible answers "positive" and "negative" for an indefinite problem are not simultaneously correct, since that will be a contradiction. The key here is that indefiniteness means that you will never know, by any possible mean, which of the answers is the right one.

If you are familiar with computability theory, you may take a look at the problem P = NP, which, up to this moment, remains unclarified, and can be considered as a example of possible "indefinite problem".

What makes me think about AI several questions: what is intelligence? Can it be simulated? There several authors (e.g. Roger Penrose) who think that several forms of human thinking are non-computable, which will imply the impossibility to simulate the human intelligence as a whole.

Best regards,

Domingo

Thank you, Domingo, for your answer. But i am not familiar with P=NP problem. Can you be more specific about how it is related to indefinite problem? Thank you.

A good point to start: http://en.wikipedia.org/wiki/P_versus_NP_problem

The possible indefiniteness in this problem is that perhaps we will never know if P=NP or P not equal to NP.

One should ask which sections of AI might be affected

by the incompleteness theorems. Surely not robotics,

neural networks, genetic algorithms, image recognition,

machine learning.

Theorem provers are of course limited by the

incompleteness theorems, if their processed language

is powerful enough. But do the incompleteness

theorems set humand and computers apart?

I don't think so. The incompleteness theorems are truths,

and they are that for computers and as well for humans.

There will remain some mathematical statements

which neither humans nor computers can prove

or know whether they are provable.

On the other hand, correctness. Humans make

mistakes, and if we encounter two proofs, one

stating S and another proof stating ~S, then one

proof must be wrong. But then our brains don't

explode.

http://www.youtube.com/watch?v=G6o881n35GU

Humans are able to deal with contradictions.

This can be done by putting dubious propositions

into some kind of quarantine. Or with adding a

dependency on context. But that doesn't make

a difference betweens humans and computers,

computer programs can be designed to

do that likewise.

Regards,

Joachim

Thank you Domingo and Joachim, for your answers. Best wishes to both of you

I'd like to point out that Godel's theorems take place in any single powerful enough logic, so we can most likely escape them by changing the logic. That may sound strange, but it is something we do often when we get stuck in hard mathematical problems. There seems to be an indefinite supply of acceptable logics to choose from, and new ones are being invented quite frequently, along with the corresponding new proof techniques. Remember, mathematical induction of the minimal counterexample style was only developed a few hundred years ago (Fermat and Hilbert both used versions of it). In short, we make up new mathematics when we cannot solve a certain kind of problem. Check out Non-Well-Founded Set Theory for an example (Peter Aczel, Jon Barwise, others)

Also, it is not always true that those indefinite statements mentioned above will never be known. For example, look up the Paris - Harrington Theorem, which demonstrates an assertion that can be stated in peano arithmetic, but it cannot be proven there. It is one of the first non-pathological examples of the kind of theorems Godel proves exist.