On 27 Aug 2012, at 11:47, Alberto G. Corona wrote:

Please don´t take my self confident style for absolute certainty. I just expose my ideas for discussion.

The fascination with which the Gödel theorem is taken may reflect the atmosphere of magic that invariably goes around anything for which there is a lack of understanding. In this case, with the addition of a supposed superiority of mathematicians over machines.

I have never really herad about a mathematician or a logician convinced by such an idea.



What Gödel discovered was that the set of true statements in mathematics, (in the subset of integer arithmetics) can not be demonstrated by a finite set of axioms. And to prove this, invented a way to discover new unprovable theorems, given any set of axioms, by means of an automatic procedure, called diagonalization, that the most basic interpreted program can perform. No more, no less. Although this was the end of the Hilbert idea.

OK.




What Penrose and others did is to find a particular (altroug quite direct) translation of the Gódel theorem to an equivalent problem in terms of a Turing machine where the machine

Translating Gödel in term of Turing machine is a well known pedagogical folklore in logic. It is already in the old book by Kleene, Davis, etc. It makes indeed things simpler, but sometimes it leads to misunderstanding, notably due to the common confusion between computing and proving.



does not perform the diagonalization and the set of axioms can not be extended.

That's the case for the enumeration of total computable functions, and is well known. I am not sure Penrose find anything new here. Penrose just assumes that Gödel's theorem does not apply to us, and he assumes in particular than humans know that they are consistent, without justification. I agree with Penrose, but not for any form of formalisable knowledge. And this is true for machines too.



By restricting their reasoning to this kind of framework, Penrose demonstrates what eh want to demonstrate, the superiority of the mind, that is capable of doing a simple diagonalization.

IMHO, I do not find the Gödel theorem a limitation for computers. I think that Penrose and other did a right translation from the Gódel theorem to a problem of a Turing machine,. But this translation can be done in a different way.

It is possible to design a program that modify itself by adding new axioms, the ones produced by the diagonalizations, so that the number of axioms can grow for any need. This is rutinely done for equivalent problems in rule-based expert systems or in ordinary interpreters (aided by humans) in complex domains. But reduced to integer aritmetics, A turing machine that implements a math proof system at the deep level, that is, in an interpreter where new axioms can be automatically added trough diagonalizations, may expand the set of know deductions by incorporating new axioms. This is not prohibited by the Gódel theorem. What is prohibited by such theorem is to know ALL true statements on this domain of integer mathematics. But this also apply to humans. But a computer can realize that a new axiom is absent in his initial set and to add it, Just like humans.

I do not see in this a limitation for human free will. I wrote about this before. The notion of free will based on the deterministc nature of the phisics or tcomputations is a degenerated, false problem which is an obsession of the Positivists.

Got the feeling I have already comment this. yes, Gödel's proof is constructive, and machines can use it to extend themselves, and John Myhill (and myself, and others) have exploited this in many ways.

Gödel's second incompleteness theorem has been generalized by Löb, and then Solovay has shown that the modal logical systems G and G* answer all the question at the modal propositional level. For example the second incompleteness theorem <>t -> ~[]<>t is a theorem of G, and <>t is a theorem of G*, etc.

Gödel's theorem is not an handicap for machine, on the contrary it prevents the world of numbers and machines from any normative or "totalitarian" (complete- theory about them. It shows that arithmetical truth, of computerland, is inexhaustible. Incompleteness is a chance for mechanism, as Judson Webb already argued.

Bruno

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



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