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

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Please don´t take my self confident style for absolute certainty. Ijust expose my ideas for discussion.The fascination with which the Gödel theorem is taken may reflectthe atmosphere of magic that invariably goes around anything forwhich there is a lack of understanding. In this case, with theaddition 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 inmathematics, (in the subset of integer arithmetics) can not bedemonstrated by a finite set of axioms. And to prove this, inventeda way to discover new unprovable theorems, given any set of axioms,by means of an automatic procedure, called diagonalization, that themost 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 quitedirect) translation of the Gódel theorem to an equivalent problem interms 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 notbe 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, Penrosedemonstrates what eh want to demonstrate, the superiority of themind, that is capable of doing a simple diagonalization.IMHO, I do not find the Gödel theorem a limitation for computers. Ithink that Penrose and other did a right translation from the Gódeltheorem to a problem of a Turing machine,. But this translation canbe done in a different way.It is possible to design a program that modify itself by adding newaxioms, the ones produced by the diagonalizations, so that thenumber of axioms can grow for any need. This is rutinely done forequivalent problems in rule-based expert systems or in ordinaryinterpreters (aided by humans) in complex domains. But reduced tointeger aritmetics, A turing machine that implements a math proofsystem at the deep level, that is, in an interpreter where newaxioms can be automatically added trough diagonalizations, mayexpand the set of know deductions by incorporating new axioms. Thisis not prohibited by the Gódel theorem. What is prohibited by suchtheorem is to know ALL true statements on this domain of integermathematics. But this also apply to humans. But a computer canrealize 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 aboutthis before. The notion of free will based on the deterministcnature of the phisics or tcomputations is a degenerated, falseproblem 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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.