Is it true that real numbers are complete?
Richard

On Mon, Aug 27, 2012 at 9:11 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

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