Is it true that real numbers are complete? Richard On Mon, Aug 27, 2012 at 9:11 AM, Bruno Marchal <[email protected]> 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/> > > > > -- > 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<[email protected]> > . > To unsubscribe from this group, send email to everything-list+unsubscribe@ > **googlegroups.com <everything-list%[email protected]>. > For more options, visit this group at http://groups.google.com/** > group/everything-list?hl=en<http://groups.google.com/group/everything-list?hl=en> > . > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
