On 27 Aug 2012, at 15:15, Richard Ruquist wrote:

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Is it true that real numbers are complete?

It is true that the first order theory of the real numbers, is complete. This has been proven by Tarski.

`Now add some trigonometric function axioms, and you are incomplete`

`again, as the trigonometric functions will re-instantiate the natural`

`numbers, by equation like sin(2PI*x) = 0. To be short.`

`yes, from a first order logic perspective: the real numbers (R, + x)`

`are simpler than (N, +, x), as the second are Turing universal, the`

`first are not.`

Bruno

RichardOn 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. 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 logicianconvinced 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 machineTranslating Gödel in term of Turing machine is a well knownpedagogical folklore in logic. It is already in the old book byKleene, Davis, etc. It makes indeed things simpler, but sometimes itleads to misunderstanding, notably due to the common confusionbetween 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, andhe assumes in particular than humans know that they are consistent,without justification. I agree with Penrose, but not for any form offormalisable 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 isconstructive, and machines can use it to extend themselves, and JohnMyhill (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. Forexample the second incompleteness theorem <>t -> ~[]<>t is a theoremof G, and <>t is a theorem of G*, etc.Gödel's theorem is not an handicap for machine, on the contrary itprevents the world of numbers and machines from any normative or"totalitarian" (complete- theory about them. It shows thatarithmetical truth, of computerland, is inexhaustible.Incompleteness is a chance for mechanism, as Judson Webb alreadyargued.Bruno http://iridia.ulb.ac.be/~marchal/ --You received this message because you are subscribed to the GoogleGroups "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.--You received this message because you are subscribed to the GoogleGroups "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.

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