21. Goedel's Incompleteness Theorem - Wikipedia Other languages Deutsch. Goedel's incompleteness theorem. From Wikipedia, thefree encyclopedia. These results do not require the incompleteness theorem. http://www.wikipedia.org/wiki/Goedel's_incompleteness_theorem

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Other languages: Deutsch Goedel's incompleteness theorem From Wikipedia, the free encyclopedia. In mathematical logic are two celebrated theorems proven by Kurt Gödel in 1930. Somewhat simplified, the first theorem states that in any axiomatic system sufficiently strong to allow one to do basic arithmetic , one can construct a statement that either can be neither proven nor disproven within that system or can be both proven and disproven within that system. In the first case, we call the axiomatic system incomplete , in the second case we call it inconsistent . A short version of the first incompleteness theorem is therefore: "Any sufficiently strong consistent axiomatic system is incomplete." Examples of undecidable statements The subsequent combined work of Gödel and Paul Cohen has given concrete examples of undecidable statements (statements which can be neither proven nor disproven): both the axiom of choice and the continuum hypothesis are undecidable in the standard axiomatization of set theory . These results do not require the incompleteness theorem. Following this work, many more statements in set theory have been proven to be undecidable.

22. Gödel, The Man next up previous Next Main Idea Up Gödel's incompleteness theoremPrevious Gödel's incompleteness theorem Gödel, the man. This http://physics.colorado.edu/~clineks/godel/node1.html

Next: Main Idea Up: Previous: This man who was to revolutionize modern mathematics, to say nothing of philosophy and computer science, was born in 1906, in what is now the Czech republic. His family was well off because his father owned a textile mill, and was encouraged to study extensively by his mother, who had been educated in France. Although his first area of interest was linguistics, he moved to mathematics and physics because he found them easier to study independently. He intended on studying theoretical physics when he entered the University of Vienna in 1924, but eventually moved to mathematics. While there, he began involved with the `Vienna Circle,' a group of mathematicians who founded the philosophy of `logical positivism.' Next: Main Idea Up: Previous: Kelly Cline

23. Godel's Incompleteness Theorem More precisely, his first incompleteness theorem states that in any formal systemS of arithmetic, there will be a sentence P of the language of S such that http://www.faragher.freeserve.co.uk/godeldef2.htm

Definitions Godel's Theorem. The proof, published by Kurt Godel in 1931, of the existence of formally undecidable propositions in any formal system of arithmetic. More precisely, his first incompleteness theorem states that in any formal system S of arithmetic, there will be a sentence P of the language of S such that if S is consistent, neither P nor its negation can be proved in S . This makes it possible to show that there must be a sentence P of S which can be interpreted (very roughly) as saying 'I am not provable'. It is shown that if S is consistent, this sentence is not provable, and hence, it is sometimes argued, P must be true. It is this last step which had led people to claim that Godel's theorem demonstrates the superiority of men over machines - men can prove propositions which no machine (programmed with the axioms and rules of a formal system) can prove. But this is to overlook the point that the proof of the theorem only allows one to conclude that if S is consistent, neither

24. An Incompleteness Theorem Via. . . (ResearchIndex) FALSE (1 8) 15 not PROVABLE(~ d) (Update) Similar documents (at the sentence level)20.0% An incompleteness theorem via Abstraction Bundy, Giunchiglia.. http://citeseer.nj.nec.com/bundy97incompletenes.html

An Incompleteness Theorem via (1997) (Make Corrections) Alan Bundy, et al. Home/Search Context Related View or download: dream.dai.ed.ac.uk/group/tw/ jar97.ps Cached: PS.gz PS PDF DjVu ... Help From: dream.dai.ed.ac.uk/group papers (more) Homepages: A.Bundy HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: Proof absmetamaths:: show proof; 1 OCONS (1) 2 PROVABLE(d) (2) 3 PROVABLE(d) imp (PROVABLE(~ d) and NUMBER(k(d))) 4 PROVABLE(~ d) (2) 5 CONS imp ((not PROVABLE(d)) or (not PROVABLE(~ d))) 6 FALSE (1 2) 7 not PROVABLE(d) (1) 8 PROVABLE(~ d) (8) 9 exists n. (not (NUMBER(n) imp PROVABLE(d))) (1 8) 10 not (NUMBER(n) imp PROVABLE(d)) (10) 11 (NUMBER(n) and PROVABLE(~ d)) imp PROVABLE(d) 12 PROVABLE(d) (1 8) 13 CONS imp ((not PROVABLE(d)) or (not PROVABLE(~ d))) 14 FALSE (1 8) 15 not PROVABLE(~ d)... (Update) Similar documents (at the sentence level): An Incompleteness Theorem via Abstraction - Bundy, Giunchiglia.. (1996) (Correct) Active bibliography (related documents): More All A General Purpose Reasoner for Abstraction - Fausto Giunchiglia (1993) (Correct) ... Automating the synthesis of decision procedures - Armando, Gallagher, Smaill (1994)

25. An Incompleteness Theorem Via Abstraction (ResearchIndex) checking (Update) Active bibliography (related documents) MoreAll 0.8 An incompleteness theorem via. . . Bundy, al. (1997 http://citeseer.nj.nec.com/bundy96incompleteness.html

An Incompleteness Theorem via Abstraction (1996) (Make Corrections) Alan Bundy, Fausto Giunchiglia, Adolfo Villafiorita, Toby Walsh Home/Search Context Related View or download: dai.ed.ac.uk/pub/daidb/pa rp809.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: dai.ed.ac.uk/daidb/papers (more) Homepages: A.Bundy F.Giunchiglia A.Villafiorita T.Walsh ... (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: ion Alan Bundy 1 , Fausto Giunchiglia 2;3 , Adolfo Villafiorita 4;5 and Toby Walsh 2;5 1. Mathematical Reasoning Group, Dept of AI, University of Edinburgh 2. Mechanized Reasoning Group, IRST 3. DISA, University of Trento 4. Istituto di Informatica, University of Ancona 5. Mechanized Reasoning Group, DIST, University of Genoa April 13, 1996 Abstract We demonstrate the use of abstraction in aiding the construction of an interesting and difficult example in a proof checking... (Update) Active bibliography (related documents): More All An Incompleteness Theorem via. . . - Bundy, al. (1997) (Correct) ... (Correct) Similar documents based on text: More All Theories of Abstraction - Giunchiglia, Villafiorita, Walsh (1997)

26. III. Gödel's Proof Of His Incompleteness Theorem III. Gödel's Proof of his incompleteness theorem. Synopsis. Discussesa LISP run exhibiting a fixed point, and a LISP run which illustrates http://www.umcs.maine.edu/~chaitin/unknowable/ch3.html

Synopsis How can we enable a LISP expression to know itself? Well, it's very easy once you've seen the trick! Consider the LISP function f(x) that takes x into (('x)('x)). In other words, f assumes that its argument x is the lambda expression for a one-argument function, and it forms the expression that applies x to x. It doesn't evaluate it, it just creates it. It doesn't actually apply x to x, it just creates the expression that will do it. You'll note that if we were to evaluate this expression (('x)('x)), in it x is simultaneously program and data, active and passive. Okay, so let's pick a particular x, use f to make x into (('x)('x)), and then run/evaluate the result! And to which x shall we apply f? Why, to f itself! So f applied to f yields what? It yields f applied to f, which is what we started with!! So f applied to f is a self-reproducing LISP expression! You can think of the first f, the one that's used as a function, as the organism, and the second f, the one that's copied twice, that's the genome. In other words, the first f is an organism, and the second f is its DNA! I think that that's the best way to remember this, by thinking it's biology. Just as in biology, where a organism cannot copy itself directly but needs to contain a description of itself, the self-reproducing function f cannot copy itself directly (because it cannot read itself-and neither can you ). So f needs to be given a (passive) copy of itself. The biological metaphor is quite accurate!

27. About "Gödel's Incompleteness Theorem" Gödel's incompleteness theorem. Library Home Full Table of Contents Suggesta Link Library Help Visit this site http//www.miskatonic.org/godel.html. http://mathforum.org/library/view/12250.html

Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.miskatonic.org/godel.html Author: William Denton Description: Levels: High School (9-12) Early College Languages: English Resource Types: Quotations Math Topics: Axiomatic Systems Philosophy Suggestion Box Home ... Search http://mathforum.org/ webmaster@mathforum.org

28. [HM] Canonical List Of Proofs For Godel's Incompleteness Theorem By Jim Nicholso HM Canonical List of Proofs for Godel's incompleteness theoremby Jim Nicholson. reply to this message post a message on a new http://mathforum.org/epigone/historia_matematica/spurbrarfyr

[HM] Canonical List of Proofs for Godel's Incompleteness Theorem by Jim Nicholson reply to this message post a message on a new topic Back to Historia-Matematica Discussion Group Subject: [HM] Canonical List of Proofs for Godel's Incompleteness Theorem Author: nicholson671a@covad.net Date: 1 Mar 02 21:42:01 -0500 (EST) I have been searching for a complete list of published proofs of Godel's theorem. I remember reading somewhere that the count was up to 38. Does anyone know where to look? Jim Nicholson The Math Forum

29. Goedel's Incompleteness Theorem From FOLDOC Goedel's incompleteness theorem. completeness . Try this searchon OneLook / Google. Nearby terms Godproofs of the existence of http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Goedel's incompleteness theorem

30. Goedel From FOLDOC Goedel's incompleteness theorem. completeness . When one hears simply Goedel'stheorem it usually refers to the first incompleteness theorem. http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Goedel

31. Gödel On The Net Every day, Gödel's incompleteness theorem is invoked on the net to support someclaim or other, or just to whack people over the head with it in a general way http://www.sm.luth.se/~torkel/eget/godel.html

Gödel on the net Every day, Gödel's incompleteness theorem is invoked on the net to support some claim or other, or just to whack people over the head with it in a general way. In news, we find such invocations not only in sci.logic, sci.math, comp.ai.philosophy, sci.philosophy.tech and other such places where one might expect them, but with equal frequency in groups dealing with politics or religion, and indeed in alt.cuddle, soc.culture.malaysia, rec.music.hip-hop, and what have you. In short, whenever a bunch of people get together on the net, sooner or later somebody will invoke Gödel's incompleteness theorem. Unsurprisingly, the bulk of these invocations covers a range from the nonsensical to the merely technically inaccurate, and they often give rise to a flurry of corrections and more or less extended technical or philosophical disputes. My purpose in these pages is to provide a set of responses to many such invocations, couched in non-confrontational and hopefully helpful and intelligible terms. There are few technicalities, except in connection with a couple of technical (and less frequently raised) issues. All of my comments and explanations are intended to be non-controversial, in the sense that people who are familiar with the incompleteness theorem can be expected to agree with them. (Thus, for example, I don't present any criticism of so-called Gödelian arguments in the philosophy of mind, but only a couple of technical observations relevant for the discussion of such arguments.)

32. Gödel's Theorem Gödel's second incompleteness theorem. Gödels first incompletenesstheorem proves that formal systems T satisfying certain conditions http://www.sm.luth.se/~torkel/eget/godel/second.html

Gödel's second incompleteness theorem Gödels first incompleteness theorem proves that formal systems T satisfying "certain conditions" are incomplete, i.e. that there is a sentence A in the language of the T which can neither be proved, nor disproved in T. Among the "certain conditions" must be some condition implying that T is consistent. Gödel's second incompleteness theorem proves that formal systems T satisfying certain other conditions "cannot prove their own consistency", in the sense that a suitable formalization in the language of T of the statement "T is consistent" cannot be proved in T. Again one necessary condition is that T is in fact consistent, since otherwise everything is provable in T. The second incompleteness theorem applies in particular to those formal systems that can be used to develop all of the ordinary mathematics that one finds in textbooks. One such system is the axiomatic set theory called ZFC. Since all the theorems ordinarily proved in mathematics can be proved in ZFC, and since the consistency of ZFC cannot be proved in ZFC (unless ZFC is inconsistent), it is often concluded that we cannot expect to prove, and therefore can't know, that ZFC is consistent. "We can't know that mathematics is consistent." This is the conclusion discussed in this section. "Different" doesn't mean "stronger" In commenting on this, first let me mention a widespread misconception. Clearly, for any theory T, there is another theory T' in wich "T is consistent" can be proved. For example, we can trivially define such a theory T' obtained by adding "T is consistent" as a new axiom to T. The misconception consists in the notion that any such theory T' in which "T is consistent" is provable must be

33. Kurt Godel And His Incompleteness Theorem, And Martin Luther Understanding Of Tr Kurt Godel's incompleteness theorem had some profound impacts on generalthought, but Martin Luther had him beat by a few centuries http://www.abarim-publications.com/artctsuspects.html

12. Children of the Primes A logical system can not liberate. But that doesn't mean that rules are bad. Rules help create order and understanding. A society based on rules is far better off than a society not based on rules. And Math put a man on the moon. Math gave us the Internet. Math gave us this whole wild global culture. Yea, Math may run with the wind and the free range chickens as long as not the whole Truth is addressed. A system of logic can not cover Truth; Truth can not be expressed in logic. The Grand Unified Theory (GUT, or GUTH as a certain somebody demands) will not be written in Math. This is also the reason why you never hear anyone solemnly swear to tell the Truth, the whole Truth and nothing but the Truth, so help me Mathematics. Still, she is beautiful. Do thy best old Math, despite thy wrong. A logical system (scientific, philosophical, religious, legal) Departs from: And then: Which leads to: an axiomatic platform wrought from the present insight of the observer. starts concluding and forms a body of deriviations nothing; must remain incomplete. Hence a consensus is not possible. Hence confusion abounds.

34. Damjan Bojadziev: Mind Versus Gödel Compares the mind to advanced computers and artificial intelligence, as Goedel would possibly view them.Category Science Math Logic and Foundations...... Gödel's (first) incompleteness theorem can be expressed in the form a sufficientlyexpressive formal system cannot be both consistent and complete. http://nl.ijs.si/~damjan/g-m-c.html

Damjan Bojadziev in M. Gams , M. Paprzycki and X. Wu (eds.), Mind Versus Computer IOS Press 1997, pp. 202-210 Slightly edited version, with added links, derived from , published in the special issue of Informatica , vol. 19, no. 4, Nov. 1995, pp. 627-34, MIND not equal COMPUTER Introduction Self-reference in Implications Non-implications Formal models of the mind The basic incompleteness argument Mind over machine omega:1 Reflexive sequences of reflexive theories Self-reference in computers Self-reference in minds Conclusion References Introduction At first sight, the designation of the topic of this special issue, "MIND <> COMPUTER", also transcribed as "Mind NOT EQUAL Computer", looks like a piece of computer ideology, a line of some dogmatic code. But there are as yet no convincing artificial animals, much less androids, and computers are not yet ready for the unrestricted Turing test. Although they show a high degree of proficiency in some very specific tasks, computers are still far behind humans in their general cognitive abilities. Much more, and in much more technical detail, is known about computers than about humans and their minds. Thus, the required comparison between minds and computers does not even seem possible, much less capable of being stated in such a simple formula. incompleteness might be called, following [Hofstadter 79, p. 468] and [Mendelson 64, p. 147], essential incompleteness.

35. Gödel's Incompleteness Theorem Rokumeikan Rokumeikan Lectures on mathematics Gödel's incompleteness theoremsand Proofs. Japanese/English. Contents. Definitions, Propositions http://www.rinku.zaq.ne.jp/suda/incomplete/index_e.html

Rokumeikan Lectures on mathematics Japanese /English Contents Definitions, Propositions and Proofs Interpretations and Variations Russell's Paradox links Rokumeikan Web site of science, philosophy and music etc. Japanese. General Relativity and Cosmology Lectures on General Relativity and Cosmology. Takao Suda E-Mail: tyc@rinku.zaq.ne.jp

36. Godel Incompleteness Theorem Meme Name Godel incompleteness theorem. Category mathematics, RelatedConcepts Related Links Core Concept. No complete truth exists. http://www.agentsmith.com/memento/g/godel incompleteness theorem.html

37. Godel Incompleteness Theorem.xml Back %} % http://www.agentsmith.com/memento/view.php?sourceMeme=godel incompleteness theor

38. Godel's Incompleteness Theorem And The Limits Of Human Knowledge Gödel's incompleteness theorem and the Limitsof Human Knowledge. Ms. Natalie Baeza. http://www.math.uah.edu/mathclub/talks/12-7-2001.html

UAH Math Math Club Talks Gödel's Incompleteness Theorem and the Limits of Human Knowledge Ms. Natalie Baeza Department of Mathematical Sciences University of Alabama in Huntsville December 7, 2001 For Mathematicians, the theorem means that any formal system trying to capture all mathematical truths in a finite set of axioms and rules is doomed to failure.For philosophers, it means that truth is elusive and ultimately unattainable. For all of us, Gödel’s incompleteness theorem results in the realization of the limitations of the human mind, and of the nobleness of our pursuit of knowledge. “Down how many roads among the stars must man propel himself in search of the final secret? The journey is difficult, immense, at times impossible, yet that will not deter some of us from attempting it ”

39. Comments On Hilbert's Program And Gödel's Incompleteness Theorem Comments on Hilbert's Program and Gödel's incompleteness theorem. David M. Burton,The History of Mathematics, McGrawHill, New York, 1997, pp. 560-563. http://www.ms.uky.edu/~lee/ma502/notes2/node9.html

Next: Practice With Induction Up: The Natural Numbers Previous: Some Theorems Derivable from David M. Burton, The History of Mathematics , McGraw-Hill, New York, 1997, pp. 560-563. Douglas R. Hofstadter, , Vintage Books, New York, 1979. Carl Lee Wed Sep 16 09:26:16 EDT 1998

40. Gödel's Incompleteness Theorem - Www.ezboard.com Author, Comment. Mystic Eyz Registered User Posts 27 (10/1/02 25052pm) Reply, Gödel's incompleteness theorem Pastor Ahyh writes http://pub83.ezboard.com/fthechurchofyahwehfrm1.showMessage?topicID=68.topic

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