Geometry.Net - Theorems_And_Conjectures: Incompleteness Theorem

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

Extractions: 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 Other languages: Deutsch 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 or 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." 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.
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

Extractions: 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.'
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

Extractions: 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
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

Extractions: 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)
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

Extractions: 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)
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

Extractions: 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!
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.
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[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
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Extractions: 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
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
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
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

Extractions: 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.)
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

Extractions: 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. 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
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

Extractions: 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.
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

Extractions: 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 Formal models of the mind Self-reference in computers Self-reference in minds Conclusion References 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.
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
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
Godel Incompleteness Theorem.xml
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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

Extractions: University of Alabama in Huntsville 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, Gödels 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
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
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
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