1. Godel's Theorems Godel s incompleteness theorem by Dale Myers. http://www.math.hawaii.edu/~dale/godel/godel.html

Godel's Incompleteness Theorem By Dale Myers Cantor's Uncountability Theorem Richard's Paradox The Halting Problem ... Godel's Second Incompleteness Theorem Diagonalization arguments are clever but simple. Particular instances though have profound consequences. We'll start with Cantor's uncountability theorem and end with Godel's incompleteness theorems on truth and provability. In the following, a sequence is an infinite sequence of 0's and 1's. Such a sequence is a function f Thus 10101010... is the function f with f f f A sequence f is the characteristic function i f i If X has characteristic function f i ), its complement has characteristic function 1 - f i Cantor's Uncountability Theorem. There are uncountably many infinite sequences of 0's and 1's. Proof . Suppose not. Let f f f , ... be a list of all sequences. Let f be the complement of the diagonal sequence f i i Thus f i f i i For each i f differs from f i at i Thus f f f f This contradicts the assumption that the list contained all sequences.

2. Society For Philosophy And Technology - Volume 2, Numbers 3-4 Article on a much debated subject by John Sullins III published in Philosophy and Technology. http://scholar.lib.vt.edu/ejournals/SPT/v2n3n4/sullins.html

ETDs ImageBase Ejournals News ... Special Collections Announcements: The Special Collections Reading Room will be opening on October 4, 2005. Society for Philosophy and Technology Current Editor: Davis Baird db@sc.edu Current Editorial Assistant: A Bryant aubreybryant@hotmail.com Number 3-4 Spring-Summer 1997 Volume 2 DLA Ejournal Home SPT Home Table of Contents for this issue Search SPT and other ejournals John P. Sullins III, San Jose State University 1. INTRODUCTION It is not my purpose to rehash these argument in terms of Cognitive Science. Rather my project here is to look at the two incompleteness theorems and apply them to the field of AL. This seems to be a reasonable project as AL has often been compared and contrasted to AI ( Sober, 1992 Keeley, 1994 ); and since there is clearly an overlap between the two studies, criticisms of one might apply to the other. We must also keep in mind that not all criticisms of AI can be automatically applied to AL; the two fields of study may be similar but they are not the same ( Keeley, 1994

3. The Berry Paradox Transcript of a lecture by Gregory Chaitin on how the Berry Paradox ( the smallest number that needs at least n words to specify it, where n is large ) illuminates Godel's incompleteness theorem. http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html

The Berry Paradox G. J. Chaitin, IBM Research Division, P. O. Box 704, Yorktown Heights, NY 10598, chaitin@watson.ibm.com Complexity 1:1 (1995), pp. 26-30 Lecture given Wednesday 27 October 1993 at a Physics - Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa Fe Institute 24-26 May 1994. What is the paradox of the liar? Well, the paradox of the liar is ``This statement is false!'' Why is this a paradox? What does ``false'' mean? Well, ``false'' means ``does not correspond to reality.'' This statement says that it is false. If that doesn't correspond to reality, it must mean that the statement is true, right? On the other hand, if the statement is true it means that what it says corresponds to reality. But what it says is that it is false. Therefore the statement must be false. So whether you assume that it's true or false, you must conclude the opposite! So this is the paradox of the liar. Now let's look at the Berry paradox. First of all, why ``Berry''? Well it has nothing to do with fruit! This paradox was published at the beginning of this century by Bertrand Russell. Now there's a famous paradox which is called Russell's paradox and this is not it! This is another paradox that he published. I guess people felt that if you just said the Russell paradox and there were two of them it would be confusing. And Bertrand Russell when he published this paradox had a footnote saying that it was suggested to him by an Oxford University librarian, a Mr G. G. Berry. So it ended up being called the Berry paradox even though it was published by Russell.

4. Gödel's Incompleteness Theorem Gödel s incompleteness theorem. This theorem is one of the most important The proof of Gödel s incompleteness theorem is so simple, and so sneaky, http://www.miskatonic.org/godel.html

Miskatonic University Press wtd@pobox.com Burton and Gordon Fictional Footnotes FRBR Blog ... available in English translation and also in a modernized translation . It's also in print from Dover in a nice, inexpensive edition. Jones and Wilson, An Incomplete Education outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. Boyer, History of Mathematics Nagel and Newman, Principia , or any other system within which arithmetic can be developed, is essentially incomplete . In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set. Rucker

5. The Troublesome Paradox - Per Lundgren Online version of book seeking publication by Per Lundgren. Author attempts to argue that a consequence of Goedel's incompleteness theorem is that we should overturn our current approach to scientific method. http://www.yesgoyes.com/

This page uses frames, but your browser doesn't support them.

6. G Del's Incompleteness Theorem G del's incompleteness theorem. This theorem is one of the most important proven in the twentieth century. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Gödel's Incompleteness Theorem -- From MathWorld Informally, Gödel s incompleteness theorem states that all consistent axiomatic This is sometimes called Gödel s first incompleteness theorem, http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Logic Decidability ... consistent axiomatic formulations of number theory Hilbert's problem asking whether mathematics is "complete" (in the sense that every statement in the language of number theory can be either proved or disproved). Formally, Gödel's theorem states, "To every -consistent recursive class of formulas , there correspond recursive class-signs such that neither ( Gen ) nor Neg( Gen ) belongs to Flg( ), where is the free variable of " (Gödel 1931). number theory is consistent, then a proof of this fact does not exist using the methods of first-order predicate calculus . Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

8. Andrew Burbanks Home Page (Mathematics, University Of Bristol) Brief introductions to combinatory logic, the incompleteness theorems and independence results, by Andrew D Burbanks. http://www.maths.bris.ac.uk/~maadb/research/topics/logic/

You Are Here: University of Bristol Mathematics / Dr A Burbanks home research topics publications Dr. Andrew Burbanks Research Current research topics include the Dynamical Systems approach to Transition State Theory and visualization of high-dimensional structures. more "I read it all, every word. And I still don't understand a thing." -Fran Healy. You Are Here: University of Bristol Mathematics / Dr A Burbanks [Updated Fri, 5 Aug 2005 16:29:17 +0100]

9. Godel's Incompleteness Theorem Godel's incompleteness theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Goedel's Incompleteness Theorem. Liar's Paradox. Self Reference . 5.3. Goedel's incompleteness theorem. It seems that SelfReference Lemma allows formulating the Liar's paradox in PA. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. Gödel's Incompleteness Theorem - Wikipedia, The Free Encyclopedia Gödel s first incompleteness theorem is perhaps the most celebrated result in Solomon Feferman showed that Gödel s second incompleteness theorem goes http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorem

G¶del's incompleteness theorem From Wikipedia, the free encyclopedia. In mathematical logic G¶del's incompleteness theorems are two celebrated theorems proven by Kurt G¶del in Contents First incompleteness theorem Second incompleteness theorem Gentzen's theorem Meaning of G¶del's theorems ... edit First incompleteness theorem G¶del's first incompleteness theorem is perhaps the most celebrated result in mathematical logic. It basically says that: For any formal theory in which basic arithmetical facts are provable, either the theory is inconsistent or it is possible to construct an arithmetical statement which is true but not provable or refutable in the theory. Here, "theory" means a set of statements closed under certain logical inference rules . A theory is "consistent" if it contains no contradictions The meaning of "it is possible to construct" is that there is a mechanical procedure which when given a description of the theory, e.g. in form of a computer program listing its axioms or a formula in the language of arithmetic defining the set of axioms of the theory, produces a sentence in the language of arithmetic which has the stated property. That this sentence is true if the theory is consistent simply means that what it says about natural numbers is true, in a mathematically defined sense, if no contradiction can be derived in the theory. Specifically, the sentence is equivalent to the claim that there does not exist a natural number coding a proof of contradiction in the theory, and its being true just means that there really is no such natural number. The sentence produced by the procedure is often referred to as "the" G¶del sentence for that theory.

12. On Computable Numbers, With An Application To The Entscheidungsproblem - A. M. T Turing's paper which discusses the halting problem in the context of G¶del's incompleteness theorem. HTML. http://www.abelard.org/turpap2/turpap2.htm

site map A. M. Turing  [ NOV. 12 1936.] ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM By A. M. TURING On computable numbers, with an application to the Entscheidungsproblem was written by Alan Turing in 1936. Computing machinery and intelligence (Turing) the Turing test and intelligence (abelard) On computable numbers, with an application to the Entscheidungsproblem (Turing) Decision processes (abelard) The document, decision processes by abelard, gives an empiric analysis of the Entscheidungsproblem.. Computing machinery and intelligence was published by Alan Turing in 1950. The document, the Turing test and intelligence by abelard, gives further analysis. Web abelard.org This document uses advanced technology. Does this embedded character z match this character? You will need to use a Microsoft Internet Explorer browser (version 4 or above) to see this document in full. If, on your screen, the embedded character (above on the left) does not appear similar to the character on the right, your browser is unable to display these embedded characters. A suitable browser is available to download (free) from Microsoft Such browsers (Microsoft browsers version 4 and higher) can also be found on many software CD-ROMs.

13. What Is Mathematics G Del's Theorem And Around. Incompleteness . reference lemma 5.3. Goedel's incompleteness theorem 5.4. Goedel's second theorem 6. Around Goedel's theorem 6.1 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. Gödel's Incompleteness Theorem - Wikipedia, The Free Encyclopedia Gödel s first incompleteness theorem is perhaps the most celebrated result in Gödel s second incompleteness theorem also implies that a theory T1 http://en.wikipedia.org/wiki/Incompleteness_theorem

G¶del's incompleteness theorem From Wikipedia, the free encyclopedia. (Redirected from Incompleteness theorem In mathematical logic G¶del's incompleteness theorems are two celebrated theorems proven by Kurt G¶del in Contents First incompleteness theorem Second incompleteness theorem Gentzen's theorem Meaning of G¶del's theorems ... edit First incompleteness theorem G¶del's first incompleteness theorem is perhaps the most celebrated result in mathematical logic. It basically says that: For any formal theory in which basic arithmetical facts are provable, either the theory is inconsistent or it is possible to construct an arithmetical statement which is true but not provable or refutable in the theory. Here, "theory" means a set of statements closed under certain logical inference rules . A theory is "consistent" if it contains no contradictions The meaning of "it is possible to construct" is that there is a mechanical procedure which when given a description of the theory, e.g. in form of a computer program listing its axioms or a formula in the language of arithmetic defining the set of axioms of the theory, produces a sentence in the language of arithmetic which has the stated property. That this sentence is true if the theory is consistent simply means that what it says about natural numbers is true, in a mathematically defined sense, if no contradiction can be derived in the theory. Specifically, the sentence is equivalent to the claim that there does not exist a natural number coding a proof of contradiction in the theory, and its being true just means that there really is no such natural number. The sentence produced by the procedure is often referred to as "the" G¶del sentence for that theory.

15. Godel's Theorems Godel's incompleteness theorem by Dale Myers. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Godel's Incompleteness Theorem Godel s incompleteness theorem. Zillion s Philosophy Pages. First let me try tostate in clear terms exactly what he proved, since some of us may have sort http://www.myrkul.org/recent/godel.htm

Godel's Incompleteness Theorem Zillion's Philosophy Pages First let me try to state in clear terms exactly what he proved, since some of us may have sort of a fuzzy idea of his proof, or have heard it from someone with a fuzzy idea of the proof.. The proof begins with Godel defining a simple symbolic system. He has the concept of a variables, the concept of a statement, and the format of a proof as a series of statements, reducing the formula that is being proven back to a postulate by legal manipulations. Godel only need define a system complex enough to do arithmetic for his proof to hold. Godel then points out that the following statement is a part of the system: a statement P which states "there is no proof of P". If P is true, there is no proof of it. If P is false, there is a proof that P is true, which is a contradiction. Therefore it cannot be determined within the system whether P is true. As I see it, this is essentially the "Liar's Paradox" generalized for all symbolic systems. For those of you unfamiliar with that phrase, I mean the standard "riddle" of a man walking up to you and saying "I am lying". The same paradox emerges. This is exactly what we should expect, since language itself is a symbolic system. Godel's proof is designed to emphasize that the statement P is *necessarily* a part of the system, not something arbitrary that someone dreamed up. Godel actually numbers all possible proofs and statements in the system by listing them lexigraphically. After showing the existence of that first "Godel" statement, Godel goes on to prove that there are an infinite number of Godel statements in the system, and that even if these were enumerated very carefully and added to the postulates of the system, more Godel statements would arise. This goes on infinitely, showing that there is no way to get around Godel-format statements: all symbolic systems will contain them.

17. G Del On The Net By G del's second incompleteness theorem, we can't know that mathematics is consistent. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. INCOMPLETENESS THEOREM incompleteness theorem. Goedel s thesis initially about number theory but nowfound applicable to all formal systems that include the arithmetic of natural http://pespmc1.vub.ac.be/ASC/INCOMP_THEOR.html

P RINCIPIA C YBERNETICA ... EB Parent Node(s): Web Dictionary of Cybernetics and Systems INCOMPLETENESS THEOREM Goedel's thesis initially about number theory but now found applicable to all formal systems that include the arithmetic of natural numbers: "any consistent axiomatic system does include propositions whose truth is undecidable within that system and its consistency is, hence, not provable within that system". The self-reference involved invokes the paradox: "a formal system of some complexity cannot be both consistent and decidable at the same time". The theorem rendered Frege, Russell and Whitehead's ideals of finding a few axions of mathematics from which all and only true statements can be deduced non-achievable. It has profound implications for theories of human cognition, computational linguistics and limits artificial intelligence in particular. ( Krippendorff Next Previous Index ... Help URL= http://pespmc1.vub.ac.be/ASC/INCOMP_THEOR.html

19. Godel's Incompleteness Theorem the first incompleteness proof can be formalized in S allows one to derive Godel's second incompleteness theorem as a corollary. This theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. Goedel's Incompleteness Theorem The Undecidability of Arithmetic, Goedel's incompleteness theorem, and the class of Arithmetical Languages http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

Page 1     1-20 of 95    1  | 2  | 3  | 4  | 5  | Next 20