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. Gödel's Incompleteness Theorem Gödel s incompleteness theorem. The proof of Gödel s incompleteness theoremis so simple, and so sneaky, that it is almost embarassing to relate. http://www.miskatonic.org/godel.html

in HTML or as a PDF . It's also in print from Dover in a nice, inexpensive edition. Other web pages of interest are and 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

3. Incompleteness Theorem incompleteness theorem. This discussion of Gödel 's proof does not follow Gödel 's constructions or formulation. http://www.mtnmath.com/book/node56.html

New version of this book Next: Physics Up: Set theory Previous: Recursive functions Incompleteness theorem Recursive functions are good because we can, at least in theory, compute them for any parameter in a finite number of steps. As a practical matter being recursive may be less significant. It is easy to come up with algorithms that are computable only in a theoretical sense. The number of steps to compute them in practice makes such computations impossible. Just as recursive functions are good things decidable formal systems are good things. In such a system one can decide the truth value of any statement in a finite number of mechanical steps. Hilbert first proposed that a decidable system for all mathematics be developed. and that the system be proven to be consistent by what Hilbert described as `finitary' methods.[ ]. He went on to show that it is impossible for such systems to decide their own consistency unless they are inconsistent. Note an inconsistent system can decide every proposition because every statement and its negation is deducible. When I talk about a proposition being decidable I always mean decidable in a consistent system. S he is working with a statement that says ``I am unprovable in S''(128)[ ]. Of course if this statement is provable in

4. 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

Editor: Davis Baird bairdd@gwm.sc.edu Spring-Summer 1997 Volume 2 Numbers 3-4 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 Wang, 1987, pg. 197 ). In fact there is an interesting argument posed by Rudy Rucker where he shows that it is possible to construct a Lucas style argument using the incompleteness theorems which actually suggests the possibility of creating machine minds ( Rucker, 1983, pp. 315-317

5. Goedel's Incompleteness Theorem The Undecidability of Arithmetic, Goedel's incompleteness theorem, and the class of Arithmetical Languages The proof of Goedel's incompleteness theorem given here rests heavily on http://kilby.stanford.edu/~rvg/154/handouts/incompleteness.html

The Undecidability of Arithmetic, Goedel's Incompleteness Theorem, and the class of Arithmetical Languages First-order arithmetic is a language of terms and formulas. Terms or (positive) polynomials are built from variables x,y,z,..., the constants and 1 and the operators + and x of addition and multiplication. The multiplication operator is normally suppressed in writing. The simplest formulas are the equations, obtained by writing an = between two terms, for instance y+2x+xy+2x z=5y , which is an abbreviation for y+x+x+xy+(1+1)xxxz = yy+yy+yy+yy+yy. More complicated formulas can be build from equations by means of connectives and quantifiers: if P and Q are formulas, then P is a formula, P Q is a formula, P Q is a formula, P Q is a formula, and P Q is a formula. if P is a formula and x a variable, then x: P and x: P are formulas. Arithmetic is interpreted in terms of the natural numbers. Every formula is either true or false (if there are free variables a formula is considered equivalent to its universal closure). Theorem: It is undecidable whether an arithmetical formula is true.

6. 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.

7. Gödel's Incompleteness Theorem Gödel s incompleteness theorem. In this section we lay the groundwork for asimplified version of Gödel s theorem that we prove in the next section. http://www.mtnmath.com/whatth/node30.html

Completed second draft of this book PDF version of this book Next: The Halting Problem Up: Mathematical structure Previous: Cardinal numbers Contents All formal systems that humans can write down are finite. However the idea of an arbitrary real number seems so obvious that mathematicians claim as formal systems a finite set of axioms plus an axiom for each real number that asserts the existence of that number. They assert the existence of other infinite formal systems including ones that could solve the Halting Problems. We now informally prove that if we could solve the Halting Problem we could solve the consistency problem for finite formal systems. The idea of the proof is simple. A finite formal system is a mechanistic process for deducing theorems. This means we can construct a computer program to generate all the theorems deducible from the axioms of the system. We add to this program a check that tests each theorem as it is generated to see if it is inconsistent with any theorem previously generated. If we find an inconsistency we cause the program to halt. Such a program will halt if and only if the original formal system is inconsistent. For the program will eventually generate and check every theorem that can be deduced from the system against every other theorem to insure no theorem is proven to be both true and false.

8. Goedel's Incompleteness Theorem. Liar's Paradox. Self Reference. By K.Podnieks Kurt Goedel invented the argument used in the proof of SelfReference Lemma to provehis famous incompleteness theorem in 1930. Goedel s incompleteness theorem. http://www.ltn.lv/~podnieks/gt5.html

fGoedel, incompleteness theorem, liar paradox, liar, self reference, second, incompleteness, paradox, theorem, Rosser, Godel, Bernays Back to title page Left Adjust your browser window Right 5. Incompleteness Theorems 5.1. Liar's Paradox Epimenides (VI century BC) was a Cretan angry with his fellow-citizens who suggested "All Cretans are liars". Is this statement true or false? a) If Epimenides' statement is true, then Epimenides also is a liar, i.e. he is lying permanently, hence, his statement about all Cretans is false (and there is a Cretan who is not a liar). We have come to a contradiction. b) If Epimenides' statement is false, then there is a Cretan, who is not a liar. Is Epimenides himself a liar? No contradiction here. Hence, there is no direct paradox here, only an amazing chain of conclusions: if a Cretan says that "All Cretans are liars", then there is a Cretan who is not a liar. Still, do not allow a single Cretan to slander all Cretans. Let us assume that Epimenides was speaking about himself only: "I am a liar". Is this true or false? a) If this is true, then Epimenides is lying permanently, and hence, his statement "I am a liar" also is false. I.e. Epimenides is not a liar (i.e. sometimes he does not lie). We have come to a contradiction.

9. 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/

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10. Gödel's Incompleteness Theorem -- From MathWorld Gödel s incompleteness theorem. Informally, Gödel s incompletenesstheorem states that all consistent axiomatic formulations of http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook 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 r such that neither ( v Gen r ) nor Neg( v Gen r ) belongs to Flg( ), where v is the free variable of r " (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.

11. Gödel's Incompleteness Theorem - Wikipedia, The Free Encyclopedia Gödel s incompleteness theorem. There are many statements that sound similarto Gödel s first incompleteness theorem, but are in fact not true. http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorem

Gödel's incompleteness theorem From Wikipedia, the free encyclopedia. In mathematical logic Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in . Somewhat simplified, the first theorem states: In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers , one can construct a statement that can be neither proved nor disproved within that system. This theorem is one of the most famous outside of mathematics, and one of the most misunderstood. It is a theorem in formal logic , and as such is easy to misinterpret. There are many statements that sound similar to Gödel's first incompleteness theorem, but are in fact not true. We will explore this further in Misconceptions about Gödel's theorems Gödel's second incompleteness theorem, which is proved by formalizing part of the proof of the first within the system itself, states: Any consistent system cannot be used to prove its own consistency. This result was devastating to a philosophical approach to mathematics known as Hilbert's program David Hilbert proposed that the consistency of more complicated systems, such as

12. On Computable Numbers (decision Problem), A.M. Turing, 1936 - Entry Page At Abel 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

ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM by A. M. TURING site map This page has been retired. Click here to go to On Computable Numbers advertising Published on the abelard site by permission of the London Mathematical Society. Originally published by the London Mathematical Society in Proceedings of the London Mathematical Society Series 2, Vol.42 (1936 - 37) pages 230 to 265, with corrections from Proceedings of the London Mathematical Society Series 2, Vol.43 (1937) pages 544 to 546. email abelard@abelard.org the address for this document is http://www.abelard.org/turpap2/turpap2.htm complete paper: approx 14,000 words; this page: 100 words prints as (complete document) 40 A4 pages; (this page) 1 A4 page (on my printer and set-up)

13. Gödel's Incompleteness Theorem From MathWorld Gödel's incompleteness theorem from MathWorld Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions ( http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/GoedelsIncompleten

14. INCOMPLETENESS THEOREM Parent Node(s) Web Dictionary of Cybernetics and Systems. incompleteness theorem incompleteness theorem. Goedel's thesis initially about number theory but now found applicable to all at the 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

15. Talk:Gödel's Incompleteness Theorem - Wikipedia, The Free Encyclopedia TalkGödel's incompleteness theorem. ( Redirected from TalkGoedel's incompleteness theorem) Walt Pohl is making some important http://www.wikipedia.org/wiki/Talk:Goedel's_incompleteness_theorem

Talk:Gödel's incompleteness theorem From Wikipedia, the free encyclopedia. (Redirected from Talk:Goedel's incompleteness theorem Walt Pohl is making some important rearrangements to organize the material in a nice progression that also includes identified sections on the import on the theorem, what people make of it, and how it is misinterpreted. I am not going to tromp on that, but it looks like this Talk needs new organization too. So I will make a start. Orcmid 17:39, 20 Mar 2004 (UTC) My main goal is to break this into manageable sections, and I may have put some material into awkward places. But I did not lose anything, it is only rearranged. Orcmid 17:39, 20 Mar 2004 (UTC) Table of contents 1 Introductory Material 2 Meaning of Gödel's theorems 3 Discussion and Implications 4 Philosophical Implications and Interpretations ... edit Introductory Material As I recall the incompleteness theorm is not quite as stated in the opening paragraph and I do consider the paragraph to be misleading. Mindyou - I will need to verify what I am saying here. The issue stems from the word "or". This can be interpreted in two contexts: (1) one can read the paragraph as saying that (a) one can find unprovable true statements as well as (b)one can show that the axiomatic system is incomplete. the other context is: (2) one can do either (a) or (b) above but not both.

16. What Is Mathematics: Goedel's Theorem And Around. Incompleteness. By K.Podnieks what is mathematics, logic, mathematics, foundations, incompleteness theorem,mathematical, Gödel, Godel, book, Goedel, tutorial, textbook, methodology http://www.ltn.lv/~podnieks/gt.html

what is mathematics, logic, mathematics, foundations, incompleteness theorem, mathematical, Gödel, Godel, book, Goedel, tutorial, textbook, methodology, philosophy, nature, theory, formal, axiom, theorem, incompleteness, online, web, free, download, teaching, learning, study, student, Podnieks, Karlis My personal page - click here Any comments are welcome - e-mail to Karlis.Podnieks@mii.lu.lv This web-site presents 100% of a hyper-textbook for students. Try joining the Foundations of Mathematics (FOM) e-mail list Visit FOM Archives Try joining the Historia Mathematica (HM) forum Visit HM Archives New! Section 6.7. Consistent universal statements are provable Section 6.8. Berry's paradox and incompleteness In preparation: Large cardinal axioms. Quo vadis? Diploma What is Mathematics: Goedel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Associate Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book " Around Goedel's theorem " published in 1992 in Russian.

17. Goedel's Theorem And Information At the time of its discovery, Kurt Gödel s incompleteness theorem was a great shockand caused much uncertainty and depression among mathematicians sensitive http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html

International Journal of Theoretical Physics 22 (1982), pp. 941-954 Gregory J. Chaitin IBM Research, P.O. Box 218 Yorktown Heights, New York 10598 Abstract 1. Introduction To set the stage, let us listen to Hermann Weyl (1946), as quoted by Eric Temple Bell (1951): We are less certain than ever about the ultimate foundations of (logic and) mathematics. Like everybody and everything in the world today, we have our ``crisis.'' We have had it for nearly fifty years. Outwardly it does not seem to hamper our daily work, and yet I for one confess that it has had a considerable practical influence on my mathematical life: it directed my interests to fields I considered relatively ``safe,'' and has been a constant drain on the enthusiasm and determination with which I pursued my research work. This experience is probably shared by other mathematicians who are not indifferent to what their scientific endeavors mean in the context of man's whole caring and knowing, suffering and creative existence in the world. And these are the words of John von Neumann (1963): ... there have been within the experience of people now living at least three serious crises... There have been two such crises in physics-namely, the conceptual soul-searching connected with the discovery of relativity and the conceptual difficulties connected with discoveries in quantum theory... The third crisis was in mathematics. It was a very serious conceptual crisis, dealing with rigor and the proper way to carry out a correct mathematical proof. In view of the earlier notions of the absolute rigor of mathematics, it is surprising that such a thing could have happened, and even more surprising that it could have happened in these latter days when miracles are not supposed to take place. Yet it did happen.

18. Incompleteness Theorem The incompleteness theorem is a pair of logical proofs that revolutionizedmathematics. The first result was published by Kurt Godel http://www.whatis.com/definition/0,,sid9_gci835123,00.html

Search our IT-specific encyclopedia for: or jump to a topic: Choose a topic... CIO CRM Databases Domino Enterprise Linux Exchange IBM S/390 IBM AS/400 Mobile Computing Networking Oracle SAP Security Storage Visual Basic Web Services Windows 2000 Advanced Search Browse alphabetically: A B C D ... General Computing Terms Incompleteness Theorem The First Incompleteness Theorem states that any contradiction-free rendition of number theory (a branch of mathematics dealing with the nature and behavior of numbers and number systems) contains propositions that cannot be proven either true or false on the basis of its own postulates. The Second Incompleteness Theorem states that if a theory of numbers is contradiction-free, then this fact cannot be proven with common reasoning methods. set theory . He was a friend of Albert Einstein during the time they were both at the Institute for Advanced Study at Princeton University Read more about it at: William Denton provides insight into the nature of incompleteness as applied to mathematical theories. Last updated on: Jun 25, 2002

19. The Concept Of Completeness Captivates Mankind Because Of Its Infinite Implicati Gödel, and his incompleteness theorem. Provability is a weaker notion than truth . Douglas R. Hofstadter. Mark Wakim. 802-686-127. Honor’s Collegium 41. http://www.math.ucla.edu/~rfioresi/hc41/Goedel.html

G del, and his Incompleteness Theorem "Provability is a weaker notion than truth " - Douglas R. Hofstadter Mark Wakim Honor’s Collegium 41 Professor Fioresi The concept of Completeness captivates mankind because of its infinite implications. Completeness bestows upon a body of knowledge a stigma of high aptitude, but more importantly illustrates a final state incapable of being improved upon. Completeness, in a conventional, non-technical sense, simply means: to make whole with all necessary elements or parts. The finality of any work that is "complete" should be the goal of every creative individual. In 1931, Kurt Gödel ’s Incompleteness Theorem illustrated that in a mathematical system there are propositions that cannot be proved or disproved from axioms within the system. Moreover, the consistency of axioms cannot be proved. Such a shattering theorem wrought havoc within the mathematical community. Partially due to its disturbing consequences, Gödel’s Incompleteness Theorem has remained one of the lesser known (though most profound) advancements of this century. With its 1931 publication, Principia Mathematica und verwandter Systeme showed that a sense of "completeness" for the mathematical community was out of reach in certain respects. That is to say, "It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules."

20. An Incompleteness Theorem For Calculating The Future Email the webmaster. SFI Working Paper Abstract 1996. Title An IncompletenessTheorem for Calculating the Future. Author(s) David H. Wolpert. http://www.santafe.edu/sfi/publications/Abstracts/96-03-008abs.html

Publications Santa Fe Institute Quick Search Publications Santa Fe Institute Bulletin Santa Fe Institute Books (alphabetical) Stanislaw Ulam Memorial Lecture Series ... Bibliography Home Page Santa Fe Institute Questions? Email the webmaster SFI Working Paper Abstract Title: An Incompleteness Theorem for Calculating the Future Author(s): David H. Wolpert Files: postscript pdf Paper #: Abstract: Keywords: uncompatibility, incompleteness, time-series analysis, indecidability Return to the 1996 working papers list. SFI Home Page Business Network Education Employment ... Find Us

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