1. 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

This theorem is one of the most important proven in this century, ranking with Einstein's Theory of Relativity and Heisenberg's Uncertainty Principle. A related page with some interesting links is . You can also look at available on-line . It's also in print in a nice but inexpensive edition from Dover. Jones and Wilson, An Incomplete Education outside the system in order to come up with new rules an 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

2. INCOMPLETENESS THEOREM Parent Node(s) Web Dictionary of Cybernetics and Systems. INCOMPLETENESSTHEOREM. Goedel's thesis initially about number theory but 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

3. Godel's Incompleteness Theorem Godel's incompleteness theorem. Zillion's Philosophy Pages. Firstlet me try to state in clear terms exactly what he proved, since 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.

4. Mathematical Logic. Around Goedel's Theorem. By K.Podnieks Peruse an online book about Godel's incompleteness theorems, as well as axiomatic set theory, first order calculus and Hilbert's tenth problem. logic, mathematical, what is mathematics, incompleteness theorem, Gödel, online, web, Godel, book, Internet, http://www.ltn.lv/~podnieks

mathematical logic, foundations, mathematics, logic, mathematical, what is mathematics, incompleteness theorem, Gödel, online, web, Godel, book, Internet, Goedel, tutorial, textbook, teaching, learning, study, student, Podnieks, Karlis, philosophy, free, download LU studentiem Karlis.Podnieks@mii.lu.lv My best mathematical paper My book about probabilities ... Picture (first row, third) Hegel, Marx, and Goedel's theorem Digital mathematics and non-digital mathematics Trying to understand non-formalists " Let X = X But Not Necessarily " by William J. Greenberg This web-site presents 100% of two hyper-textbooks for students. Read online, follow links all over the world. Feel free to download any parts. My favorite (printed) textbook in mathematical logic, since many years: "Introduction to Mathematical Logic", by Elliot Mendelson Diploma Around Goedel's Theorem Hyper-textbook for students by Karlis Podnieks Associate Professor University of Latvia Institute of Mathematics and Computer Science English version Russian version ... Diploma Left Adjust your browser window Right Introduction to Mathematical Logic Hyper-textbook for students by Vilnis Detlovs and Karlis Podnieks University of Latvia Four provably equivalent definitions of mathematics: Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.

5. Incompleteness Theorem incompleteness theorem. If we could we could derive and this would imply thatis false. This result is Gödel 's second incompleteness theorem . 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

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. Godel's Theorems Gödel's incompleteness theorem Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions (Hofstadter 1989). 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.

8. Gödel's Incompleteness Theorem Gödel's incompleteness theorem. Kelly S. Cline http://physics.colorado.edu/~clineks/godel

Next: Kelly S. Cline Ever since Euclid, mathematicians have worked toward the axiomization of mathematics. By describing mathematics with a set of logical rules and axioms, many have believed that a total understanding of the number system, geometry, and all other fields of mathematics could be gained. Further, they attempted to not only axiomize the number system, but logic and proof technique itself, reducing all mathematics to the level of algorithm. This was hoped to be the ultimate in the systemization of mathematics, and lead to the resolution of every mathematical question. Main Idea A Generalized Mathematical Language Sets The Set ... Expressible Sets and Provability Great Theorem: There is a Statement Which is True, but Unprovable T Back to Kelly's Page Kelly Cline

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. A Burbanks / Research / Topics / Logic And Proof Theory 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: Dr A Burbanks research topics Home ... Artwork e.g. Posters e.g. Miscellany Photos Links Contact Research topic summaries Logic and proof theory Topics Combinatory logic Incompleteness theorems Natural independence phenomena including a related seminar available on the web Algorithmic chaos My interest in this area is mainly recreational and expository; I have always been fascinated by certain aspects of logic and would like to help make these results accessible to a wider audience. Combinatory logic: Combinatory logic involves the study of abstract objects called combinators which may be applied to each other to yield new combinators; it is not important what the combinators `are', only how they act upon each other. (This set-up is sometimes referred to as an `applicative system'.) In his marvellous book, To Mock a Mocking Bird Raymond Smullyan introduces combinatory logic by choosing the combinators as songbirds, living in a forest: if we call-out the name of a bird B to another bird A, then A will answer by singing the name of a third bird, which we call A's response to B, written AB. We then say, for example, that bird A is fond of bird B if AB=B. By looking at different forests (with different sets of axioms) many of the important results in the theory are revealed. (An interesting bird is the Mocking Bird, M, such that for any bird x we have Mx=xx, i.e. the Mocking bird's response to x is the same as x's response to itself.) Incompleteness theorems: Part of Hilbert's famous programme for mathematics

11. 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

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

12. On Computable Numbers (decision Problem) ... - Entry Page At Abelard.org 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 This document uses advanced technology. Does this embedded character z match this character? You will need to use a CCS2-enabled browser to see this document in full. If, on your screen, the embedded character (on the left) does not appear similar to the character on the right, your browser is not CCS2-enabled. A suitable browser is available to down load (free) from Microsoft Such browsers (Microsoft browsers version 4 and higher ) can also be found on many software CD-ROMs. If the characters do look similar, or if you don't care, click here to go to this Turing document on this site, at http://www.abelard.org/turpap2/tp2-ie.asp. For those interested in the reason for this, the document includes embedded text that can only be viewed with a browser enabled to interpret CSS2 (cascading style sheets). 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

13. 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

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.

14. Incompleteness Theorem - A Whatis Definition The incompleteness theorem is a pair of logical proofs that revolutionized mathematics. The first result was published by Kurt Godel (19061978) in 1931 when he was 24 years old. http://whatis.techtarget.com/definition/0%2C%2Csid9_gci835123%2C00.html

Search our IT-specific encyclopedia for: or jump to a topic: Choose a topic... CIO CRM Databases Domino Enterprise Linux IBM S/390 IBM AS/400 Networking SAP Security Solaris Storage Systems Management Visual Basic Web Services Windows 2000 Windows Manageability 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

15. Goedel's Incompleteness Theorem. Gödel's Theorem. Liar's Paradox Goedel, incompleteness theorem, Gödel, liar, paradox, self reference, second, theorem, Rosser, Godel, Bernays, http://www.ltn.lv/~podnieks/gt5.html

Goedel, incompleteness theorem, Gödel, liar, paradox, self reference, second, theorem, Rosser, Godel, Bernays, incompleteness 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.

16. Goedel's Theorem And Information A GJ.Chaitin proof of Gödel's Theorem using arguments having an algorithmic information theory flavor.Category Science Math Applications Information Theory...... 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.

17. Goedel's Incompleteness Theorem The Undecidability of Arithmetic, Goedel's incompleteness theorem,and the class of Arithmetical Languages. Firstorder arithmetic 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.

18. Incompleteness Theorem - A Whatis Definition 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 IBM S/390 IBM AS/400 Networking SAP Security Solaris Storage Systems Management Visual Basic Web Services Windows 2000 Windows Manageability 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. 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

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. Gerhard Gentzen showed that the consistency and completeness of arithmetic can be proved if transfinite induction is used. However, this approach does not allow proof of the consistency of all mathematics. Consistency Hilbert's Problems Kreisel Conjecture Natural Independence Phenomenon ... Undecidable References Barrow, J. D.

20. Goedel's Incompleteness Theorem - Wikipedia Goedel's incompleteness theorem. (Redirected from Goedels IncompletenessTheorem). These results do not require the incompleteness theorem. http://www.wikipedia.org/wiki/Goedels_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 (Redirected from Goedels Incompleteness Theorem 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.

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