21. Goedel's Incompleteness Theorem. Liar's Paradox. Self Reference. By K.Podnieks What is Mathematics? Goedel s Theorem and Around. Textbook for students. Section 5.By K.Podnieks. http://www.ltn.lv/~podnieks/gt5.html

Goedel, 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.

22. G Del's Theorem On Formally Undecidable Propositions ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS 11 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

23. Goedel's Incompleteness Theorem The Undecidability of Arithmetic, Goedel s incompleteness theorem, Goedel sincompleteness theorem If a proof system for arithmetic is sound (meaning 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.

24. CNN.com - Students Filibuster Against Frist At His Alma Mater - "Cloudy with a Chance of Meatballs" and "Godel's incompleteness theorem" were also featured during filibusters. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

25. Incompleteness Theorem This result is Gödel s second incompleteness theorem . Another way to proveGödel s first incompleteness theorem. is to use the result that we cannot 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

26. Gödel's Incompleteness Theorem Gödel s incompleteness theorem. Gödel s incompleteness theorem. In thissection we lay the groundwork for a simplified version of Gödel s theorem that 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.

27. Incompleteness Theorem - A Whatis.com Definition The incompleteness theorem is a pair of logical proofs that revolutionized The First incompleteness theorem states that any contradictionfree rendition http://whatis.techtarget.com/definition/0,,sid9_gci835123,00.html

Search our IT-specific encyclopedia for: or jump to a topic: Choose a topic... CIO CRM Data Center Domino Enterprise Linux Enterprise Voice Exchange IBM S/390 IBM AS/400 Mobile Computing Networking Oracle SAP Security Small Medium Business SQL Server Storage Visual Basic Web Services Windows 2000 Windows Security Windows Systems 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: Nov 26, 2004

28. Byunghan Kim S Homepage Complete proofs of Gödel s incompleteness theorems. Course lecture on Gödel sincompleteness theorems. Hyperimaginaries and canonical bases http://www-math.mit.edu/~bkim/

Byunghan Kim Mailing Address: Room 2-171 Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139 Telephone: (617) 253-4385 Fax: (617) 253-4358 E-mail: bkim@math.mit.edu I am an Assistant Professor in Mathematics Department at the Massachusetts Institute of Technology Brief CV Recent papers Stability, and simplicity in there. Around stable forking , (with A. Pillay). The notions around stable forking are studied. The geometry of 1-based minimal types , (with T. de Piro). The first chapter of geometric simplicity theory. A note on weak dividing , preprint (with N. Shi). Constructiong the hyperdefinable group from the group configuration , preprint (with T. de Piro and J. Young). Examples around stable definability , preprint (with R. Moosa). Non-stable definability of ACF equipped with a generic relation, pseudo-finite fields and ACFA is shown. "Generalized type-amalgamation and n-simplicity" , in preparation (with A. Kolesnikov and A. Tsuboi).

29. Gödel On The Net Every day, Gödel s incompleteness theorem is invoked on the net to support By Gödel s second incompleteness theorem, we can t know that mathematics is 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.)

30. Gödel's Theorem Gödel s second incompleteness theorem proves that formal systems T satisfying The second incompleteness theorem applies in particular to those formal 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

31. Does Gödel Matter? - The Romantic's Favorite Mathematician Didn't Prove What Goldstein calls Gödel s incompleteness theorem the third leg, In 2001,Timothy Noah discussed how Gödel s incompleteness theorem applied to http://slate.msn.com/id/2114561/

placeAd(1,'slate.news/slate') Print E-mail Discuss Newsletters ... About Us Search Slate Advanced Search placeAd(3,'slate.homepage/slate') placeAd(6,'slate.homepage/slate'); do the math A mathematician's guide to the news. Does G¶del Matter? The romantic's favorite mathematician didn't prove what you think he did. By Jordan Ellenberg Posted Thursday, March 10, 2005, at 4:27 AM PT The reticent and relentlessly abstract logician Kurt G¶del might seem an unlikely candidate for popular appreciation. But that's what Rebecca Goldstein aims for in her new book Incompleteness , an account of G¶del's most famous theorem, which was announced 75 years ago this October. Goldstein calls G¶del's incompleteness theorem "the third leg, together with Heisenberg's uncertainty principle and Einstein's relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the 'exact sciences.' " What is this great theorem? And what difference does it really make? Mathematicians, like other scientists, strive for simplicity; we want to boil messy phenomena down to some short list of first principles called axioms , akin to basic physical laws, from which everything we see can be derived. This tendency goes back as far as Euclid, who used just

32. The Concept Of Completeness Captivates Mankind Because Of Its Infinite Implicati In 1931, Kurt Gödel ’s incompleteness theorem illustrated that in a The proofof Gödel’s incompleteness theorem hinges upon the writing of a 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."

33. Gödel's Incompleteness Theorem: Information From Answers.com Gödel s incompleteness theorem In mathematical logic , Gödel s incompletenesstheorems are two celebrated theorems proved by Kurt Gödel in 1931. http://www.answers.com/topic/g-del-s-incompleteness-theorem

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping G¶del's incompleteness theorem Wikipedia G¶del's incompleteness theorem In mathematical logic G¶del's incompleteness theorems are two celebrated theorems proved by Kurt G¶del in First incompleteness theorem G¶del's first incompleteness theorem is perhaps the most celebrated result in mathematical logic. It basicly says that For any formal theory in which basic arithmetical facts are provable, it's possible to construct an arithmetical statement which, if the theory is consistent, is true but not provable or refutable in the theory The meaning of "it's 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 it's 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.

34. Language Log: A New Incompleteness Theorem A new incompleteness theorem. Or is it just a new proof by talkpage diagonalizationof the same old result? No web forum sufficiently powerful to express http://itre.cis.upenn.edu/~myl/languagelog/archives/002081.html

Language Log Main April 20, 2005 A new incompleteness theorem Or is it just a new proof by talk-page diagonalization of the same old result? "No web forum sufficiently powerful to express interesting things can be established as coherent by arguments within its own format..." [Note: a couple of literal-minded readers have emailed to clue me in that the linked page is not in fact an example of the proof technique known as diagonalization . I do know this: it's supposed to be a joke, not a theorem...] Posted by Mark Liberman at April 20, 2005 04:19 PM var site="sm7languagelog"

35. PlanetMath: Gödel's Incompleteness Theorems Gödel s first and second incompleteness theorems are perhaps the most celebrated The second version of Gödel s first incompleteness theorem suggests a http://planetmath.org/encyclopedia/GodelsIncompletenessTheorems.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About G¶del's incompleteness theorems (Theorem) logic , one can formulate properties of theories and sentences as arithmetical properties of the corresponding , thus allowing 1st order arithmetic to speak of its own consistency, provability of some sentence and so forth. On Formally Undecidable Propositions in Principia Mathematica and Related Systems can be stated as Theorem No theory axiomatisable in the type system of PM (i.e., in Russell's theory of types ) which contains Peano-arithmetic and is -consistent proves all true theorems of arithmetic (and no false ones). Stated this way, the theorem is an obvious corollary of Tarski's result on the undefinability of truth formula , and by Tarski's result it isn't definable by any arithmetic formula. But assume there's a theory

36. PlanetMath: Gödel's Incompleteness Theorems The second version of Gödel s first incompleteness theorem suggests a Gödel s second incompleteness theorem concerns what a theory can prove about its http://planetmath.org/encyclopedia/IncompletenessTheorem.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About G¶del's incompleteness theorems (Theorem) logic , one can formulate properties of theories and sentences as arithmetical properties of the corresponding , thus allowing 1st order arithmetic to speak of its own consistency, provability of some sentence and so forth. On Formally Undecidable Propositions in Principia Mathematica and Related Systems can be stated as Theorem No theory axiomatisable in the type system of PM (i.e., in Russell's theory of types ) which contains Peano-arithmetic and is -consistent proves all true theorems of arithmetic (and no false ones). Stated this way, the theorem is an obvious corollary of Tarski's result on the undefinability of truth formula , and by Tarski's result it isn't definable by any arithmetic formula. But assume there's a theory

37. Gödel\'s Incompleteness Theorem - Definition Of Gödel\'s Incompleteness Theore Searchword not found in the selected dictionary, but you can try the following http://encyclopedia.laborlawtalk.com/Gödel's_incompleteness_theorem

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39. 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 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

40. Gödel’s Incompleteness Theorem Gödel’s incompleteness theorem Gödel s incompleteness result doesn t touchdirectly on the most important sense of completeness and incompleteness, http://www.daviddarling.info/encyclopedia/G/Godels_incompleteness.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Gödel’s incompleteness theorem In a nutshell: All consistent axiomatic systems contain undecidable propositions. What does this mean? An axiomatic system consists of some undefined terms, a number of axioms referring to those terms and partially describing their properties, and a rule or rules for deriving new propositions from already existing propositions. There are a couple of main reasons why axiomatic systems are so useful: first, they're compact descriptions of the whole field of propositions derivable from the axioms, so large bodies of math can be compressed down into a very small compass; second, because they're so abstract, these systems let us derive all, and only, the results that follow from things having the formal properties specified by the axioms. An axiomatic system is said to be consistent if, given the axioms and the derivation rules, it doesn't lead to any contradictory propositions. One of the first modern axiomatic systems was a formalization of simple arithmetic (adding and multiplying whole numbers), achieved the great logician Giuseppe

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