61. GödelÂ’s Theorems (PRIME) urt Gödel is most famous for his second incompleteness theorem, and many people are unaware that, important as it was and is within the field of http://www.mathacademy.com/pr/prime/articles/godel/index.asp

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Physics Statistics Topology Trigonometry logic and beyond, this result is only the middle movement, so to speak, of a metamathematical symphony of results stretching from 1929 through 1937. These results are: (1) the Completeness Theorem; (2) the First and Second Incompleteness Theorems; and (3) the consistency of the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) with the other axioms of Zermelo-Fraenkel set theory . These results are discussed in detail below. THE COMPLETENESS THEOREM (1929) In 1928, David Hilbert and Wilhelm Ackermann published , a slender but potent text on the foundations of logic. In this text they posed the question of whether a certain system of axioms for the first-order predicate calculus is complete, i.e., whether every logically valid sentence in first-order logic can be derived from the

62. PHYS771 Lecture 3: Gödel, Turing, And Friends The Incompleteness Theorem says that, given any consistent, computable set of axioms, there s a true statement about the integers that can never be proved http://www.scottaaronson.com/democritus/lec3.html

Lecture 3: Gödel, Turing, and Friends Scott Aaronson On Thursday, I probably should've told you explicitly that I was compressing a whole math course into one lecture. On the one hand, that means I don't really expect you to have understood everything. On the other hand, to the extent you did understand hey! You got a whole math course in one lecture! You're welcome. But I do realize that in the last lecture, I went too fast in some places. In particular, I wrote an example of logical inference on the board. The example was, if all A's are B's, and there is an A, then there is a B. I'm told that the physicists were having trouble with that? Hey, I'm just ribbin' ya. If you haven't seen this way of thinking before, then you haven't seen it. But maybe, for the benefit of the physicists, we should go over the basic rules of logic? Propositional Tautologies: A or not A not(A and not A) , etc. are valid. Modus Ponens: If A is valid and A implies B is valid then B is valid. Equality Rules: x=x x=y implies y=x x=y and y=z implies x=z , and x=y implies f(x)=f(y) are all valid.

63. Incompleteness Theorem Incompleteness theorem General Math discussion The proof of G del's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. http://www.physicsforums.com/showthread.php?t=116758

64. Gödel's Incompleteness Theorem | Facebook Welcome to the Facebook Community Page about Gödel s Incompleteness Theorem, a collection of shared knowledge concerning Gödel s Incompleteness Theorem. http://www.facebook.com/pages/Godels-Incompleteness-Theorem/106329082735594

7 people like this. to connect with Wall Info Fan Photos Just Others joined Facebook. April 3 at 11:56pm See More Posts English (US) Español More… Download a Facebook bookmark for your phone. Login Facebook ©2010

65. Godel Incompleteness Theorem | Define Godel Incompleteness Theorem At Dictionary –noun Logic, Mathematics . 1. the theorem that states that in a formal logical system incorporating the properties of the natural numbers , there exists at least one formula http://dictionary.reference.com/browse/godel incompleteness theorem

66. 2nd Paradox In Godels Incompleteness Theorem That Makes Invalid - FrostCloud For 4 posts 2 authors - Last post Oct 21, 2007Arguments why Godels incompleteness theorem is invalid, nightdreamer Godels incompleteness theorem ends in absurdity meaninglessness http://www.frostcloud.com/forum/showthread.php?t=15016

67. Peter Suber, "Gödel's Proof" Hunter's proof of G del's first incompleteness theorem differs from G del's in several ways. Here is a sketch of G del's own method. We'll prove G del's theorem for a http://www.earlham.edu/~peters/courses/logsys/g-proof.htm

Peter Suber Philosophy Department Earlham College Preliminaries ... Reminders Preliminaries We need only three preliminary notions. Proof pairs With the predicate Pxy we can also say that some wff A is not x)Pxa says that there is no sequence that proves A, or that there is no proof pair with a as its second member, or simply that ~ A. We could also say (x)~Pxa. These expressions are about numbers in one interpretation, but they are about the proof theory of S in another. We cannot say that the number theory interpretation is "primary" and the metatheory interpretation "secondary" or nonstandard except in reference to human intentions. As meanings supported by the syntax, they are on a par. Self-Reference Of coures the closed wff, Nn, is talking about the open wff, Nx, which makes the self-reference even more oblique. I will break the proof into four steps: (1) formulating G and understanding how it can simultaneously make an assertion about numbers and about its own provability, (2) showing that G is undecidable, (3) showing that G is true, and (4) drawing the consequences for the incompleteness of S. Formulating G Here's how to construct G. First we make the following wff:

68. Peter Suber, "Kurt Gödel In Blue Hill" Aug 27, 1992 The first incompleteness theorem showed that some perfectly wellformed arithmetical statements could never be proved true or false. http://www.earlham.edu/~peters/writing/godel.htm

This essay originally appeared in the Ellsworth American , August 27, 1992, Section I, p. 2. Peter Suber Ellsworth American serves the Blue Hill area.) For this HTML version I restore the footnotes , which I did not submit to the newspaper, and a sidebar , which the newspaper omitted perhaps for being too technical. 50 Years Later, The Questions Remain Peter Suber Philosophy Department Earlham College The first incompleteness theorem showed that some perfectly well-formed arithmetical statements could never be proved true or false. Worse, it showed that some arithmetical truths could never be proved true. More precisely, for every axiomatic system designed to capture arithmetic, there will be arithmetic truths which cannot be derived from its axioms, even if we supplement the original set of axioms with an infinity of additional axioms. This shattered the assumption that every mathematical truth could eventually be proved true, and every falsehood disproved, if only enough time and ingenuity were spent on them. The second incompleteness theorem showed that axiomatic systems of arithmetic could only be proved consistent by other systems. This made the proof conditional on the consistency of the second system, which in turn could only be validated by a third, and so on. No consistency proof for arithmetic could be final, which meant that our confidence in arithmetic could never be perfect.

69. Goedels Incompleteness Theorem s of the theorem have been merged in from GoedelsTheorem. Discussion of the implications and consequences of the theorem have been extracted to...... http://c2.com/cgi/wiki?GoedelsIncompletenessTheorem

70. Edge: GÖDEL IN A NUTSHELL By Verena Huber-Dyson May 14, 2006 The essence of Gödel s incompleteness theorem is that you cannot have both completeness and consistency. A bold anthropomorphic conclusion http://www.edge.org/3rd_culture/hd06/hd06_index.html

Home About Edge Features Edge Editions ... Subscribe By Verena Huber-Dyson Introduction JB VERENA HUBER-DYSON is emeritus professor of the Philosophy department of the University of Calgary, Alberta Canada, where she taught graduate courses on the Foundations of Mathematics, the Philosophy and Methodology of the sciences. Before the Vietnam war she was an associate professor in the Mathematics department of the University of Illinois. She taught in the Mathematics department at the University of California in Berkeley. She is the author or a monograph, She lives in Berkeley, California. VERENA HUBER-DYSON's Edge Bio Page The first kind are prone to refer to authorities; religion, bureaucracy, governments and their own prejudices. They postulate a Supreme Being that knows all the answers because everything must have an answer. With inconsistencies they deal by hopping over them, brushing them aside, sweeping them under a rug, ignoring them or making fun of them. These people are unpredictable and exasperating to deal with, though often disarmingly charming. The second kind are the more heroic and independent thinkers. They are not afraid of vast expanses of the unknown; they forge ahead and rejoice over every new question opened up by questions answered. But when up against the walls of inconsistencies they go berserk. These claustrophobics are in fact the scientific minds.

71. Godel's Second Incompleteness Theorem Explained In Words Of One Syllable@Everyth Godel's Second Incompleteness Theorem says, officially, that given a set of axiom s A and rules by which you can deduce (prove) theorem s from the axioms, if you can deduce all http://www.everything2.com/index.pl?node_id=1189604

72. Atheism: Common Arguments Gödel s Incompleteness Theorem demonstrates that it is impossible for (This is Gödel s Second Incompleteness Theorem, which is rather tricky to prove. http://www.infidels.org/library/modern/mathew/arguments.html

Library Modern Documents mathew : Common Arguments Common Arguments (1997) mathew Hebrew translation Introduction This document contains responses to points which were brought up repeatedly in Usenet newsgroups and on discussion boards devoted to discussion of atheism. Points covered here are ones which are not covered in the document " An Introduction to Atheism ." Note: It is highly recommended that you read that document first. These answers are not intended to be exhaustive or definitive. The purpose of FAQ documents is not to stifle debate, but to raise its level. Overview of contents: Adolf Hitler was an atheist! The Bible proves it Pascal's Wager (Why God is a safe bet) Lord, Liar or Lunatic? ... Aren't atheists Satanic? Adolf Hitler was an atheist! "Hitler was an atheist, and look at what he did!" Adolf Hitler was emphatically not an atheist. As he said himself: The folkish-minded man, in particular, has the sacred duty, each in his own denomination, of making people stop just talking superficially of God's will, and actually fulfill God's will, and not let God's word be desecrated. [original italics] For God's will gave men their form, their essence, and their abilities. Anyone who destroys His work is declaring war on the Lord's creation, the divine will. Therefore, let every man be active, each in his own denomination if you please, and let every man take it as his first and most sacred duty to oppose anyone who in his activity by word or deed steps outside the confines of his religious community and tries to butt into the other.

73. The Popular Impact Of G Del's Incompleteness Theorem, Vo.ume 53 The Popular Impact of G del's Incompleteness Theorem Torkel Franz n 440 N OTICES OF THE AMS V OLUME 53, N UMBER 4 A mong G del's celebrated results in logic, there are two http://www.ams.org/notices/200604/fea-franzen.pdf

74. Godels Incompleteness Theorem Are Invalid Ie Illegitimate - Science Forums it is argued by colin leslie dean that no matter how faultless godels logic is Godels incompleteness theorem are invalid ie illegitimate for 5 reaso http://www.scienceforums.net/topic/49655-godels-incompleteness-theorem-are-inval

75. Godel's Incompleteness Theorem System Statement Axioms Numbers Godel's Incompleteness Theorem System Statement Axioms Numbers Economy. http://www.economicexpert.com/a/Godel:s:incompleteness:theorem.htm

76. Godels Incompleteness Theorem Invalid-illegitimate Godels incompleteness theorems are invalid ie illegitimate for 5 reasons he uses the axiom of reducibility which is invalid ie illegitimate he constructs http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegit

77. James R Meyer: Gödel's Incompleteness Theorem What is G del's Incompleteness Theorem? Up to now, G del's proof of his Incompleteness Theorem has been perhaps one of the most celebrated proofs in the entire history of http://jamesrmeyer.com/godel_theorem.html

James R Meyer Towards greater understanding of logic and language (Fallback Menu) [Home] [Thoughts..] [Books] [Site Map] ... [Contact] Well, it can be a bit confusing, since the phrase "Gödel's Incompleteness Theorem" has been used for many years to mean one of two things: formal language (subject to certain conditions), there are sentences that cannot be proved to be true or false. The conditions are that the formal language is consistent (consistent: means that the language cannot ever make a contradictory statement) and that it includes sentences about numbers. Usually it is quite obvious which meaning is intended, but see Pedantic Objections formal language "What does it mean to say that a sentence is 'true'?" It also begs the question: "If you can say that a sentence must be 'true', then surely you must have proved it to be 'true'?" "For every formal language, there is a sentence of that formal language which is not provable by the formal language, but it is provable." It can be shown (see below) that Gödel's proof cannot be considered to express some sort of indispensable universal fundamental '"truth"', since the assumptions involved in generating Gödel's result are completely unacceptable by any commonly accepted standards of logic.

78. Gödel's Incompleteness Theorem Completed second draft of this book. PDF version of this book Next The Halting Problem Up Mathematical structure Previous Cardinal numbers Contents http://www.mtnmath.com/whatth/node30.html

Mountain Math Software home consulting videos ... 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.

79. The Romantic's Favorite Mathematician. - By Jordan Ellenberg - Slate Magazine Mar 10, 2005 Gödel s incompleteness theorem says she likens the impact of Gödel s incompleteness theorem to that of relativity and quantum mechanics http://www.slate.com/id/2114561/

80. Gödel S Incompleteness Theorem File Format PDF/Adobe Acrobat Quick View http://www.math.tamu.edu/~berko/other/goedel1.pdf

Page 4     61-80 of 95    Back | 1  | 2  | 3  | 4  | 5  | Next 20