Godel's Incompleteness Theorem

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	The Eternal Quest - Rebooted: Einstein and Gödel's Freinship
		With his incompleteness theorem he had shaken the foundations of mathematics, prompting the great mathematician Hilbert to propose a new law of logic just to refute Gödel's results.
		The Gödel universe, correctly understood, shares with his incompleteness theorem an underlying methodology and purpose.
		The friendship between Einstein and Gödel was one of the most fruitful in the history of science and yet also the most neglected notes Pale Yourgrou in the Chronicle of Higher education.
	theeternalquest.blogspot.com /2004/12/einstein-and-gdels-freinship.html (471 words)

	Waldo & Karin's Perceptions: Proof of Gödel's Incompleteness Theorem
		Waldo and Karin's Perceptions: Proof of Gödel's Incompleteness Theorem
		Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms of that mathematical branch itself.
		Now Gödel laughs his high laugh and asks UTM whether G is true or not.
	www.gamma.za.net /blog/archives/2004/12/proof_of_gdels.html (399 words)

	3quarksdaily: What Gödel's Incompleteness Thoerem means and doesn't mean
		The first is that Gödel's theorem imposes some some of profound limitation on knowledge, science, mathematics.
		Now, as to science, this ignores in the first place that Gödel's theorem applies to deduction from axioms, a useful and important sort of reasoning, but one so far from being our only source of knowledge it's not even funny.
		This brings us to the other, and possibly even more common fallacy, that Gödel's theorem says artificial intelligence is impossible, or that machines cannot think.
	3quarksdaily.blogs.com /3quarksdaily/2005/06/what_gdels_inco.html (260 words)

	Goedels impact on AI
		Gdel 6: Kurt Gdel: ber eine bisher noch nicht bentzte Erweiterung
		Gdel's contributions to logic and his engagement for this discipline influenced a lot of young people in their decision to study logic.
		Gdel is not the father of A.I.##, but he can be considered as a grandfather (together with Alan Turing and possibly John von Neumann).
	www.univie.ac.at /bvi/jimmy/Artikel/goedelai.htm (3390 words)

	The Australian National University Undergraduate Handbook 1999
		Syllabus: Foundations: First order logic, Turing machines, Gdels incompleteness theorem, axiomatization of set theory, model theory.
		Elementary conformal mapping, integration, theorem and formula, convergence of holomorphic function, Cauchy integral theorems, representation of a holomorphic function by its Taylor series, isolated singularities, residues, residue theorem and its application to real integration.
		Argument principle, Runge’s Theorem, monodromy theorem, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.
	www.anu.edu.au /sas/handbook/1999/Mathematics.htm (4616 words)

	Philosophy
		Topics discussed include the compactness theorem, the logic of identity, names and descriptions, second-order logic, type theory, the ancestral, the Frege-Russell definition of natural number, and Gdels incompleteness results.
		Topics include schemata and interpretation, models, satisfiability, normal forms, expressive completeness, proof procedures, metalogical laws, soundness and completeness theorems.
		Critical consideration of recent philosophical work from a variety of points of view on the question of what exists, for example: matter, mind, time, space, universal properties, causes, and essences.
	registrar.fsu.edu /9899general/philosop.htm (2108 words)

	PHIL 2340: Incompleteness
		This is a quick overview of the high points of Gödel's incompleteness theorems.
		(In a way, the precise details of the axioms do not matter all that much, because Gödel's incompleteness theorem gives a recipe for finding a true but unprovable sentence of arithmetic for any set of axioms that attempts to capture the truths of arithmetic.)
		Gödel's proof depends on the fact that we can associate with every sentence, and sequence of sentences of FOL (plus '0' and 's') a natural number, its "Gödel number," in such a way that we can also, given a Gödel number, determine the sentence it is the Gödel number (GN) of.
	www.trinity.edu /cbrown/logic/incompleteness.html (1245 words)

	Gödel’s Incompleteness Theorems hold vacuously
		In this essay, we now argue that Theorem XI of Gödel’s paper [Go31a], commonly referred to as “Gödel’s Second Incompleteness Theorem”, also holds vacuously.
		Gödel’s Theorem XI essentially states that, if there is a P-formula [Con(P)] whose standard interpretation is equivalent to the assertion “P is consistent”, then [Con(P)] is not P-provable.
		Meta-theorem 2, Gödel’s Thesis 3 is an invalid implicit assumption, we conclude that Gödel’s Theorem XI is essentially the vacuous meta-assertion:
	alixcomsi.com /CTG_02.htm (1312 words)

	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.
		I have included a sketch of how the incompleteness theorems can be proved using the so-called Gödel sentence for a theory.
		By Gödel's second incompleteness theorem, we can't know that mathematics is consistent.
	www.sm.luth.se /~torkel/eget/godel.html (501 words)

	Gödel’s Incompleteness Theorems hold vacuously
		In this essay, we now argue that Theorem XI of Gödel’s paper [Go31a], commonly referred to as “Gödel’s Second Incompleteness Theorem”, also holds vacuously.
		Gödel’s Theorem XI essentially states that, if there is a P-formula [Con(P)] whose standard interpretation is equivalent to the assertion “P is consistent”, then [Con(P)] is not P-provable.
		Meta-theorem 2, Gödel’s Thesis 3 is an invalid implicit assumption, we conclude that Gödel’s Theorem XI is essentially the vacuous meta-assertion:
	alixcomsi.com /CTG_02.htm (501 words)

	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.
		I have included a sketch of how the incompleteness theorems can be proved using the so-called Gödel sentence for a theory.
		By Gödel's second incompleteness theorem, we can't know that mathematics is consistent.
	www.sm.luth.se /~torkel/eget/godel.html (501 words)

	SSP 11si Gödel, Escher, Bach
		Gödel on the net and a sketch of the incompleteness theorems, as presented by Torkel Franzén of Sweden.
		Home page of J. Lucas, containing his paper "Minds, machines and Gödel" arguing that Gödel's incompleteness theorem indicates minds cannot be explained as (Turing) machines.
		Within that, one group of links is devoted to Godel's theorem and AI.
	www.stanford.edu /class/symbsys11s (705 words)

	93.html
		I don't know of a single professional mathematician who disbelieves Godel's theorem, and that includes cryptographic experts who work professionally with combinatorics and USE incompleteness and related concepts in their profession of attacking and/or producing cryptosystems.
		I am merely saying that an attack on Godel, especially one that states or even implies that his theorem is "stupid" and "phony", will be met with a maelstorm of criticism.
		[...] I say the ^^^^^^ > incompleteness theorems are phony.
	keithlynch.net /cryonet/41/93.html (623 words)

	pal-courses-s05.txt
		The famous incompleteness, undecidability and undefinability results of Godel and Tarski are presented, along with Lob's Theorem about the sentence which says "I am provable".
		With this background, we are going to investigate, whether the proofs of Gödel's incompleteness theorems can be found mechanically.
		We carry out case studies for Gödel's incompleteness theorems and an elementary part of set theory.
	www.cs.cmu.edu /afs/cs/project/pal/www/pal-courses-s05.txt (1827 words)

	On Formally Undecidable Propositions of Principia Mathematica and Related Systems
		Smullyan's "Godel's Incompleteness Theorems" is more difficult, but not impossible and amounts to what would serve as the textbook of a solid mathematical logic course or two at an elite university.
		If you want to study Godel's incompleteness theorems there are other books out there that prove his theorems in a much more refined, shorter, and easier fasion.
		Godel's incompleteness theorem's are without a doubt genious.
	www.mountainstatestech.com /mststore/item_00486669807P.html (1128 words)

	Dan Willard
		I have also developed generalizations of Godel's Second Incompleteness Theorem that closely complement the boundary-case exceptions to it that I have discovered.
		Godel's Incompleteness Theorem states such an axiom system can not prove all the theorems of Peano Arithmetic.
		My JSL-2001 article (below) is the first paper on the historical record to document a type of boundary-case exception for Godel's Second Incompleteness Theorem.
	www.cs.albany.edu /profiles/willard.htm (1128 words)

	Gödel's Incompleteness Theorem
		The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate.
		All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally.
		Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths...
	www.meta-religion.com /Mathematics/Articles/godel_theorem.htm (1128 words)

	Model Theoretic Proofs of the Incompleteness Theorems (ResearchIndex)
		Abstract: Introduction Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or finitary viewpoint.
		1 A new proof of the Godel incompleteness theorem (context) - Boolos - 1989
		3 the incompleteness theorems (context) - Kotlarski - 1994
	citeseer.ist.psu.edu /517582.html (323 words)

	Butterflies and Wheels Article
		He had meant his incompleteness theorems to prove the philosophical position to which he was, heart and soul, committed: mathematical Platonism, which is, in short, the belief that there is a human-independent mathematical reality that grounds our mathematical truths; mathematicians are in the business of discovering, rather than inventing, mathematics.
		And then there’s the more philosophical fallout from his theorems, the light they shed not only on the nature of mathematical knowledge - the fact that it can’t be captured in a formal system - but also on the nature of the mathematical knower herself.
		I came to feel extremely close to my subject while I wrote “Incompleteness.” Of course it wasn’t that all-penetrating closeness that a writer feels with her characters, but there was something sometimes approximating it.
	www.butterfliesandwheels.com /articleprint.php?num=116 (2978 words)

	Review of Rodriguez-Consuegra
		Since, by Tarski's theorem, the set of truth sentences of the language of first-order arithmetic is not even arithmètic, it seems likely that incompleteness theorems hold for any plausible mechanistic model, though the undecidable sentences may be of greater logical complexity than those Gödel and Jeroslow provide for formal systems and experimental logics respectively.
		The incompleteness theorems show that we do not, even in principle, thereby come to know what theorems are derivable from our postulates.
		His formal argument appears to make use only of considerations arising out of the incompleteness theorems, but the discussion of set-theoretic axioms perhaps renders his conclusions more plausible by providing a concrete example of a mathematical intuition obtainable from reflection on abstract concepts.
	www.philosophy.unimelb.edu.au /handouts/161042/gibbs.html (1599 words)

	Stop Smiling Magazine: The magazine for high-minded lowlifes
		The Incompleteness Theorems are a pair of mathematical ideas, rigorously proven by Gödel in a completely new way, that are significant less for what they say than for their broad implications for many other fields.
		It is possible to understand the Incompleteness Theorems without a math degree, and the degree to which the general intelligent reader can be made to comprehend the man’s work is the real test of any book about Gödel. Goldstein does an admirable job, presenting the information stripped of all but the most necessary technical lingo.
		By the age of twenty-three, he completed his work on The Incompleteness Theorems, which has since made him known as “the most important logician since Aristotle.” But that is only the beginning.
	www.stopsmilingonline.com /books_detail.html?id1=257 (789 words)

	theorems.html
		Now if ~G were a theorem of T, then (by the defining property of G) it would be provable in T that there is a y such that Prf(y,#G).
		However, since G is not in fact a theorem of T if T is consistent, there is no proof in T of A, and therefore ~Prf(n,#G) is provable in T for every numeral n.
		Thus []A is a formula which says that A is a theorem of T. This notation can be iterated, so that [][]A is a formula which says that []A is a theorem of T. We will see later how this iteration is useful.
	www.sm.luth.se /~torkel/eget/godel/theorems.html (1319 words)

	Isitpossible.html
		Godel's theorems and Penrose's subsequent applications are difficult and confusing for non-mathemeticians and logicians.
		Immediately after Penrose's book was published many logicians and mathemeticians criticized his ideas saying that Godel's theorems don't apply to the field of artificial intelligence.
		Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulae which cannot {44} be proved-in-the-system, but which we can see to be true.
	dubinserver.colorado.edu /prj/cca/Isitpossible.html (1138 words)

	712.txt
		It begins with the Incompleteness Theorems of Godel and continues with other undecidability results, including Hilbert's 10th Problem.
		Math 712 begins with an introduction to first-order languages and a proof of Godel's Completeness Theorem.
		Math 713 treats the topics of incompleteness, undecidability and computability.
	www.math.umd.edu /~dwk/712.txt (97 words)

	Deconstructing Mathematics and Mind
		Finally, Godel's First Incompleteness Theorem can be proved to hold for a finitely axiomatizable part (subtheory of) of F known as Q. So Q, like F, is undecidable.
		Penrose attempts to derive from Godel's First Incompleteness Theorem an argument to the effect that our own reasoning powers, those for example, possessed collectively by the mathematical community in its search for mathematical truth, necessarily exceed the reasoning powers of any conceivable computer.
		The notes that follow contain an outline of the proofs of the incompleteness theorems, the unsolvability of the halting problem, and some of the arguments of the Penrose versus AI debate.
	www.ptproject.ilstu.edu /godel01.htm (1934 words)

	Topics in Logic Syllabus (Spring 2002)
		It includes proofs of soundness and completeness theorems for first-order logic, the compactness theorem, the upward and downward Löwenheim-Skolem theorems, and culminates in proofs of Gödel's incompleteness theorems.
		We will also consider relations between these metalogical results and results in computability theory, and will discuss arguments that these results have significant philosophical implications, notably Penrose's argument that the incompleteness theorems show that artificial intelligence is impossible and Hilary Putnam's argument that the Löwenheim-Skolem theorem shows that metaphysical realism is untenable.
		proofs of Gödel's first and second incompleteness theorems
	www.trinity.edu /cbrown/topics_in_logic/syllabus.html (524 words)

	Model Theoretic Proofs of the Incompleteness Theorems (ResearchIndex)
		Abstract: Introduction Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or finitary viewpoint.
		1 A new proof of the Godel incompleteness theorem (context) - Boolos - 1989
		3 the incompleteness theorems (context) - Kotlarski - 1994
	citeseer.ist.psu.edu /517582.html (524 words)

	Inconsistent Mathematics
		Hilbert's program was widely held to have been seriously damaged by Gödel's Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself.
		Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		This is because the natural account of the negation of a proposition in such a space says that it holds on the largest open set contained in the Boolean complement of the set of points on which the original proposition held, which is in general smaller than the Boolean complement.
	www.braungardt.com /Mathematica/inconsistent_mathematics.htm (524 words)

	The Eternal Quest - Rebooted: Einstein and Gödel's Freinship
		The Gödel universe, correctly understood, shares with his incompleteness theorem an underlying methodology and purpose.
		With his incompleteness theorem he had shaken the foundations of mathematics, prompting the great mathematician Hilbert to propose a new law of logic just to refute Gödel's results.
		The friendship between Einstein and Gödel was one of the most fruitful in the history of science and yet also the most neglected notes Pale Yourgrou in the Chronicle of Higher education.
	theeternalquest.blogspot.com /2004/12/einstein-and-gdels-freinship.html (524 words)

	Deconstructing Mathematics and Mind
		Finally, Godel's First Incompleteness Theorem can be proved to hold for a finitely axiomatizable part (subtheory of) of F known as Q. So Q, like F, is undecidable.
		Penrose attempts to derive from Godel's First Incompleteness Theorem an argument to the effect that our own reasoning powers, those for example, possessed collectively by the mathematical community in its search for mathematical truth, necessarily exceed the reasoning powers of any conceivable computer.
		The notes that follow contain an outline of the proofs of the incompleteness theorems, the unsolvability of the halting problem, and some of the arguments of the Penrose versus AI debate.
	www.ptproject.ilstu.edu /godel01.htm (524 words)

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