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Topic: Incompleteness theorems

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	PlanetMath: Gödel's incompleteness theorems
		This second form of the theorem is the one usually proved, although the theorem is usually stated in a form for which the nonconstructive proof based on Tarski's result would suffice.
		The second version of Gödel's first incompleteness theorem suggests a natural way to extend theories to stronger theories which are exactly as sound as the original theories.
		This is version 9 of Gödel's incompleteness theorems, born on 2003-08-12, modified 2005-03-08.
	planetmath.org /encyclopedia/GodelsIncompletenessTheorems.html (722 words)

	Penrose's Goedelian Argument: A review of Roger Penrose's "Shadows of the Mind".
		Theorem 2: If F is a formal system (in the general sense) which is sound for the predicate P then it is not complete for it.
		Conversely, to obtain Theorem 1 from Theorem 2, simply take the "formula" phi(q,n) to be the pair (q,n) and the set of "provable formulas" of F to be the set of pairs on which A halts.
		All that the Gödel incompleteness theorem requires of F is the former, since that is equivalent to the consistency of F. But Penrose tends to emphasize the global notion of soundness and to tie it to his Platonistic philosophy of mathematics.
	psyche.cs.monash.edu.au /v2/psyche-2-07-feferman.html (4659 words)

	Society for Philosophy and Technology - Volume 2, numbers 3-4
		Since Gödel's incompleteness theorems have been used to argue against certain mechanistic theories of the mind, it seems natural to attempt to apply the theorems to certain strong mechanistic arguments postulated by some AL theorists.
		I believe that Gödel's incompleteness theorems have some bearing on the question of the validity of the strong claim in AL since the second premise just listed makes a claim to a level of formal completeness that may be subject to the limitations of formal systems described by Gödel.
		In order to show that Gödel's incompleteness theorems have a bearing on AL, we have to prove that it is necessary for strong AL to hold to postulate number 2 as I have stated it above.
	scholar.lib.vt.edu /ejournals/SPT/v2n3n4/sullins.html (5617 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Incompleteness, like religion, stands in an easily-defended bunker: 1) The issues involved are difficult to express with precision and are often counterintuitive.
		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.
	keithlynch.net /cryonet/41/93.html (623 words)

	Gödel’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 mathematical logic and beyond, this result is only the middle movement, so to speak, of a metamathematical symphony of results stretching from 1929 through 1937.
		Gödel was born in 1906 in the Austro-Hungarian province of Moravia.
		It is this second incompleteness result which is generally meant when people talk about “the Gödel Incompleteness Theorem.” It is widely believed to have important repercussions beyond mathematics, since it is really a statement about the limits of formal knowledge, that is, knowledge which depends upon the rigors of logic.
	www.mathacademy.com /pr/prime/articles/godel/index.asp (2041 words)

	Godel Theorems (Site not responding. Last check: 2007-10-20)
		Around Gödel's theorem by Karlis Podnieks: this is an on-line book devoted to the exposition of incompleteness theorems (be careful with his philosophical viewpoints).
		This is not the original paper on incompleteness theorem (which dates back to 1931) but it is a pleasant, clear and definitive exposition; here Gödel works in the second order logic, which makes thing more intuitive and appealing for a mathematician.
		Gödel theorem is proved in his booklet; an introduction is given in Crossley's book and a complete proof in Cohen's book and Manin's book.
	www.math.unifi.it /~caressa/math/godel.html (1138 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.
		On the other hand, the completeness theorem asserts that a sentence is not derivable from a theory if and only if there exists a model of the theory which does not satisfy the sentence, and it suggests the existence of model-theoretic...
		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)

	The History and Kinds of Logic: LOGIC SYSTEMS: Metalogic: DISCOVERIES ABOUT FORMAL MATHEMATICAL SYSTEMS: The two ...
		The proof of this incompleteness may be viewed as a modification of the liar paradox, which shows that truth cannot be defined in the language itself.
		The first half leads to Gödel's theorem on consistency proofs, which says that if a system is consistent, then the arithmetic sentence expressing the consistency of the system cannot be proved in the system.
		The proof of this theorem consists essentially of a formalization in arithmetic of the arithmetized version of the proof of the statement, "If a system is consistent, then p is not provable"; i.e., it consists of a derivation within number theory of p itself from the arithmetic sentence that says that the system is consistent.
	www.cs.auc.dk /~luca/FS2/41.html_bold=on_sw=pincomp.html (566 words)

	Review of Rodriguez-Consuegra
		Experimental logics generalize formal systems: the theorems of any consistent formal system can be included among those of an experimental logic, but experimental logics can have theorems asserting their own consistency and the set of theorems of an experimental logic need not be recursively enumerable.
		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.
	www.philosophy.unimelb.edu.au /handouts/161042/gibbs.html (1599 words)

	Incompleteness Theorems. Consequences. Related Results
		This can be done, indeed, and as a result, we would obtain the first ("Goedelian") version of the double incompleteness theorem: if theories T, M are both w-consistent, then the formula H is undecidable it T, yet this cannot be established in M (see Podnieks [1975]).
		Incompleteness theorems were additional evidence that no stable list of axioms can be sufficient to solve all problems that can appear in mathematical theories.
		The Unsolvability theorem establishes for formal theories essentially the same phenomenon that is well known from the history of (the traditional) mathematics: no particular set of ideas and/or methods allows to solve all problems that arise in mathematics (even when our axioms remain stable and "sufficient").
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt6.html (5956 words)

	Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) (Site not responding. Last check: 2007-10-20)
		His monumental theorem of incompleteness demonstrated that in every formal system of arithmetic there are true statements that nevertheless cannot be proved.
		Her story does not omit the irony that both early and late, his work was misconstrued to support one version or another of "Man is the Measure of all things", which completely denied the point of his work.
		Once you have Turing machines, the proof of the first incompleteness theorem is maybe one sentence ("If arithmetic was complete, then we could solve the halting problem, but we can't").
	www.mountainstatestech.com /mststore/item_00393051692P.html (1334 words)

	American Scientist Online - The Incomplete Gödel (Site not responding. Last check: 2007-10-20)
		Gödel's two incompleteness theorems were his crowning achievement.
		So it would be more accurate to describe Gödel's theorems as having to do with "incompletability" rather than "incompleteness." The second incompleteness theorem states that, given the same hypothesis as the first, one such true but unprovable sentence expresses the consistency of the system.
		She even gets the year in which Gödel first proved his incompleteness theorem wrong—it was 1930, not 1929 as she suggests on page 156.
	www.americanscientist.org /template/BookReviewTypeDetail/assetid/45922 (1927 words)

	Incompleteness Theorems. Consequences. Unsolvability. Creativity in Mathematics. Size of Proofs. Related Results. By ...
		The set of theorems is only a secondary aspect of a real theory: it is the set of assertions that can be proved by using the axioms and rules of theory.
		The double incompleteness theorem shows that the principle of stable safety also is incomplete.
		In mathematics all theorems are being proved by using a stable list of axioms (for example, the axioms of ZFC).
	www.ltn.lv /~podnieks/gt6.html (6321 words)

	On Formally Undecidable Propositions of Principia Mathematica and Related Systems (Site not responding. Last check: 2007-10-20)
		Godel's incompleteness theorem's are without a doubt genious.
		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.
		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.
	www.mountainstatestech.com /mststore/item_00486669807P.html (1128 words)

	Untitled
		Gödel's incompleteness theorems make the philosophical position of naturalism untenable because they imply that human rationality is forever out of reach of complete scientific explanation.
		Gödel's Theorem in particular has nothing at all to tell us about whether there might be algorithms that could do an impressive job of "producing as true" or "detecting as true or false" candidate sentences of arithmetic.
		Second, since Gödel's theorems show that human rationality is scientifically inexplicable, and since there seem to be clear reasons why God might well create beings with our kind of rationality, human rationality is more to be expected with the theistic hypothesis than without it.
	www.utexas.edu /cola/depts/philosophy/faculty/koons/ntse/papers/Baird.html (6214 words)

	Gödel’s Incompleteness Theorems hold 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.
		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.
		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)

	Goedel's Incompleteness Theorems
		I argue that, given the widely accepted understanding of scientific explanation as explanation of phenomena in terms of general laws and specific boundary conditions, Goedel's theorems show that, given certain highly plausible assumptions, humans can never discover a complete scientific explanation of the phenomena of human mathematical ability, and consequently of human psychology in general.
		In two recent books, Roger Penrose has developed an intricate argument that 1) Goedel's theorems do indeed seem to have this implication for science as currently understood, and that therefore 2) the current understanding of science will have to be modified to include real processes which are essentially non-lawlike.
		Having already shown that this kind of psychology almost certainly has no scientific explanation, we are left with a good probability that it has a personal explanation and a reasonably high probability that it has an explanation in terms of the personal God of theism.
	www.leaderu.com /offices/koons/docs/baird.html (304 words)

	[No title]
		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.
		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.
	www.sm.luth.se /~torkel/eget/godel/theorems.html (1319 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.
		Goldstein accurately calls them “the most prolix theorems in the history of mathematics.” They basically say that arithmetic — and, by extension, all mathematics — is an incomplete system, that there are statements in it that are true yet cannot be proven.
		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.
	www.stopsmilingonline.com /books_detail.html?id1=257 (789 words)

	MT5582 Syllabus (Site not responding. Last check: 2007-10-20)
		The course will be centred on proofs of Gödel's two incompleteness theorems and will examine some of their principal applications.
		To introduce the student to recursion theory and to Gödel's two incompleteness theorems and to their most important corollaries.
		Applications of the incompleteness theorem to show the undecidability of the predicate calculus and other axiom systems.
	www.ma.man.ac.uk /pg/logic/OldFile_mt5582.htm (236 words)

	[No title]
		The theorem of Church and Turing asserts that there is no mechanical procedure deciding whether or not a sentence in the language of first-order logic is a logical truth.
		With this background, we are going to investigate, whether the proofs of Gödel's incompleteness theorems can be found mechanically.
		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".
	www.cs.cmu.edu /afs/cs/project/pal/www/pal-courses-s05.txt (1827 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		In the 1980's, Roger Penrose applied these theorems to the field of artificial intelligence, saying they proved that it was impossible to achieve (The idea had actually come about much earlier but Penrose resurrected it for modern interpretation).
		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.
		Godel's theorems and Penrose's subsequent applications are difficult and confusing for non-mathemeticians and logicians.
	dubinserver.colorado.edu /prj/cca/Isitpossible.html (1138 words)

	The two incompleteness theorems (from metalogic) -- Encyclopædia Britannica
		More results on "The two incompleteness theorems (from metalogic)" when you join.
		In geometry, a proposition is commonly considered as a problem (a construction to be effected) or a theorem (a statement to be proved).
		The Pythagorean Theorem is used to calculate the relationship between the legs and angles of a triangle.
	www.britannica.com /eb/article-65869?tocId=65869 (921 words)

	Meta-Mathematics and the Foundations of Mathematics
		The first part is retrospective, and presents a beautiful antique, Gödel's proof; the first modern incompleteness theorem, Turing's halting problem; and a piece of postmodern metamathematics, the halting probability Ω.
		More precisely, we have the following incompleteness result: You need an N-bit formal axiomatic theory (that is, one that has an N-bit algorithm to generate all the theorems) in order to be able to determine the first N bits of Ω, or, indeed, the values and positions of any N bits of Ω.
		I think that incompleteness cannot be dismissed and that mathematicians should occasionally be willing to add new axioms that are justified by experience, experimentally, pragmatically, but are not at all self-evident.
	www.cs.umaine.edu /~chaitin/italy.html (4657 words)

	Provability Logic
		As a reminder, Gödel's first incompleteness theorem states that, for a sufficiently strong formal theory like Peano Arithmetic, any sentence asserting its own unprovability is in fact unprovable.
		Solovay's theorem is so significant because it shows that an interesting fragment of an undecidable formal theory like Peano Arithmetic -- namely that which arithmetic can express in propositional terms about its own provability predicate -- can be studied by means of a decidable modal logic, GL, with a perspicuous possible worlds semantics.
		Discussions of the consequences of Gödel's incompleteness theorems sometimes involve confusion around the notion of provability, giving rise to claims that humans could beat formal systems in “knowing” theorems (see Davis (1990, 1993) for good discussions of such claims).
	plato.stanford.edu /entries/logic-provability (4840 words)

	Mathematics Department 2000-01 Colloquia
		Goedel's incompleteness theorems, like the uncertainty principle in physics, are among the most profound and surprising "negative" scientific discoveries of the twentieth century.
		This talk will explain the ingenious ideas used by Goedel to prove these theorems, and will show how the same ideas were used to prove some of the most important results in recursion theory, the branch of mathematics that studies computers and their capabilities.
		This fact is the content of Goedel's famous incompleteness theorem - one of the deepest theorems in mathematics.
	www.calpoly.edu /~math/colloquia0001.html (4521 words)

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