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	Soundness theorem - Wikipedia, the free encyclopedia
		Soundness theorems are among the most fundamental results in mathematical logic.
		The weak soundness theorem for a deductive system is the result that any sentence that is provable in that deductive system is a true on all interpretations or models of the semantic theory for the language upon which that theory is based.
		However, a soundness theorem is generally considered a minimal requisite to have an interesting deductive system at all.
	en.wikipedia.org /wiki/Soundness_theorem (433 words)

	NationMaster - Encyclopedia: Soundness
		The soundness theorem is a theorem in mathematical logic stating for a given system of inference rules and system of axioms satisfying certain conditions, any first-order formula that is provable is universally valid.
		Sound is characterized by the properties of sound waves which are frequency, wavelength, period, amplitude and velocity or speed.
		Since the velocity of sound is approximately the same for all wavelengths, frequency is often used to better describe the effects of the different wavelengths.
	www.nationmaster.com /encyclopedia/Soundness (494 words)

	Soundness - Wikipedia, the free encyclopedia
		For soundness in mathematical logic see the entry on the soundness theorem.
		Suppose we have a sound argument (in this case a syllogism):
		The argument is valid and since the premises are in fact true, the argument is sound.
	en.wikipedia.org /wiki/Soundness (134 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)

	ipedia.com: Original proof of Gödel's completeness theorem Article (Site not responding. Last check: 2007-10-30)
		The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalism that are outdated and terminology that is o...
		The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) is not easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure.
		We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove "φ is either refutable or satisfiable".
	www.ipedia.com /original_proof_of_goedel_s_completeness_theorem.html (1424 words)

	Original proof of Gödel's completeness theorem : Goedels completeness theorem/Original Proof
		The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) isn't easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure.
		If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ isn't satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.
		The Normal Form theorem[?] proves that such a ψ exists for every φ, and the construction of ψ from φ adds no new function or constant symbols.
	www.fastload.org /go/Goedels_completeness_theorem___Original_Proof.html (1486 words)

	Classical Logic
		The theorem clearly holds if θ is atomic, since in those cases only the values of the variable-assignments at the variables in θ figure in the definition.
		A converse to Soundness (Theorem 18) is a straightforward corollary:
		Soundness and completeness together entail that an argument is deducible if and only if it is valid, and a set of formulas is consistent if and only if it is satisfiable.
	plato.stanford.edu /entries/logic-classical (11911 words)

	Original proof of goedel s completeness theorem - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-30)
		Start the Original proof of goedel s completeness theorem article or add a request for it.
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	www.sciencedaily.com /encyclopedia/original_proof_of_goedel_s_completeness_theorem (212 words)

	Mathematics, Metamathematics and Computer Science (Site not responding. Last check: 2007-10-30)
		Theorem (Soundness): If F is a theorem, then F is valid (If all branches of the tableau proof of A from Ss are closed then Ss = A).
		Comment: This theorem shows that the method of proof preserves the truth of statements and may be restated as: If a formula is a theorem, then it is valid.
		The question is the converse of the soundness theorem: If a formula F is valid, then it is a theorem. The other direction is developed in the next theorem.
	cs.wwc.edu /~aabyan/CII/MetaMath.html (1374 words)

	Between The Motion And The Act...:A review of "Shadows of the Mind" by Roger Penrose
		And this conclusion is certainly correct: since the soundness of an algorithm (or at least its consistency) is a mathematical fact, mathematicians who only believe the theorems proved by an algorithm will only believe that the algorithm is sound if it proves itself to be sound.
		But that is not being "knowably sound" in the sense that Gödel's theorem requires, since it is not a matter of establishing the soundness by any mathematical or formal considerations.
		As soon as we switch from the idea of an algorithm that manipulates mathematical symbols to one that manipulates representations of physical states, it becomes inescapable that the soundness of the algorithm (in terms of the sentences it eventually produces) is necessarily beyond the grasp of the person whose brain is being modeled.
	psyche.cs.monash.edu.au /v2/psyche-2-02-maudlin.html (5288 words)

	Ebbinghaus, Flum, Thomas. Mathematical Logic. (Site not responding. Last check: 2007-10-30)
		The idea behind Godel's Completeness Theorem is explained, with an intuitive idea of "propositions," (semantic) "consequences," and "proofs." Also, an outline of the material after Godel's Completeness Theorem is given.
		The "correctness" [soundness] of each rule is shown as it is introduced, leading to the "Correctness Theorem" [Soundness Theorem].
		Tarski's Theorem that "Arithmetic truth is not representable in arithmetic" follows (since complete number theory is complete and representability in arithmetic is equivalent to definability in arithmetic).
	www.andrew.cmu.edu /~cebrown/notes/ebbinghaus.html (1936 words)

	Prof. Nipkow: Semantik WS 2001/2002
		A rephrased type safety theorem based on substitutions for type variables: type correct statements guarantee type safe execution starting from any state that is type correct w.r.t.
		Definition of the set of start states that guarantee termination using the Knaster-Tarski fixpoint theorem to obtain the least fixpoint of a monotone function (in the case of loops).
		Soundness of Hoare logic: proof by rule induction.
	www4.in.tum.de /lehre/vorlesungen/semantik/WS0102 (638 words)

	Soundness and Completeness (Site not responding. Last check: 2007-10-30)
		Informally, soundness means that if we can show by equational logic that t = t' is a theorem then the the realisations of t and t' within the algebra will be identified.
		The only way that this can be a theorem of the logic is if it is added it to the set E of equations, in which case it is axiomatically true.
		Gödel's Theorem states formally that it is not possible to construct a finitely axiomatized theory of numbers which is complete (though we could do a lot better than this).
	homepages.feis.herts.ac.uk /~comqejb/algspec/node11.html (415 words)

	AMIL-commentary
		Theorem 17G tells us that whenever we have a decidable (or at least semidecidable) set Sigma, then Cn Sigma is sure to be semidecidable.
		The theorem implies that all of high school algebra and geometry is included in a certain decidable theory.
		Theorem 34A on page 232 tells that every recursive relation (i.e., a relation representable in some finitely axiomatizable consistent theory) is in fact representable in the theory given by A_E.
	www.math.ucla.edu /~hbe/amil/commentary.html (9633 words)

	[No title]
		An alternative approach, justified by Herbrand's theorem, is to generate the ground instances of such a problem and use a propositional decision system to determine the satisfiability of the resulting propositional problem.
		This theorem cannot be expressed in ZF, and its proof requires reasoning at the meta-level.
		The emphasis of the paper is on the derivation of an adequate and sound set of fairness requirements (both weak and strong) which enable proofs of liveness properties of the abstract system, from which we can safely conclude a corresponding liveness property of the original parameterized system.
	floc02.diku.dk /floc02.bib (8233 words)

	First Class Test -- Section N
		Once one reaches a recognizable theorem at the end of such a proof, with equal signs down the left margin, it follows by Transitivity and Equanimity that all lines in the proof were theorems.
		Then by definition of theorem, there is a proof of P starting from the axioms of E', using the inference rules of E', i.e., the inference rules of E.
		If P is a theorem in the new system, then a proof of it in the new system is automatically a proof of it in our original system (a proof that avoids using a certain axiom).
	math.yorku.ca /Courses/9798/Math2090/rg_t1_answers/rg_t1_answers.html (1636 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		There are two very important theorems of Boolean logic that we do not have time for in this course: the Soundness Theorem and the Completeness Theorem.
		Soundness and Completeness in pdf format is by our text's authors Gries and Schneider.
		The Soundness Theorem asserts that every theorem of predicate logic is a valid formula.
	www.math.yorku.ca /Who/Faculty/dhpell/1090/SoundComplete.html (173 words)

	Godel's Completeness Theorem (Site not responding. Last check: 2007-10-30)
		Hence, if you believe the Soundness Theorem, we should not expect to be able to prove either S or (not S) from F because there is one model of F in which S is true, and one where S is false.
		However, there are damn good reasons why it is incomplete because there are statements which can be either true or false depending on which model of F you are currently working.
		The Completeness Theorem basically says that this is the only way a system can be incomplete.
	www.math.uchicago.edu /~mileti/museum/complete.html (492 words)

	Axiomatic Theories of Truth
		Tarski's theorem on the undefinability of the truth predicate shows that a definition of such a predicate requires resources that go beyond those of the formal language for which truth is going to be defined.
		The reductions of second-order theories (i.e., theories of properties or sets) to axiomatic theories of truth may be conceived as forms of reductive nominalism, for they replace existence assumptions for sets or properties (e.g., comprehension axioms) by ontologically innocuous assumptions, in the present case by assumptions on the behaviour of the truth predicate.
		According to Gödel's incompleteness theorems, the statement that Peano Arithmetic (PA) is consistent, in its guise as a number-theoretic statement (given the technique of Gödel numbering), cannot be derived in PA itself.
	plato.stanford.edu /entries/truth-axiomatic (6002 words)

	Geometry.Net - Theorems_And_Conjectures: Completeness Theorem
		The vision behind the notion of first order languages is centered on the so-called "domain" - a collection of "objects" that you wish to "describe" by using the language.
		Theorem 3.2.2 completeness theorem in R. Let be a Cauchy sequence of realnumbers.
		Define the set S R j Since the sequence is bounded (by part one of the theorem), say by a constant M, we know that every term in the sequence is bigger than -M.
	www.geometry.net /detail/theorems_and_conjectures/completeness_theorem.html (1939 words)

	Soundness Theorem (Site not responding. Last check: 2007-10-30)
		NEW YORK -- Before his musical won three Tony Awards, before Barbra Streisand's people were calling him to write her a song or he was collaborating with the "South Park" boys, Jeff Marx was just another law school grad knocking around Manhattan, wondering what to do with his life.
		Soundness theorem,bayes theorem,binomial theorem,brushes,calculus,central limit theorem,fermats last theorem,geometry...
		The strong soundness theorem for real closed fields and Hilbert's Nullstellensatz in second order arithmetic...
	www.soundnesstheorem.info (346 words)

	Paper Abstracts
		This results in a flexible framework for studying and improving practical programming languages where the type of an object gives certain implementation guarantees, such as would be needed to statically determine the offset of a function in a method lookup table or safely implement binary operations without exposing the internal representation of objects.
		Type soundness is proved using operational semantics and an analysis of typing derivations.
		Type soundness is proved using operational semantics and examples illustrating the expressiveness of the pure calculus are presented.
	www.research.att.com /~kfisher/abstracts.html (1123 words)

	Logic II
		For the incompleteness theorems the following results are essential: Sigma-completeness of Robinson's arithmetic, recursiveness of logical syntax and definability of all RE sets in the standard model of arithmetic.
		Every sound recursively axiomatizable extension of Robinson's arithmetic is incomplete and undecidable (Goedel's 1st incompleteness theorem).
		Using effectively recursively enumerable sets one can replace the assumption of soundness in Goedel's 1st incompleteness theorem by an assumption of mere consistency; this is the Rosser's generalisation.
	www.cuni.cz /%7Esvejdar/courses/logic_ii.html (672 words)

	Inductive Theorem Prover INKA 5.0
		Abstract: Originally developed as an automatic inductive theorem prover, we describe a new version INKA 5.0, which is a result from the long experience made in formal software development.
		In order to ensure soundness within a reasoning level, we use techniques developed in the context of matrix characterization relying on the notion of indexed formulas.
		In this scenario, user interaction traditionally is restricted to the mode in which the user decides which tactic to apply on the top-level, without being able to interact with the tactic once it has begun running.
	www.dfki.de /vse/systems/inka/inka5.html (956 words)

	Review of Stephen G. Simpson: Subsystems of Second Order Arithmetic
		Thence, if a theorem has been proven from the right (set existence) axioms, the axioms themselves can be proven from the theorem.
		Hence the theorem and the axioms used to prove it are equivalent (over a weak base theory).
		, is equivalent for example to Baire's category theorem, the intermediate value theorem, the soundness theorem and a weak form of Gödel's incompleteness theorem, and the existence of the algebraic closure of a countable field.
	www.math.psu.edu /simpson/sosoa/moellerfeld (1238 words)

	AMCA: A Model Based Cut Elimination Proof by Olivier Hermant
		In order to prove the cut-elimination theorem, we prove the completness of the cut-free sequent calculus modulo, that is, Gödel's completness theorem.
		The proof of a sequent is made with the help of deduction rules and the cut rule is the only rule where we have a proposition P in premises that doesn't appears in the conclusion.
		This theorem implies the consistency of sequent calculus.
	at.yorku.ca /c/a/j/y/09.htm (1124 words)

	03: Mathematical logic and foundations
		For example, the Löwenheim-Skolem-Tarski Theorem asserts that if there are any models, then there are models of every infinite cardinality, unless there is a (finite) upper bound on the cardinalities of the models (assuming the language is countable).
		We are interested in the "theorems" of T, which is the smallest set S of formulas which includes T and is closed under certain operations (the "rules of inference"), such as modus ponens (if both "A \arrow B" and "A" are in S, then "B" must also be in S).
		The theorems are defined in a purely procedural way, yet they should be related to those statements which are (semantically) "true", that is, statements which are valid in every model of those axioms.
	www.math.niu.edu /~rusin/known-math/index/03-XX.html (2050 words)

	Classical Two-valued Logic (Site not responding. Last check: 2007-10-30)
		The question is the converse of the soundness theorem: If a formula F is valid, then it is a theorem.
		Theorem (Completeness I - Godel): For every sentence S and set of sentences Ss, the tableau method establishes either Ss = S or constructs a model of {Ss, ~S}.
		Theorem (Skolem-Lowenheim): If a countable set of sentences Ss is satisfiable, (that is, it has a model), then it has a countable model.
	cs.wwc.edu /~aabyan/Logic/Classical.html (1759 words)

	Proof Theory as an Alternative to Model Theory
		The design and analysis of declarative programming languages require that computation be described in two separate ways: by an interpreter that extracts values from a program and by a declarative specification of what values should be attributed to a program.
		A soundness theorem for a programming language states that its interpreter computes only attributed values and a completeness theorem states that the interpreter extracts all such values.
		Since these two concepts -- resolution and model theory -- are very different, soundness and completeness theorems give us some confidence that Horn clause programming is not "hacky" and that they will have useful meta-level properties.
	www.cs.rice.edu /~taha/ProofTheoryAsAlternative.html (752 words)

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