Some will be provable and some will be impossible to corroborate.
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Commissioned by the Washington-based strategic intelligence firm Fusion GPS, the Steele dossier was never intended for public consumption—much of what it assert may not be provable.
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It's called "puffing," and that is the official term that legally protects salespeople and businesses from making boastful claims about their products and services that no one really expects to be provable by empirical facts.
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But we have just proved that is not provable. Since either or must be provable, we conclude that, for all natural numbers is provable. Suppose the negation of , , is provable. Proving both , and , for all natural numbers , violates ω-consistency of the formal theory.
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In other words, the fact that an algorithm listing all total functions in sequence cannot be coded up is not captured by classical axioms regarding set and function existence. We see that, depending on the axioms, subcountability may be more likely to be provable than countability.
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This more general theorem is used implicitly, for example, when a sentence is shown to be provable from the axioms of group theory by considering an arbitrary group and showing that the sentence is satisfied by that group. Gödel's original formulation is deduced by taking the particular case of a theory without any axiom.
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Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved.
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Judicial Commissioner Sundaresh Menon thought that there was a real difference between the reasonable suspicion and real likelihood tests. In his opinion, suspicion suggests a belief that something that may not be provable could still be possible. Reasonable suggests that the belief cannot be fanciful. Here the issue is whether it is reasonable for the one to harbour the suspicions in the circumstances even though the suspicious behaviour could be innocent.
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As just seen, is provable for each natural number , and is thus true in the model ℕ. Therefore, within this model, : P(G(P)) = \forall y\,q(y, G(P)) holds. This is what the statement " is true" usually refers to--the sentence is true in the intended model. It is not true in every model, however: If it were, then by Gödel's completeness theorem it would be provable, which we have just seen is not the case.
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Thus, the common law rule applied to the situation without alteration, and she took away from the relationship and the household what she brought to it. The Court went on to explain that while the state abolished common law marriage in 1896, California law recognizes non- marital relationship contracts. These contracts may be express or implied, oral or written—but they must be provable in any case. The contract may also provide for a sexual relationship as long as it is not a contract for sexual services.
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Hilbert's goal of proving the consistency of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems. However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means. Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work with Paul Bernays some experts such as William Tait have argued that the primitive recursive arithmetic can be considered an upper bound on what Hilbert considered finitistic mathematics.
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There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of F as a formula in the language of F. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons(F) from the second incompleteness theorem is a particular expression of consistency. Other formalizations of the claim that F is consistent may be inequivalent in F, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent subset of PA" is consistent.
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When used, Step II involves showing that each of the axioms is a (semantic) logical truth. The Basis steps demonstrate that the simplest provable sentences from are also implied by , for any . (The proof is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable—by considering each case where we might reach a logical conclusion using an inference rule—and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "" we can derive " or ".
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The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be GF itself. In this way, the Gödel sentence GF indirectly states its own unprovability within F (Smith 2007, p. 135). To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system.
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They may be provable, even if they cannot all be derived from a single consistent set of axioms."Platonism in the Philosophy of Mathematics", (Stanford Encyclopedia of Philosophy) Set-theoretic realism (also set-theoretic Platonism)Ivor Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, 2002, p. 681. a position defended by Penelope Maddy, is the view that set theory is about a single universe of sets.Naturalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of Paul Benacerraf's epistemological problem.
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Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate. Eventually it was realized that this postulate may not be provable from the other four. According to this opinion about the parallel postulate (Postulate 5) does appear in print: > Apparently the first to do so was G. S. Klügel (1739-1812), a doctoral > student at the University of Gottingen, with the support of his teacher A. > G. Kästner, in the former's 1763 dissertation Conatuum praecipuorum theoriam > parallelarum demonstrandi recensio (Review of the Most Celebrated Attempts > at Demonstrating the Theory of Parallels). In this work Klügel examined 28 > attempts to prove Postulate 5 (including Saccheri's), found them all > deficient, and offered the opinion that Postulate 5 is unprovable and is > supported solely by the judgment of our senses.
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"Digboi, 100, still alive and kicking", Santanu Sanyal, The Hindu Business Line, 17 December 2001, retrieved online April 2008 This is possibly the most distilled – though fanciful – version of the legend explaining the siting and naming of Digboi. Two events separated by seven years have become fused, but although neither is likely to be provable, such evidence that does exist appears sufficiently detailed to be credible. Various web sites offer variations on the elephant’s foot story, a consensus of which would be that engineers extending the Dibru-Sadiya railway line to Ledo for the Assam Railways and Trading Company (AR&TC;) in 1882 were using elephants for haulage and noticed that the mud on one pachyderm’s feet smelled of oil. Retracing the trail of footprints, they found oil seeping to the surface. One of the engineers, the Englishman Willie Leova Lake, was an ‘oil enthusiast’ and persuaded the company to drill a well. Oil India Ltd makes no reference to elephants’ feet in its company history,"Heritage", retrieved online September 2009 although on its previous web site the company noted that Lake had noticed "the oil seepages around Borbhil".
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