Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive.
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For example, a Euclidean straight line has no width, but any real drawn line will. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, which usually required a statement such as "Find the greatest common measure of ..." Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space.
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In proof theory, a branch of mathematical logic, proof mining (or proof unwinding) is a research program that analyzes formalized proofs, especially in analysis, to obtain explicit bounds or rates of convergence from proofs that, when expressed in natural language, appear to be nonconstructive. This research has led to improved results in analysis obtained from the analysis of classical proofs.
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In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle . The LPO and LLPO axioms are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the sense of Brouwer.
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Such theorems provide no indication as to how to construct (or exhibit) the object whose existence is being claimed. From a constructivist viewpoint, such approaches are not viable as it lends to mathematics losing its concrete applicability,See the section on nonconstructive proofs of the entry "Constructive proof". while the opposing viewpoint is that abstract methods are far-reaching (meaning what?), in a way that numerical analysis cannot be.
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Note that, as is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs. However, algorithmic versions of the local lemma with stronger preconditions are also known (Beck 1991; Czumaj and Scheideler 2000). More recently, a constructive version of the local lemma was given by Robin Moser and Gábor Tardos requiring no stronger preconditions.
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References to Wikipedia in popular culture have increased as more people learn about and use the online encyclopedia project. Many parody Wikipedia's openness, with individuals vandalizing or modifying articles in nonconstructive ways. Still, others feature individuals using Wikipedia as a reference work, or positively comparing their intelligence to Wikipedia. In some cases, Wikipedia is not used as an encyclopedia at all, but instead serves more as a character trait or even as a game.
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As discussed above, in ZFC, the axiom of choice is able to provide "nonconstructive proofs" in which the existence of an object is proved although no explicit example is constructed. ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed. The status of the axiom of choice varies between different varieties of constructive mathematics.
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There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.
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The changes have been seen as unwelcome and nonconstructive by passengers and taxi drivers, with some saying they will boycott the airport. The change has also attracted criticism from local Councillors in Cheshire, who point out that many places directly under Manchester Airport's flight paths don't have a direct public transport link to the airport. While some other UK airports also have drop off charges Manchester Airport's charges are overall higher than those at any other airport.
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Although new technology has created more leisure time for all people, most of this new time is passed in "passive" (or nonconstructive) recreation. The introduction of the radio and automobile are considered the largest changes. Listening to radio shows and taking drives are now the most popular leisure activities. Many working-class families formerly never strayed more than a few miles from town; with the automobile, they are able to take vacations across the United States.
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The answer is, "not necessarily"; indeed, every infinite-dimensional normed space admits linear operators that are not closable. As in the case of discontinuous operators considered above, the proof requires the axiom of choice and so is in general nonconstructive, though again, if X is not complete, there are constructible examples. In fact, there is even an example of a linear operator whose graph has closure all of X ×Y. Such an operator is not closable.
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Adaptive behavior refers to behavior that enables a person (usually used in the context of children) to get along in their environment with greatest success and least conflict with others. This is a term used in the areas of psychology and special education. Adaptive behavior relates to everyday skills or tasks that the "average" person is able to complete, similar to the term life skills. Nonconstructive or disruptive social or personal behaviors can sometimes be used to achieve a constructive outcome.
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The function giving the sum of a convergent series is linear, and it follows from the Hahn–Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This is called the Banach limit. This fact is not very useful in practice, since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma. They are therefore nonconstructive.
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In mathematics and computer science, the probabilistic method is used to prove the existence of mathematical objects with desired combinatorial properties. The proofs are probabilistic — they work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probability. Consequently, they are nonconstructive — they don't explicitly describe an efficient method for computing the desired objects. The method of conditional probabilities , converts such a proof, in a "very precise sense", into an efficient deterministic algorithm, one that is guaranteed to compute an object with the desired properties.
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In some individuals, there is evidence that depressed or anxious mood may increase sexual interest or arousal. In general, men were more likely than women to report increased sexual drive during negative mood states. Negative moods are labeled as nonconstructive because it can affect a person's ability to process information; making them focus solely on the sender of a message, while people in positive moods will pay more attention to both the sender and the context of a message. This can lead to problems in social relationships with others.
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Williams' views on development of the colony included close association with European ideas and institutions, and he was against thoughtless and nonconstructive criticism of the administration. However, although Williams in many ways accepted European concepts and values, in October 1896 he sponsored an Egungun dance, a traditional ceremony. Moves such as this by one of the leaders of the Ekitis in Lagos were welcomed by the traditional rulers of the Yoruba. Williams once said: "A lawyer lives for the direction of his people and the advancement of the cause of his country".
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EHAA uses a ten to twelve notrump range; the distribution must be 4-3-3-3 or 4-4-3-2. On occasion, a 5-3-3-2 hand with a long minor can be opened 1NT if the values are concentrated in the short suits. Opposite the mini notrump, there is no reason to use transfers or many of the other accoutrements of standard notrump systems. Accordingly, 2, 2, 2, and 3 are nonconstructive improvements of the contract. Holding minimal values and a five card suit, responder should take out 1NT almost automatically.
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For example, a particular statement may be shown to imply the law of the excluded middle. An example of a Brouwerian counterexample of this type is Diaconescu's theorem, which shows that the full axiom of choice is non-constructive in systems of constructive set theory, since the axiom of choice implies the law of excluded middle in such systems. The field of constructive reverse mathematics develops this idea further by classifying various principles in terms of "how nonconstructive" they are, by showing they are equivalent to various fragments of the law of the excluded middle. Brouwer also provided "weak" counterexamples.
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The axiom of choice, first stated by Zermelo (1904), was proved independent of ZF by Fraenkel (1922), but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive.
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An early proponent of predicativism was Hermann Weyl, who showed it is possible to develop a large part of real analysis using only predicative methods (Weyl 1918). Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as the Gödel–Gentzen negative translation show that it is possible to embed (or translate) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs.
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For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.Dauben 1979, p. 266.
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Some problems are known to be solvable in polynomial-time, but no concrete algorithm is known for solving them. For example, the Robertson–Seymour theorem guarantees that there is a finite list of forbidden minors that characterizes (for example) the set of graphs that can be embedded on a torus; moreover, Robertson and Seymour showed that there is an O(n3) algorithm for determining whether a graph has a given graph as a minor. This yields a nonconstructive proof that there is a polynomial-time algorithm for determining if a given graph can be embedded on a torus, despite the fact that no concrete algorithm is known for this problem.
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Responses to the EHAA two bid fall into three categories. All jump bids are forcing to game and slam invitational, because any hand which could force to game opposite a minimum EHAA two-bid must be interested in slam opposite a maximum. (However, a direct jump to game is a two-way bid, and opener is expected to pass.) A nonjump bid in a new suit is a nonforcing and nonconstructive attempt to improve the contract. Finally, 2NT or a raise of the opening bid shows 14 to 17 high card points, and is invitational to game; a raise shows three card support or better, and 2NT denies three card support.
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For example, the axiom AC11 can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R(x,y) holds, then there is actually a function F such that R(x,F(x)) holds for all real numbers. Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number x there is a real number y such that R(x,y) holds". According to the BHK interpretation, this proof itself is essentially the function F that is desired.
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The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s. In his biography of Kurt Gödel, John W. Dawson, Jr summarizes the result as follows: "At issue in the sometimes bitter disputes was the relation of mathematics to logic, as well as fundamental questions of methodology, such as how quantifiers were to be construed, to what extent, if at all, nonconstructive methods were justified, and whether there were important connections to be made between syntactic and semantic notions."Dawson 1997:48 Dawson observes that "partisans of three principal philosophical positions took part in the debate" – the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his "school" of collaborators), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L.E.J. Brouwer.
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