# About:Proof of impossibility

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In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often put to rest decades or centuries of work attempting to find a solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic (see universal quantification for more).

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• In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often put to rest decades or centuries of work attempting to find a solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic (see universal quantification for more). The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as the ratio of two integers. Another proof of impossibility was the 1882 proof of Ferdinand von Lindemann, which showed that the ancient problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century. A problem that arose in the 16th century was that of creating a general formula using radicals expressing the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible, using concepts such as solvable groups from Galois theory—a new subfield of abstract algebra. Among the most important proofs of impossibility found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm at all, with the most famous one being the halting problem. Gödel's incompleteness theorems are other examples that uncover some fundamental limitations in the provability of formal systems. In computational complexity theory, techniques like relativization (see oracle machine) provide "weak" proofs of impossibility, excluding certain proof techniques. Other techniques, such as proofs of completeness for a complexity class, provide evidence for the difficulty of problems by showing them to be just as hard to solve as other known problems that have proved intractable. (en)
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• Heath (en)
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• Principia Mathematica, 2nd edition 1927, p. 61 (en)
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• Richard's paradox ... is as follows. Consider all decimals that can be defined by means of a finite number of words ([“words” are symbols; boldface added for emphasis]); let E be the class of such decimals. Then E has ([an infinite number of]) terms; hence its members can be ordered as the 1st, 2nd, 3rd, ... Let X be a number defined as follows ([Whitehead & Russell now employ the Cantor diagonal method]). (en)
• If the n-th figure in the n-th decimal is p, let the n-th figure in X be p + 1 . Then X is different from all the members of E, since, whatever finite value n may have, the n-th figure in X is different from the n-th figure in the n-th of the decimals composing E, and therefore X is different from the n-th decimal. Nevertheless we have defined X in a finite number of words ([i.e. this very definition of “word” above.]) and therefore X ought to be a member of E. Thus X both is and is not a member of E. (en)
• It is unknown when, or by whom, the "theorem of Pythagoras" was discovered. The discovery can hardly have been made by Pythagoras himself, but it was certainly made in his school. Pythagoras lived about 570&ndash;490 BCE. Democritus, born about 470 BCE, wrote on irrational lines and solids ... (en)
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• In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often put to rest decades or centuries of work attempting to find a solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a theory. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic (see universal quantification for more). (en)
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• Proof of impossibility (en)
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